Recent zbMATH articles in MSC 35Ahttps://zbmath.org/atom/cc/35A2022-10-04T19:40:27.024758ZWerkzeugBook review of: S. I. Repin and S. A. Sauter, Accuracy of mathematical models. Dimension reduction, homogenization, and simplificationhttps://zbmath.org/1492.000222022-10-04T19:40:27.024758Z"Madureira, Alexandre L."https://zbmath.org/authors/?q=ai:madureira.alexandre-lReview of [Zbl 1481.35011].Variational solutions to the total variation flow on metric measure spaceshttps://zbmath.org/1492.301182022-10-04T19:40:27.024758Z"Buffa, Vito"https://zbmath.org/authors/?q=ai:buffa.vito"Kinnunen, Juha"https://zbmath.org/authors/?q=ai:kinnunen.juha"Camacho, Cintia Pacchiano"https://zbmath.org/authors/?q=ai:pacchiano-camacho.cintiaSummary: We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincaré inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian spaces and functions of bounded variation to prove a necessary and sufficient condition for a variational solution to be continuous at a given point.Uniqueness and non-uniqueness of signed measure-valued solutions to the continuity equationhttps://zbmath.org/1492.350102022-10-04T19:40:27.024758Z"Bonicatto, Paolo"https://zbmath.org/authors/?q=ai:bonicatto.paolo(no abstract)Existence of solutions for a nonhomogeneous sublinear fractional Schrödinger equationhttps://zbmath.org/1492.350112022-10-04T19:40:27.024758Z"Zhang, Peng"https://zbmath.org/authors/?q=ai:zhang.peng.2|zhang.peng|zhang.peng.1"Han, Zhi-qing"https://zbmath.org/authors/?q=ai:han.zhiqingSummary: This paper is concerned with the following sublinear fractional Schrödinger equation:
\[
(-\Delta)^s u+V(x) u=a(x)|u|^{q-1} u+f(x), \quad x \in \mathbb{R}^N,
\] where \(s \in (0,1)\), \(q \in ((1/2_s^\ast -1),1)\), \(N>2s\), \(2_s^\ast=2N/(N-2s)\), \((-\Delta)^s\) is the fractional Laplacian operator, and \(V\), \(a\) both change sign in \(\mathbb{R}^N\), \(f \neq 0\) is a nonnegative perturbation such that \(f \in L^{(q+1)/q}(\mathbb{R}^N) \cap L^{2_s^\ast/(2_s^\ast-1)} (\mathbb{R}^N)\). By using the \(s\)-harmonic extension technique and suitable variational methods, we prove the existence of at least two nontrivial solutions for the problem.Multiple solutions for a coupled Kirchhoff system with fractional \(p\)-Laplacian and sign-changing weight functionshttps://zbmath.org/1492.350122022-10-04T19:40:27.024758Z"Zhen, Maoding"https://zbmath.org/authors/?q=ai:zhen.maoding"Yang, Meihua"https://zbmath.org/authors/?q=ai:yang.meihuaSummary: In this paper, we investigate the multiplicity of solutions for Kirchhoff fractional \(p\)-Laplacian system in bounded domains:
\[
\begin{cases}
\left(\sum_{i=1}^k [u_i]_{s,p}^p \right)^{\theta-1} (-\Delta)_p^s u_j(x) \\
\quad =\lambda_j f_j(x) |u_j|^{q-2} u_j + \sum_{i \neq j} \beta_{i,j} h(x) |u_i|^m |u_j|^{m-2} u_j & \text{in } \Omega, \\
u_j=0 & \text{in } \mathbb{R}^n \backslash \Omega.
\end{cases}
\] By using the Nehari manifold method, together with Ekeland's variational principle, we show that there exist two distinct solutions under suitable conditions on weight functions \(f_j\) and \(h\). Our results extend and generalize the main results in [\textit{M. Xiang} et al., Nonlinearity 29, No. 10, 3186--3205 (2016; Zbl 1349.35413)], in which \(f_j,h\) are constants.Bäcklund transformations: a tool to study abelian and non-abelian nonlinear evolution equationshttps://zbmath.org/1492.350132022-10-04T19:40:27.024758Z"Carillo, Sandra"https://zbmath.org/authors/?q=ai:carillo.sandra"Schiebold, Cornelia"https://zbmath.org/authors/?q=ai:schiebold.corneliaOn sharp Agmon-Miranda maximum principleshttps://zbmath.org/1492.350142022-10-04T19:40:27.024758Z"Kresin, Gershon"https://zbmath.org/authors/?q=ai:kresin.gershon-i"Maz'ya, Vladimir"https://zbmath.org/authors/?q=ai:mazya.vladimir-gilelevichSummary: In this survey we formulate our results on different forms of maximum principles for linear elliptic equations and systems. We start with necessary and sufficient conditions for validity of the classical maximum modulus principle for solutions of second order strongly elliptic systems. This principle holds under rather heavy restrictions on the coefficients of the systems. For instance, it fails for the Stokes and Lamé systems. Next, we turn to sharp constants in more general maximum principles due to S. Agmon and C. Miranda. We consider higher order elliptic equations, the Stokes and Lamé systems in a half-space, as well as the planar deformed state system in a half-plane.Existence of periodic solution for a class of beam equation via variational methodshttps://zbmath.org/1492.350172022-10-04T19:40:27.024758Z"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"de Araújo, Bruno S. V."https://zbmath.org/authors/?q=ai:de-araujo.bruno-s-v"Nóbrega, Alânnio B."https://zbmath.org/authors/?q=ai:nobrega.alannio-bThe paper is concerned with a class of beam equations, or generalized membrane equations, of the form
\begin{align*}
&u_{tt}+\Delta_x^2 u = f(x,t,u),\quad &&\text{in }Q=\Omega\times\mathbb{R},\\
&u(x,t) = D_{ii}u(x,t) = 0 &&\text{on }\partial\Omega,\\
&u(x,t) = u(x,t+2\pi) &&\text{in }Q,
\end{align*}
where \(\Omega=(0,2\pi)^N\) and \(f\in C^1(Q\times\mathbb{R}^2;\mathbb{R})\) is \(2\pi\)-periodic in \(t\), superlinear in \(u\). Under various growth conditions on the nonlinearity, the authors obtain the existence of a ground state, and more generally, the existence of a nontrivial solution.
Reviewer: Thomas J. Bartsch (Gießen)On asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimensionhttps://zbmath.org/1492.350452022-10-04T19:40:27.024758Z"Masaki, Satoshi"https://zbmath.org/authors/?q=ai:masaki.satoshi"Segata, Jun-Ichi"https://zbmath.org/authors/?q=ai:segata.jun-ichi"Uriya, Kota"https://zbmath.org/authors/?q=ai:uriya.kotaSummary: In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order.Hardy's inequality and the isotropic Landau equationhttps://zbmath.org/1492.350642022-10-04T19:40:27.024758Z"Gualdani, Maria"https://zbmath.org/authors/?q=ai:gualdani.maria-pia"Guillen, Nestor"https://zbmath.org/authors/?q=ai:guillen.nestorSummary: In this manuscript we establish an \(L^\infty\) estimate for the isotropic analogue of the homogeneous Landau equation. This is done for values of the interaction exponent \(\gamma\) in (a part of) the range of very soft potentials. The main observation in our proof is that the classical weighted Hardy inequality leads to a weighted Poincaré inequality, which in turn implies the propagation of some \(L^p\) norms of solutions. From here, the \(L^\infty\) estimate follows from certain weighted Sobolev inequalities and De Giorgi-Nash-Moser theory.Endpoint Strichartz estimates with angular integrability and some applicationshttps://zbmath.org/1492.350652022-10-04T19:40:27.024758Z"Kim, Jungkwon"https://zbmath.org/authors/?q=ai:kim.jungkwon"Lee, Yoonjung"https://zbmath.org/authors/?q=ai:lee.yoonjung"Seo, Ihyeok"https://zbmath.org/authors/?q=ai:seo.ihyeokIt is well known that the endpoint Strichartz estimate \[\|e^{it\Delta}f\|_{L_t^2L_x^\infty(\mathbb{R}\times\mathbb{R}^2)}\leq C\|f\|_{L^2(\mathbb{R}^2)}\] fails. Tao showed the spherically averaged endpoint Strichartz estimates \[\|e^{it\Delta}f\|_{L_t^2L_\rho^\infty L_w^2}\leq C\|f\|_{L_x^2(\mathbb{R}^2)},\] with \(x=\rho w\) and \(w\in\mathbb{S}^1\). In this paper, the authors established a weighted version of such spherically averaged estimates of the form \[\big\||x|^{-\gamma} e^{it\Delta}f\big\|_{L_t^2 L_\rho^r L_w^k} \leq C\|f\|_{L_x^2(\mathbb{R}^n)}\] with \(x=\rho w,\; w\in\mathbb{S}^{n-1}\) and \(n\geq3\). As an application, the existence of solutions for the inhomogeneous nonlinear Schrödinger equation is shown for \(L^2\) data.
Reviewer: Jiqiang Zheng (Beijing)Darboux transformation and solitonic solution to the coupled complex short pulse equationhttps://zbmath.org/1492.350832022-10-04T19:40:27.024758Z"Feng, Bao-Feng"https://zbmath.org/authors/?q=ai:feng.baofeng"Ling, Liming"https://zbmath.org/authors/?q=ai:ling.limingSummary: The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright-soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis. Breather and rogue wave solutions are constructed under non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue wave solution.Darboux transformation for the Hirota equationhttps://zbmath.org/1492.350932022-10-04T19:40:27.024758Z"Yilmaz, Halis"https://zbmath.org/authors/?q=ai:yilmaz.halisSummary: The Hirota equation is an integrable higher order nonlinear Schrödinger type equation which describes the propagation of ultrashort light pulses in optical fibers. We present a standard Darboux transformation for the Hirota equation and then construct its quasideterminant solutions. As examples, the multi-soliton, breather and rogue wave solutions of the Hirota equation are given explicitly.Well-posedness and ill-posedness for linear fifth-order dispersive equations in the presence of backwards diffusionhttps://zbmath.org/1492.350992022-10-04T19:40:27.024758Z"Ambrose, David M."https://zbmath.org/authors/?q=ai:ambrose.david-m"Woods, Jacob"https://zbmath.org/authors/?q=ai:woods.jacobSummary: Fifth-order dispersive equations arise in the context of higher-order models for phenomena such as water waves. For fifth-order variable-coefficient linear dispersive equations, we provide conditions under which the intitial value problem is either well-posed or ill-posed. For well-posedness, a balance must be struck between the leading-order dispersion and possible backwards diffusion from the fourth-derivative term. This generalizes work by the first author and Wright for third-order equations. In addition to inherent interest in fifth-order dispersive equations, this work is also motivated by a question from numerical analysis: finite difference schemes for third-order numerical equations can yield approximate solutions which effectively satisfy fifth-order equations. We find that such a fifth-order equation is well-posed if and only if the underlying third-order equation is ill-posed.Schrödinger equations in \(\mathbb{R}^2\) with critical exponential growth and concave nonlinearitieshttps://zbmath.org/1492.351022022-10-04T19:40:27.024758Z"Lin, Xiaoyan"https://zbmath.org/authors/?q=ai:lin.xiaoyan"Tang, Xianhua"https://zbmath.org/authors/?q=ai:tang.xian-huaSummary: In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form
\[
- {\Delta} u + V(| x |) u = Q(| x |) f(u) + \lambda g(| x |, u), \quad x \in \mathbb{R}^2,\tag{0.1}
\] where the nonlinear term \(f(s)\) has critical exponential growth which behaves like \(e^{\alpha s^2}\), \(g(r, s)\) is a concave term on \(s\), the radial potentials \(V, Q : \mathbb{R}^+ \to \mathbb{R}\) are unbounded, singular at the origin or decaying to zero at infinity and \(\lambda > 0\) is a parameter. Based on the known Trudinger-Moser inequality in \(H_{0, \operatorname{rad}}^1( B_1)\), we establish a new version of Trudinger-Moser inequality in the working space of the associated with the energy functional related to the above problem. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weak assumptions. Our results show that the presence of the concave term (i.e. \( \lambda > 0\)) is very helpful to the existence of nontrivial solutions for Eq. (0.1) in one sense.On the complete solutions to the Tchebychev affine Kähler equation and its geometric significancehttps://zbmath.org/1492.351062022-10-04T19:40:27.024758Z"Xu, Ruiwei"https://zbmath.org/authors/?q=ai:xu.ruiwei"Li, Xingxiao"https://zbmath.org/authors/?q=ai:li.xingxiaoSummary: In this paper we study the Tchebychev affine Kähler equation (simply, TAKE) which consists of a number of fourth-order nonlinear equations and is closely related to a few interesting partial differential equations, including some equations of affine extremal hypersurfaces and the determinant-constant equation which was studied by \text{E. Calabi} [Mich. Math. J. 5, 105--126 (1958; Zbl 0113.30104)]. By using a geometrical idea first found by E. Calabi, we are able to find all the Calabi complete (strictly convex) solutions of the TAKE by means of Calabi geometry which can be naturally realised as a special kind of relative affine geometry.On a planar Hartree-Fock type systemhttps://zbmath.org/1492.351072022-10-04T19:40:27.024758Z"Carvalho, J."https://zbmath.org/authors/?q=ai:carvalho.jean-paulo-s|carvalho.joel-c|carvalho.jonas-c|carvalho.jean-paul|carvalho.jose-a|carvalho.jose-l|carvalho.j-a-jun|carvalho.jose-f|carvalho.jonison-l|carvalho.jorge-j-c-m|carvalho.joao-l-a|carvalho.j-g-jun|carvalho.joao-c|carvalho.joao-a|carvalho.jose-raimundo|carvalho.joao-b|carvalho.joao-paulo|carvalho.jose-r-h|carvalho.j-d-a|carvalho.jean-paulo-dos-santos|carvalho.janete-s|carvalho.j-frederico"Figueiredo, G."https://zbmath.org/authors/?q=ai:figueiredo.gustavo-b|figueiredo.giovany-malcher"Maia, L."https://zbmath.org/authors/?q=ai:maia.l"Medeiros, E."https://zbmath.org/authors/?q=ai:medeiros.everaldo-s|medeiros.everton-s|medeiros.esdrasSummary: This work deals with the existence of solutions for a class of Hartree-Fock type system in the two dimensional Euclidean space. Our approach is variational and based on a minimization technique in the Nehari manifold. The main steps in the prove are some trick estimates from the sign-changing logarithm potential in an appropriate subspace of \(H^1(\mathbb{R}^2)\) introduced
by \textit{J. Stubbe} [``Bound States of Two-Dimensional Schrödinger-Newton Equations'',
Preprint, \url{arXiv:0807.4059}] (see also \textit{S. Cingolani} and \textit{T. Weth} [Anal. Non Linéaire 33, No. 1, 169--197 (2016; Zbl 1331.35126)]).Positive solutions for nonlinear Schrödinger-Poisson systems with general nonlinearityhttps://zbmath.org/1492.351082022-10-04T19:40:27.024758Z"Chen, Ching-yu"https://zbmath.org/authors/?q=ai:chen.ching-yu"Wu, Tsung-fang"https://zbmath.org/authors/?q=ai:wu.tsungfangSummary: In this paper, we study a class of Schrödinger-Poisson (SP) systems with general nonlinearity where the nonlinearity does not require Ambrosetti-Rabinowitz and Nehari monotonic conditions. We establish new estimates and explore the associated energy functional which is coercive and bounded below on Sobolev space. Together with Ekeland variational principle, we prove the existence of ground state solutions. Furthermore, when the `charge' function is greater than a fixed positive number, the (SP) system possesses only zero solutions. In particular, when `charge' function is radially symmetric, we establish the existence of three positive solutions and the symmetry breaking of ground state solutions.New radial solutions of strong competitive \(m\)-coupled elliptic system with general form in \(B_1(0)\)https://zbmath.org/1492.351092022-10-04T19:40:27.024758Z"Chen, Haixia"https://zbmath.org/authors/?q=ai:chen.haixia"Yang, Xian"https://zbmath.org/authors/?q=ai:yang.xianSummary: We construct a smooth radial positive solution for the following \(m\)-coupled elliptic system
\[
\begin{cases}
-\Delta u_i = f(u_i) - \beta\sum\limits_{j\neq i} u_iu_j^2,&\text{in}\quad B_1(0),\\
u_i = 0, i = 1, \dots, m,&\text{on}\quad \partial B_1(0),
\end{cases}
\]
for \(\beta > 0\) large enough, where \(f\in C^{2,1}(\mathbb{R})\), \(f(0)=0\), \(B_1(0)\subset \mathbb{R}^N\) is the unit ball centered at the origin, \(m\geq 3\), \(N\geq 1\) are positive integers. Our main result is an extension of \textit{J.-B. Casteras} and \textit{C. Sourdis} [J. Funct. Anal. 279, No. 8, Article ID 108674, 39 p. (2020; Zbl 1447.35161)] from \(m=2\) to general case \(m\geq 3\) under some natural and essential non-degeneracy conditions by gluing method. The way we construct is somehow different and greatly simplify the computations since we overcome the difficulties brought by too much parameters from multiple equations.Bifurcation branch and stability of stationary solutions of a predator-prey modelhttps://zbmath.org/1492.351112022-10-04T19:40:27.024758Z"Wang, Yu-Xia"https://zbmath.org/authors/?q=ai:wang.yuxia"Zuo, Hui-Qin"https://zbmath.org/authors/?q=ai:zuo.hui-qinSummary: This paper is concerned about a diffusive degenerate predator-prey model with Beddington-DeAngelis functional response subject to homogeneous Neumann boundary condition. First, the global bifurcation branches of positive stationary solutions are studied, which are quite different from those with different degeneracy or functional response. Second, the multiplicity and stability of positive stationary solutions are obtained as the parameter \(k\) or \(m\) in the Beddington-DeAngelis functional response is large enough, from which the effects of the functional response on the coexistence region are revealed. In particular, the global stability of the positive stationary solution is derived as it exists uniquely.On \(W^{2, p}\)-estimates for solutions of obstacle problems for fully nonlinear elliptic equations with oblique boundary conditionshttps://zbmath.org/1492.351122022-10-04T19:40:27.024758Z"Byun, Sun-Sig"https://zbmath.org/authors/?q=ai:byun.sun-sig"Han, Jeongmin"https://zbmath.org/authors/?q=ai:han.jeongmin"Oh, Jehan"https://zbmath.org/authors/?q=ai:oh.jehanThe main purpose of the paper is to prove existence and uniqueness and to derive a \(W^{2,p}\)-estimate for viscosity solutions to a fully nonlinear elliptic obstacle problem with oblique boundary data. Indeed, the authors discuss in particular the existence, uniqueness and \(W^{2,p}\)-regularity for approximate non-obstacle problems and then carry out the limiting process to deduce their result.
Reviewer: Said El Manouni (Riyadh)Entire vortex solutions of negative degree for the anisotropic Ginzburg-Landau systemhttps://zbmath.org/1492.351162022-10-04T19:40:27.024758Z"Kowalczyk, Michał"https://zbmath.org/authors/?q=ai:kowalczyk.michal"Lamy, Xavier"https://zbmath.org/authors/?q=ai:lamy.xavier"Smyrnelis, Panayotis"https://zbmath.org/authors/?q=ai:smyrnelis.panayotisThe authors study entire solutions \(u : \mathbb{R}^2 \to \mathbb{R}^2\) to the anisotropic Ginzburg-Landau equation
\[
\Delta u + \delta \nabla \left( \operatorname{div} u \right) + \delta \ \operatorname{curl}^* \left( \operatorname{curl} u \right) = \left( | u |^2 -1 \right) u \tag{1}
\]
for a fixed constant \(\delta \in (-1,1)\). The adjoint operator curl\(^*\) is defined as \(\operatorname{curl}^* = (\partial_2,-\partial_1)\). Equation (1) arises in the description of two-dimensional point defects in some liquid crystal configurations. The isotropic case \(\delta=0\) corresponds to the classical Ginzburg-Landau equation
\[
\Delta u = \left( | u |^2 -1 \right)u.
\]
It is known that for any prescribed degree (or winding number) \(d = 0\), the isotropic Ginzburg-Landau equation admits a unique solution of the form
\[
v_d(r e^{i \theta}) = \eta_d(r) e^{id\theta}, \quad \eta_d(0)=0, \quad \lim_{r \to +\infty} \eta_d(r)=1
\]
and \(\eta_d \geq 0\).
In this paper the authors prove that for every \(d=-1,-,2,\ldots\) there exists \(\delta_0(d)>0\) such that, for small enough anisotropy \(| \delta | < \delta_0(d)\), there are at least two distinct, smooth entire solutions \(u : \mathbb{R}^2 \to \mathbb{R}^2\) of (1) satisfying
\[
\int_{\mathbb{R}^2} \left( 1-| u |^2 \right)^2 < 2 \pi d^2, \quad \deg (u;\partial D_r)=d \text{ for }r \gg 1,
\]
where \(D_r\) is the disk of radius \(r\) in \(\mathbb{R}^2\).
Reviewer: Simone Secchi (Milano)Limit profiles for singularly perturbed Choquard equations with local repulsionhttps://zbmath.org/1492.351172022-10-04T19:40:27.024758Z"Liu, Zeng"https://zbmath.org/authors/?q=ai:liu.zeng"Moroz, Vitaly"https://zbmath.org/authors/?q=ai:moroz.vitalyThe paper deals with the Choquard type equation \[-\Delta u+\varepsilon u-(I_{\alpha}\ast |u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad\text{in}\,\,\,\mathbb{R}^N,\leqno(P_{\epsilon})\] where \(N\ge 3\), \(p>1\), \(q>2\), \(\varepsilon\ge 0\) is a small or large parameter, and \(I_{\alpha}(x)=A_{\alpha}|x|^{-(N-\alpha)}\) is the Riesz potential with \(\alpha\in (0,N)\). The authors investigate the existence and some properties (such as regularity and decay at infinity) of the positive ground states of problems \((P_{\varepsilon})\) with \(\varepsilon>0\) and \((P_0)\). They also study the asymptotic behavior of the ground states of \((P_{\varepsilon})\) when \(\varepsilon\to 0\) and \(\varepsilon\to \infty\), and give characterization of their limit profiles in each of the regimes.
Reviewer: Rodica Luca (Iaşi)Stationary solutions of Fokker-Planck equations with nonlinear reaction terms in bounded domainshttps://zbmath.org/1492.351182022-10-04T19:40:27.024758Z"Precup, Radu"https://zbmath.org/authors/?q=ai:precup.radu"Rubbioni, Paola"https://zbmath.org/authors/?q=ai:rubbioni.paolaSummary: Using an operator approach, we discuss stationary solutions to Fokker-Planck equations and systems with nonlinear reaction terms. The existence of solutions is obtained by using Banach, Schauder and Schaefer fixed point theorems, and for systems by means of Perov's fixed point theorem. Using the Ekeland variational principle, it is proved that the unique solution of the problem minimizes the energy functional, and in case of a system that it is the Nash equilibrium of the energy functionals associated to the component equations.Groundstates for magnetic Choquard equations with critical exponential growthhttps://zbmath.org/1492.351192022-10-04T19:40:27.024758Z"Wen, Lixi"https://zbmath.org/authors/?q=ai:wen.lixi"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-dThe authors prove the existence of a ground state solution for a maghentic Choquard equation with critical exponential growth. They establish an energy estimate involving convolution terms and use a non-Nehari manifold method developed in the paper of \textit{X. Tang} [Sci. China, Math. 58, No. 4, 715--728 (2015; Zbl 1321.35055)].
Reviewer: Anouar Bahrouni (Monastir)Infinitely many non-radial positive solutions for Choquard equationshttps://zbmath.org/1492.351202022-10-04T19:40:27.024758Z"Yu, Mingzhu"https://zbmath.org/authors/?q=ai:yu.mingzhu"Chen, Haibo"https://zbmath.org/authors/?q=ai:chen.haiboSummary: This paper is concerned with the following Choquard equation:
\[
- {\Delta} u + V(| y |) u = \left( \int_{\mathbb{R}^3} \frac{Q (|x|) u^p ( x )}{ | x - y |} dx \right) Q(| y |) u^{p - 2} u. \tag{0.1}
\] Here \(p \in(2, 2 + \delta)\), \(\delta \in (0, \frac{ 1}{ 3}) \), \(V(r)\) and \(Q(r)\) carry the following behavior:
\[
V(r) = V_0 + \frac{a}{r^n} + O (\frac{1}{r^{n + \rho}}), \quad Q(r) = Q_0 + \frac{b}{r^m} + O (\frac{1}{r^{m + \theta}}) \text{ as } r \to \infty,
\]where \(\frac{1}{2} \leq n, m < 1\), \(a, b \in \mathbb{R}\) and \(V_0, Q_0, \rho, \theta\) are positive constants. We prove that the equation (0.1) has infinitely many non-radial positive solutions when \(a\) and \(b\) satisfy some suitable conditions.Strongly nonlinear elliptic problem with measure data in Musielak-Orlicz spaceshttps://zbmath.org/1492.351212022-10-04T19:40:27.024758Z"Benkirane, Abdelmoujib"https://zbmath.org/authors/?q=ai:benkirane.abdelmoujib"EL Haji, Badr"https://zbmath.org/authors/?q=ai:el-haji.badr"EL Moumni, Mostafa"https://zbmath.org/authors/?q=ai:el-moumni.mostafaSummary: In this paper, we investigate the existence theorem of entropy solutions for nonlinear elliptic problem of the type \(-\operatorname{div} (a(x,u,\nabla u) + \Phi(x,u))+ g(x,u,\nabla u) = \mu\) in \(\Omega\), in the setting of Musielak-Orlicz spaces. The lower order term \(\Phi\) verifies the natural growth condition, the nonlinearity \(g\) has a natural growth with respect to its third argument and with sign condition, no \(\Delta_2\)-condition is assumed on the Musielak function, and the datum \(\mu\) is assumed to belong to \(L^1(\Omega)+W^{-1} E_{\bar{\varphi}}(\Omega)\).Nonlinear elliptic system with singular coefficient and with diffuse measure datahttps://zbmath.org/1492.351222022-10-04T19:40:27.024758Z"Eljazouli, A."https://zbmath.org/authors/?q=ai:eljazouli.a"Redwane, H."https://zbmath.org/authors/?q=ai:redwane.hichamSummary: We give an existence result of the nonlinear elliptic system of the type:
\[ \begin{cases}
-\operatorname{div}\left(A(x,v)\nabla u\right)=\mu \quad &\text{in } \Omega, \\
-\operatorname{div}\left (B(x,v)\nabla v\right)=\gamma |\nabla u|^{q_0} \quad & \text{in } \Omega ,\\
\end{cases} \]
where \(\Omega\) is a bounded open subset of \(\mathbb{R}^N\), \(N\ge 2\), \(\mu\) is a diffuse measure, \(A(x, s)\) is a Carathéodory function, the function \(B(x, s)\) blows up (uniformly with respect to \(x\)) as \(s\rightarrow m^- \) (with \(m>0\)), and \(\gamma\) is a positive constant and \(q_0\in [1, \frac{N}{N-1}[\). The main contribution of our work is to prove the existence of a renormalized solution.Existence of solutions of nonlinear elliptic equations with measure data in Musielak-Orlicz spaceshttps://zbmath.org/1492.351232022-10-04T19:40:27.024758Z"Kashnikova, Anastasiya P."https://zbmath.org/authors/?q=ai:kashnikova.anastasiya-pavlovna"Kozhevnikova, Larisa M."https://zbmath.org/authors/?q=ai:kozhevnikova.larisa-mikhailovnaExistence and multiplicity of solutions to concave-convex-type double-phase problems with variable exponenthttps://zbmath.org/1492.351242022-10-04T19:40:27.024758Z"Kim, In Hyoun"https://zbmath.org/authors/?q=ai:kim.in-hyoun"Kim, Yun-Ho"https://zbmath.org/authors/?q=ai:kim.yunho"Oh, Min Wook"https://zbmath.org/authors/?q=ai:oh.min-wook"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: This paper is devoted to the study of the \(L^\infty\)-bound of solutions to the double-phase nonlinear problem with variable exponent by the case of a combined effect of concave-convex nonlinearities. The main tools are the De Giorgi iteration method and a truncated energy technique. Applying this and a variant of Ekeland's variational principle, we give the existence of at least two distinct nontrivial solutions belonging to \(L^\infty\)-space when the condition on a nonlinear convex term does not assume the Ambrosetti-Rabinowitz condition in general. In addition, our problem admits a sequence of small energy solutions whose converge to zero in \(L^\infty\) space. To achieve this result, we apply the modified functional method and global variational formulation as the main tools.Positive solutions for a Kirchhoff-type equation with critical and supercritical nonlinear termshttps://zbmath.org/1492.351252022-10-04T19:40:27.024758Z"Lei, Chun-Yu"https://zbmath.org/authors/?q=ai:lei.chunyu"Liao, Jia-Feng"https://zbmath.org/authors/?q=ai:liao.jiafengSummary: We consider a Kirchhoff-type equation with critical and supercritical nonlinear terms in a ball. By providing a method of decomposition of energy functional and subtle analysis, we show that every Palais-Smale sequence at a level below a certain energy threshold admits a subsequence that converges strongly to a nontrivial critical point of the variational functional.Existence and multiplicity of positive solutions to a \(p\)-Kirchhoff-type equationhttps://zbmath.org/1492.351262022-10-04T19:40:27.024758Z"Li, Qi"https://zbmath.org/authors/?q=ai:li.qi.1|li.qi"Han, Yuzhu"https://zbmath.org/authors/?q=ai:han.yuzhuSummary: In this paper, a \(p\)-Kirchhoff-type elliptic equation involving a positive parameter \(\lambda\) is considered. Existence, nonexistence and multiplicity of weak solutions are obtained by the fibering maps and the mountain pass lemma. We use the nonlinear generalized Rayleigh quotient to characterize two extremal values \(\lambda_0^\ast\) and \(\lambda^\ast\). The parameter \(\lambda_0^\ast\) is characterized for which the energy functional has nonnegative energy for the local minimum when \(\lambda \ge \lambda_0^\ast\). The parameter \(\lambda^\ast\) is characterized for which the problem has no nonzero solution when \(\lambda >\lambda^\ast\). Moreover, the asymptotic behavior of the solutions is also studied when \(\lambda \downarrow 0\).Existence and concentration of ground state solutions for critical Kirchhoff-type equation with steep potential wellhttps://zbmath.org/1492.351272022-10-04T19:40:27.024758Z"Luo, Li-Ping"https://zbmath.org/authors/?q=ai:luo.liping"Tang, Chun-Lei"https://zbmath.org/authors/?q=ai:tang.chun-lei|tang.chunleiSummary: In this paper, we study the nonlinear Kirchhoff type equation
\[
\begin{cases}
\left(a+b \displaystyle\int_{\mathbb{R}^3} |\nabla u|^2 \mathrm{d}x \right) \triangle u+ \lambda V(x)u = |u|^4 u+f(u), \quad x \in \mathbb{R}^3, \\
u \in H^1(\mathbb{R}^3),
\end{cases}
\] where \(a, b\) are positive constants and \(\lambda >0\). Suppose that the nonnegative continuous potential \(V\) represents a potential well with the bottom \(V^{-1}(0)\) and \(f \in C (\mathbb{R},\mathbb{R})\) satisfies certain assumptions. We obtain the existence of ground state solutions by using the variational methods. Moreover, the concentration behavior of the ground state solutions as \(\lambda \to \infty\) is also worth considering.Degenerate fractional Kirchhoff-type system with magnetic fields and upper critical growthhttps://zbmath.org/1492.351282022-10-04T19:40:27.024758Z"Sun, Mingzhe"https://zbmath.org/authors/?q=ai:sun.mingzhe"Shi, Shaoyun"https://zbmath.org/authors/?q=ai:shi.shaoyun"Repovš, Dušan D."https://zbmath.org/authors/?q=ai:repovs.dusan-dSummary: This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth:
\[
\begin{cases}
-\mathfrak{M}(\Vert u\Vert_{s, A}^2)[(-\Delta)^s_Au+u] = G_u(|x|, |u|^2, |v|^2) \\
\quad +\left(\mathcal{I}_\mu \ast|u|^{p^*}\right)|u|^{p^*-2}u &\text{in }\mathbb{R}^N,\\
\mathfrak{M}(\Vert v\Vert_{s, A})[(-\Delta)^s_Av+v] = G_v(|x|, |u|^2, |v|^2) \\
\quad +\left(\mathcal{I}_\mu \ast|v|^{p^*}\right)|v|^{p^*-2}v &\text{in }\mathbb{R}^N,
\end{cases}
\]
where
\[
\Vert u\Vert_{s, A} = \left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}\mathrm{d}x\mathrm{d}y+\int_{\mathbb{R}^N}|u|^2\mathrm{d}x\right)^{1/2},
\]
and \((-\Delta)_A^s\) and \(A\) are called magnetic operator and magnetic potential, respectively, \(\mathfrak{M}:\mathbb{R}^+_0\rightarrow\mathbb{R}^+_0\) is a continuous Kirchhoff function, \(\mathcal{I}_\mu (x) = |x|^{N-\mu}\) with \(0<\mu <N\), \(C^1\)-function \(G\) satisfies some suitable conditions, and \(p^* =\frac{N+\mu}{N-2s} \). We prove the multiplicity results for this problem using the limit index theory. The novelty of our work is the appearance of convolution terms and critical nonlinearities. To overcome the difficulty caused by degenerate Kirchhoff function and critical nonlinearity, we introduce several analytical tools and the fractional version concentration-compactness principles which are useful tools for proving the compactness condition.Multiple solutions with compact support for a quasilinear Schrödinger equation with sign-changing potentialshttps://zbmath.org/1492.351312022-10-04T19:40:27.024758Z"Bahrouni, Anouar"https://zbmath.org/authors/?q=ai:bahrouni.anouar"Bedoui, Nizar"https://zbmath.org/authors/?q=ai:bedoui.nizar"Ounaies, Hichem"https://zbmath.org/authors/?q=ai:ounaies.hichemThe authors study the existence of solutions for the following Schrödinger equation with indefinite potential
\begin{align*}
-\Delta_pu+V(x)u=a(x)|u|^{q-1}u,\quad x\in \mathbb{R}^N,
\end{align*}
where \(N\geq 3\), \(1<p<2\), \(0<q<p-1\), \(\Delta_p\) denotes the \(p\)-Laplacian and \(a(\cdot)\), \(V(\cdot)\) are functions that may change sign in \(\mathbb{R}^N\).
Reviewer: Patrick Winkert (Berlin)On a nonlinear elliptic system involving the \((p(x),q(x))\)-Laplacian operator with gradient dependencehttps://zbmath.org/1492.351322022-10-04T19:40:27.024758Z"Nabab, D."https://zbmath.org/authors/?q=ai:nabab.d"Vélin, J."https://zbmath.org/authors/?q=ai:velin.jeanSummary: In this article, we deal with the existence of solutions for a class of nonlinear elliptic system with \((p(x),q(x))\)-Laplacian operator. The work is based on the Leray-Schauder principle with the use of the topological degree of Berkovits applied to an abstract Hammerstein equation associated to our system and presenting a certain lack of compactness. In this study, we establish existence results when the source terms depend on the gradient which are convection terms.Ambrosetti-Prodi problems for the Robin \((p, q)\)-Laplacianhttps://zbmath.org/1492.351332022-10-04T19:40:27.024758Z"Papageorgiou, Nikolaos S."https://zbmath.org/authors/?q=ai:papageorgiou.nikolaos-s"Rădulescu, Vicenţiu D."https://zbmath.org/authors/?q=ai:radulescu.vicentiu-d"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian.1|zhang.jian.7|zhang.jian.3|zhang.jian.5|zhang.jian.2|zhang.jian.4|zhang.jian|zhang.jian.6Summary: The classical Ambrosetti-Prodi problem considers perturbations of the linear Dirichlet Laplace operator by a nonlinear reaction whose derivative jumps over the principal eigenvalue of the operator. In this paper, we develop a related analysis for parametric problems driven by the nonlinear Robin \((p, q)\)-Laplace operator (sum of a \(p\)-Laplacian and a \(q\)-Laplacian). Under hypotheses that cover both the \((p - 1)\)-linear and the \((p - 1)\)-superlinear case, we prove an optimal existence, multiplicity, and non-existence result, which is global in the parameter \(\lambda > 0\).A new elliptic mixed boundary value problem with \((p,q)\)-Laplacian and Clarke subdifferential: existence, comparison and convergence resultshttps://zbmath.org/1492.351352022-10-04T19:40:27.024758Z"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengda"Migórski, Stanisław"https://zbmath.org/authors/?q=ai:migorski.stanislaw"Tarzia, Domingo A."https://zbmath.org/authors/?q=ai:tarzia.domingo-albertoNodal solutions for anisotropic \((p, q)\)-equationshttps://zbmath.org/1492.351362022-10-04T19:40:27.024758Z"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengda"Papageorgiou, Nikolaos S."https://zbmath.org/authors/?q=ai:papageorgiou.nikolaos-sSummary: We consider a nonlinear Dirichlet problem driven by the anisotropic \((p, q)\)-Laplacian and a Carathéodory reaction \(f (z, x)\) locally defined in \(x\). Using critical point theory, truncation and comparison techniques as well as critical groups, we show the existence of at least three nontrivial smooth solutions (positive, negative and nodal). If a symmetry condition on \(f (z, \cdot)\) is imposed, then we can produce a whole sequence of nodal solutions converging to zero in \(C_0^1 (\overline{\Omega})\).Rational function and time transformation of caloric morphism on semi-euclidean spaceshttps://zbmath.org/1492.351382022-10-04T19:40:27.024758Z"Shimomura, Katsunori"https://zbmath.org/authors/?q=ai:shimomura.katsunoriSummary: In this paper, we prove that any non-constant real rational function appears as a time transformation of a caloric morphism, mapping which preserves caloric functions, between semi-eucledean spaces.Classical solution of the mixed problem for the Klein-Gordon-Fock equation with nonlocal conditionshttps://zbmath.org/1492.351592022-10-04T19:40:27.024758Z"Korzyuk, Viktor Ivanovich"https://zbmath.org/authors/?q=ai:korzyuk.victor-i"Stolyarchuk, Ivan Igorevich"https://zbmath.org/authors/?q=ai:stolyarchuk.ivan-igorevichSummary: The mixed problem for the one-dimensional Klein-Gordon-Fock equation with nonlocal conditions in a halfstrip is considered. Solving this problem reduces to solving the systems of the second-type Volterra equations. The theorems of existence and uniqueness of a solution in the class of twice continuously differentiable functions were proved for these equations, when initial functions are smooth enough. It is proved that fulfillment of the matching conditions for given functions is necessary and sufficient for the existence of a unique smooth solution when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method reduces to splitting the original area of the definition into subdomains. The solution of the subproblem can be constructed with in each subdomain, the help of the initial and nonlocal conditions. The obtained solutions are then glued at common points, and these gluing conditions are the matching conditions.
This approach can be used in constructing both an analytical solution, when the solution of the systems of integral equations can be found explicitly, and an approximate solution. Moreover, approximate solutions can be constructed numerically and analytically. When the numerical solution is constructed, matching conditions are essential and need to be considered while developing numerical methods.Resolvent estimates and localization of the spectrum for certain classes of non-selfadjoint semiclassical pseudodifferential operators (after Dencker, Sjöstrand and Zworski)https://zbmath.org/1492.351742022-10-04T19:40:27.024758Z"Pravda-Starov, Karel"https://zbmath.org/authors/?q=ai:pravda-starov.karelSummary: In this talk, we shall present the works by Dencker, Sjöstrand and Zworski on the pseudospectrum of certain classes of non-selfadjoint semiclassical pseudodifferential operators. Understanding the pseudospectral properties of an operator reduces to studying the level lines of the norm of its resolvent. For non-selfadjoint operators, it is a non trivial problem even if the spectra of these operators are known. Indeed, there is no a priori control of the resolvent of a non-selfadjoint operator by its spectrum, and the resolvent of such an operator can blow up in norm in some unbounded regions of the resolvent set far from the spectrum. The works by Dencker, Sjöstrand and Zworski show how the microlocal analysis theory, and in particular some results of non solvability and subellipticity, allow one to account for these phenomena of control or blow up for the resolvents of certain classes of non-selfadjoint semiclassical pseudodifferential operators.
For the entire collection see [Zbl 1486.00041].Global existence of strong solutions for the generalized Navier-Stokes equations with dampinghttps://zbmath.org/1492.351802022-10-04T19:40:27.024758Z"Cai, Xiao-jing"https://zbmath.org/authors/?q=ai:cai.xiaojing"Zhou, Yan-jie"https://zbmath.org/authors/?q=ai:zhou.yanjieSummary: This paper mainly focus on the global existence of the strong solutions for the generalized Navier-Stokes equations with damping. We obtain the global existence and uniqueness when \(\alpha \ge \frac{5}{4}\) for \(\beta \geq 1\) and when \(\frac{1}{2} + \frac{2}{\beta} \le \alpha \le \frac{5}{4}\) for \(\frac{8}{3} \le \beta < + \infty \).Typicality results for weak solutions of the incompressible Navier-Stokes equationshttps://zbmath.org/1492.351812022-10-04T19:40:27.024758Z"Colombo, Maria"https://zbmath.org/authors/?q=ai:colombo.maria"De Rosa, Luigi"https://zbmath.org/authors/?q=ai:de-rosa.luigi"Sorella, Massimo"https://zbmath.org/authors/?q=ai:sorella.massimoSummary: In this work we show that, in the class of \(L^\infty ((0,T); L^2(\mathbb{T}^3))\) distributional solutions of the incompressible Navier-Stokes system, the ones which are smooth in some open interval of times are meagre in the sense of Baire category, and the Leray ones are a nowhere dense set.Leray's backward self-similar solutions to the 3D Navier-Stokes equations in Morrey spaceshttps://zbmath.org/1492.351882022-10-04T19:40:27.024758Z"Wang, Yanqing"https://zbmath.org/authors/?q=ai:wang.yanqing"Jiu, Quansen"https://zbmath.org/authors/?q=ai:jiu.quansen"Wei, Wei"https://zbmath.org/authors/?q=ai:wei.wei.3|wei.wei.5|wei.wei.2|wei.wei.6|wei.wei.4|wei.wei.7The problem of (non)existence of backward self-similar solutions of the three-dimensional Navier-Stokes equations is studied in the framework of homogeneous Morrey spaces. The authors obtain results on triviality of such solutions in the spaces \(\dot M^{q,1}(\mathbb R^3)\) for \(3/2<q<6\), and in \(\dot M^{q,\ell}(\mathbb R^3)\) for \(6\le q<\infty\), \(2<\ell\le q\), improving many previous results and simplifying the announcements: [\textit{J. Nečas} et al., Acta Math. 176, No. 2, 283--294 (1996; Zbl 0884.35115); \textit{T.-P. Tsai}, Arch. Ration. Mech. Anal. 143, No. 1, 29--51 (1998; Zbl 0916.35084); \textit{D. Chae} and \textit{J. Wolf}, Arch. Ration. Mech. Anal. 225, No. 1, 549--572 (2017; Zbl 1367.35104); \textit{C. Guevara} and \textit{N. C. Phuc}, SIAM J. Math. Anal. 50, No. 1, 541--556 (2018; Zbl 1391.35314)].
Reviewer: Piotr Biler (Wrocław)Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the cornerhttps://zbmath.org/1492.351922022-10-04T19:40:27.024758Z"Agrawal, Siddhant"https://zbmath.org/authors/?q=ai:agrawal.siddhant"Nahmod, Andrea R."https://zbmath.org/authors/?q=ai:nahmod.andrea-rStochastic 2D rotating Euler flows with bounded vorticity or white noise initial conditionshttps://zbmath.org/1492.352022022-10-04T19:40:27.024758Z"Gao, Hongjun"https://zbmath.org/authors/?q=ai:gao.hongjun"Gao, Xiancheng"https://zbmath.org/authors/?q=ai:gao.xianchengSticky particles and the pressureless Euler equations in one spatial dimensionhttps://zbmath.org/1492.352032022-10-04T19:40:27.024758Z"Hynd, Ryan"https://zbmath.org/authors/?q=ai:hynd.ryanSummary: We consider the dynamics of finite systems of point masses which move along the real line. We suppose the particles interact pairwise and undergo perfectly inelastic collisions when they collide. In particular, once particles collide, they remain stuck together thereafter. Our main result is that if the interaction potential is semi convex, this sticky particle property can be quantified and is preserved upon letting the number of particles tend to infinity. This is used to show that solutions of the pressureless Euler equations exist for given initial conditions and satisfy an entropy inequality.On global smooth solutions of the 3D spherically symmetric Euler equations with time-dependent damping and physical vacuumhttps://zbmath.org/1492.352092022-10-04T19:40:27.024758Z"Pan, Xinghong"https://zbmath.org/authors/?q=ai:pan.xinghongChaos in the incompressible Euler equation on manifolds of high dimensionhttps://zbmath.org/1492.352102022-10-04T19:40:27.024758Z"Torres de Lizaur, Francisco"https://zbmath.org/authors/?q=ai:torres-de-lizaur.franciscoSummary: We construct finite dimensional families of non-steady solutions to the Euler equations, existing for all time, and exhibiting all kinds of qualitative dynamics in the phase space, for example: strange attractors and chaos, invariant manifolds of arbitrary topology, and quasiperiodic invariant tori of any dimension. The main theorem of the paper, from which these families of solutions are obtained, states that for any given vector field \(X\) on a closed manifold \(N\), there is a Riemannian manifold \(M\) on which the following holds: \(N\) is diffeomorphic to a finite dimensional manifold in the phase space of fluid velocities (the space of divergence-free vector fields on \(M)\) that is invariant under the Euler evolution, and on which the Euler equation reduces to a finite dimensional ODE that is given by an arbitrarily small smooth perturbation of the vector field \(X\) on \(N\).The initial-value problem to the modified two-component Euler-Poincaré equationshttps://zbmath.org/1492.352112022-10-04T19:40:27.024758Z"Yan, Kai"https://zbmath.org/authors/?q=ai:yan.kai"Liu, Yue"https://zbmath.org/authors/?q=ai:liu.yueWeak solutions to the time-fractional \(g\)-Navier-Stokes equations and optimal controlhttps://zbmath.org/1492.352142022-10-04T19:40:27.024758Z"Aadi, Sultana Ben"https://zbmath.org/authors/?q=ai:ben-aadi.sultana"Akhlil, Khalid"https://zbmath.org/authors/?q=ai:akhlil.khalid"Aayadi, Khadija"https://zbmath.org/authors/?q=ai:aayadi.khadijaThe authors consider a modification of the Navier-Stokes equations with a time-fractional derivative and the condition \(\nabla\cdot(gu)=0\) for some function \(g\) replacing the usual incompressibility condition. Besides existence and uniqueness of weak solutions, an optimal control problem is discussed.
Reviewer: Piotr Biler (Wrocław)On well-posedness of generalized Hall-magneto-hydrodynamicshttps://zbmath.org/1492.352212022-10-04T19:40:27.024758Z"Dai, Mimi"https://zbmath.org/authors/?q=ai:dai.mimi"Liu, Han"https://zbmath.org/authors/?q=ai:liu.hanSummary: We obtain local well-posedness result for the generalized Hall-magneto-hydrodynamics system in Besov spaces \(\dot{B}^{-(2\alpha_1 -\gamma)}_{\infty, \infty} \times \dot{B}^{-(2\alpha_2 -\beta)}_{\infty, \infty}(\mathbb{R}^3)\) with suitable indexes \(\alpha_1, \alpha_2, \beta\) and \(\gamma\). As a corollary, the hyperdissipative electron magneto-hydrodynamics system is globally well-posed in \(\dot{B}^{-(2\alpha_2 -2)}_{\infty, \infty} (\mathbb{R}^3)\) for small initial data.Steady flow with unilateral and leak/slip boundary conditions by the Stokes variational-hemivariational inequalityhttps://zbmath.org/1492.352282022-10-04T19:40:27.024758Z"Migórski, Stanisław"https://zbmath.org/authors/?q=ai:migorski.stanislaw"Dudek, Sylwia"https://zbmath.org/authors/?q=ai:dudek.sylwiaSummary: The stationary Stokes equations for a generalized Newtonian fluid with nonlinear unilateral, and slip and leak boundary conditions are investigated. Boundary conditions include the generalized Clarke gradient and the convex subdifferential, and the variational formulation of the problem is the variational-hemivariational inequality for the velocity field. Existence and uniqueness result for weak solution is proved by using a surjectivity theorem for a pseudomonotone perturbation of a maximal monotone operator.Well-posedness and convergence results for the 3D-Lagrange Boussinesq-\(\alpha\) systemhttps://zbmath.org/1492.352322022-10-04T19:40:27.024758Z"Sboui, Abir"https://zbmath.org/authors/?q=ai:sboui.abir"Selmi, Ridha"https://zbmath.org/authors/?q=ai:selmi.ridhaSummary: In this paper, we study the three-dimensional Lagrangian averaged Boussinesq-\(\alpha\) system which is a regularized version of the three-dimensional Boussinesq system. We prove the existence of a weak solution to the 3D-Lagrangian averaged Boussinesq-\(\alpha\) system, in Sobolev spaces. Unlike preceding works, this solution is global in time and depends continuously on the initial data, in particular, it is unique. More importantly, it converges to a weak solution of the three-dimensional Boussinesq system, as the regularizing parameter \(\alpha\) vanishes.Oskolkov models and Sobolev-type equationshttps://zbmath.org/1492.352332022-10-04T19:40:27.024758Z"Sukacheva, Tamara Gennad'evna"https://zbmath.org/authors/?q=ai:sukacheva.tamara-gennadevnaSummary: This article is a review of the works carried out by the author together with her students and devoted to the study of various Oskolkov models. Their distinctive feature is the use of the semigroup approach, which is the basis of the phase space method used widely in the theory of Sobolev-type equations. Various models of an incompressible viscoelastic fluid described by the Oskolkov equations are presented. The degenerate problem of magnetohydrodynamics, the problem of thermal convection, and the Taylor problem are considered as examples. The solvability of the corresponding initial-boundary value problems is investigated within the framework of the theory of Sobolev-type equations based on the theory for \(p\)-sectorial operators and degenerate semigroups of operators. An existence theorem is proved for a unique solution, which is a quasi-stationary semitrajectory, and a description of the extended phase space is obtained. The foundations of the theory of solvability of Sobolev-type equations were laid by Professor G.A. Sviridyuk. Then this theory, together with various applications, was successfully developed by his followers.Time-decay estimates for the linearized water wave type equationshttps://zbmath.org/1492.352342022-10-04T19:40:27.024758Z"Tesfahun, Achenef"https://zbmath.org/authors/?q=ai:tesfahun.achenefSummary: Recently, A. Bulut showed that the free waves \(S_\alpha (t) f:=\exp \left( it |\nabla |^{\alpha}\right) f\) in 1D for \(\alpha \in (1/3, 1/2]\), which are known to be associated with the linearized gravity water wave equations, decay at time scale of order \(|t|^{-1/2}\) for large \(t\), provided that the \(H^1_x(\mathbb{R})\)-norm of \(f\) and the \(L^2_x(\mathbb{R})\)-norm of \(x\partial_x f\) are bounded. In this note we derive a decay estimate of order \((1-\alpha)^{-1/2} (\alpha |t|)^{-d/2}\) on \(S_\alpha (t)f\) for all \(\alpha \in (0, 1)\) and \(d\ge 1\), assuming a bound only on the \(\dot{B}_{1, 1}^{d(1-\alpha /2)} (\mathbb{R}^d)\)-norm of \(f\). Our estimate extends to any dimension, a wider range of \(\alpha\) and describes well the behaviour of the decay near \(\alpha =0\) and \(\alpha =1\), without requiring a spatial-decay assumption on \(f\) or its derivative.On the well-posedness of the compressible Navier-Stokes-Korteweg system with special viscosity and capillarityhttps://zbmath.org/1492.352382022-10-04T19:40:27.024758Z"Yu, Yanghai"https://zbmath.org/authors/?q=ai:yu.yanghai"Zhou, Mulan"https://zbmath.org/authors/?q=ai:zhou.mulanSummary: In this paper we consider the Cauchy problem to the compressible Navier-Stokes-Korteweg system with a specific choice on the viscosity and capillarity in the whole space \(\mathbb{R}^d\), and establish the local well-posedness of strong solutions for large initial data in the framework of Besov spaces. Furthermore, we construct the global-in-time small solutions in \(L^2\) type critical Besov spaces, where the vertical component of the divergence-free part of the velocity could be arbitrarily large initially.The Cauchy problem for the Klein-Gordon equation under the quartic potential in the De Sitter spacetimehttps://zbmath.org/1492.352422022-10-04T19:40:27.024758Z"Nakamura, Makoto"https://zbmath.org/authors/?q=ai:nakamura.makotoSummary: The Cauchy problem for the Klein-Gordon equation under the quartic potential is considered in the de Sitter spacetime. The existence of global solutions for small rough initial data is shown based on the mechanism of the spontaneous symmetry breaking for the small positive Hubble constant. The effects of the spatial expansion and contraction on the problem are considered.
{\copyright 2021 American Institute of Physics}Nonlinear nonisospectral differential coverings for the hyper-CR equation of Einstein-Weyl structures and the Gibbons-Tsarev equationhttps://zbmath.org/1492.352482022-10-04T19:40:27.024758Z"Morozov, Oleg I."https://zbmath.org/authors/?q=ai:morozov.oleg-iSummary: We apply the technique based on twisted extensions of symmetry algebras to construct new nonlinear four-dimensional differential coverings for the hyper-CR equation of Einstein-Weyl structures and for the associated integrable hierarchy. We expose related multi-component three-dimensional covering. By the symmetry reduction of the hyper-CR equation of Einstein-Weyl structures we derive nonlinear three-dimensional differential covering for the Gibbons-Tsarev equation.Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equationhttps://zbmath.org/1492.352522022-10-04T19:40:27.024758Z"Chen, Mingjuan"https://zbmath.org/authors/?q=ai:chen.mingjuan"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.boling"Han, Lijia"https://zbmath.org/authors/?q=ai:han.lijiaSummary: In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation \(\partial_t u - \epsilon \partial_x^2 u + \mathcal{H}\partial_x^2 u + uu_x = 0\), where \(\mathcal{H}\) denotes the Hilbert transform operator. We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space \(\widetilde{H}^{\sigma}(\mathbb{R})\;(\sigma \geqslant 0)\), which is a subspace of \(L^2 (\mathbb{R})\). It is worth noting that the low-frequency part of \(\widetilde{H}^{\sigma} (\mathbb{R})\) is scaling critical, and thus the small data is necessary. The high-frequency part of \(\widetilde{H}^{\sigma}(\mathbb{R})\) is equal to the Sobolev space \(H^{\sigma} (\mathbb{R}) \;(\sigma \geqslant 0)\) and reduces to \(L^2 (\mathbb{R})\). Furthermore, we also obtain its inviscid limit behavior in \(\widetilde{H}^{\sigma}(\mathbb{R})\; (\sigma \geqslant 0)\).Center manifold for the third-order nonlinear Schrödinger equation with critical lengthshttps://zbmath.org/1492.352532022-10-04T19:40:27.024758Z"Chen, Mo"https://zbmath.org/authors/?q=ai:chen.moSummary: This paper is concerned with the third-order nonlinear Schrödinger equation, a version of the center manifold theorem is established which is suitable for the third-order nonlinear Schrödinger equation with critical lengths.The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbationhttps://zbmath.org/1492.352542022-10-04T19:40:27.024758Z"Chen, Shuting"https://zbmath.org/authors/?q=ai:chen.shuting"Du, Zengji"https://zbmath.org/authors/?q=ai:du.zengji"Liu, Jiang"https://zbmath.org/authors/?q=ai:liu.jiang"Wang, Ke"https://zbmath.org/authors/?q=ai:wang.ke.4|wang.ke|wang.ke.1|wang.ke.2|wang.ke.3Summary: In this paper, we are concerned with the existence of solitary waves for a generalized Kawahara equation, which is a model equation describing solitary-wave propagation in media. We obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the generalized Kawahara equation without delay and perturbation by employing the phase space analysis. Furthermore the existence of solitary wave solutions for the equation with two types of special delay convolution kernels is proved by combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality. We also discuss the asymptotic behaviors of traveling wave solutions by means of the asymptotic theory. Finally, some examples are given to illustrate our results.Existence and stability of traveling waves of the fifth-order KdV equationhttps://zbmath.org/1492.352602022-10-04T19:40:27.024758Z"Esfahani, Amin"https://zbmath.org/authors/?q=ai:esfahani.amin"Levandosky, Steven"https://zbmath.org/authors/?q=ai:levandosky.steven-paulSummary: We consider the existence and stability of traveling waves of the Fifth-Order KdV equation for a general class of nonlinearities that satisfy power-like scaling relations. This class of nonlinearities includes sums and differences of powers. For such nonlinearities we use variational methods to show that there exist ground state traveling wave solutions and use the variational properties of the ground states to analyze their stability.Initial-boundary value problems on a half-strip for the generalized Kawahara-Zakharov-Kuznetsov equationhttps://zbmath.org/1492.352612022-10-04T19:40:27.024758Z"Faminskii, Andrei V."https://zbmath.org/authors/?q=ai:faminskii.andrei-vSummary: Initial-boundary value problems on a half-strip with different types of boundary conditions for the generalized Kawahara-Zakharov-Kuznetsov equation with nonlinearity of higher order are considered. In particular nonlinearity can be quadratic and cubic. Results on global existence and uniqueness in classes of weak and strong solutions and large-time decay of small solutions are established. The solutions are considered in weighted at infinity Sobolev spaces. The use of weighted spaces is crucial for the study. To this end new interpolating inequalities in weighted anisotropic Sobolev spaces are established. Both exponential and power weights are admissible.The existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equationhttps://zbmath.org/1492.352632022-10-04T19:40:27.024758Z"Guo, Yingying"https://zbmath.org/authors/?q=ai:guo.yingying"Yin, Zhaoyang"https://zbmath.org/authors/?q=ai:yin.zhaoyangSummary: This paper is devoted to studying the existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation without any additional assumptions. Firstly, introducing a new set of variables, we transform the single-cycle pulse equation into an equivalent semilinear system. Using the standard ordinary differential equation theory, the global solution of the semilinear system is studied. Secondly, returning to the original coordinates, we get a global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, choosing some vital test functions which are different from [\textit{A. Bressan} et al., Discrete Contin. Dyn. Syst. 35, No. 1, 25--42 (2015; Zbl 1304.35017); \textit{J. C. Brunelli}, Phys. Lett., A 353, No. 6, 475--478 (2006; Zbl 1181.37094)], we find a equation to single out a unique characteristic curve through each initial point. Moreover, the uniqueness of global admissible conservative weak solution is obtained.The Neumann and Robin problems for the Korteweg-de Vries equation on the half-linehttps://zbmath.org/1492.352642022-10-04T19:40:27.024758Z"Himonas, A. Alexandrou"https://zbmath.org/authors/?q=ai:himonas.a-alexandrou"Madrid, Carlos"https://zbmath.org/authors/?q=ai:madrid.carlos"Yan, Fangchi"https://zbmath.org/authors/?q=ai:yan.fangchiSummary: The well-posedness of the Neumann and Robin problems for the Korteweg-de Vries equation is studied with data in Sobolev spaces. Using the Fokas unified transform method, the corresponding linear problems with forcing are solved and solution estimates are derived. Then, using these, an iteration map is defined, and it is proved to be a contraction in appropriate solution spaces after the needed bilinear estimates are derived.
{\copyright 2021 American Institute of Physics}The Korteweg-de Vries equation on the half-line with Robin and Neumann data in low regularity spaceshttps://zbmath.org/1492.352652022-10-04T19:40:27.024758Z"Himonas, A. Alexandrou"https://zbmath.org/authors/?q=ai:himonas.a-alexandrou"Yan, Fangchi"https://zbmath.org/authors/?q=ai:yan.fangchiSummary: The well-posedness of the initial-boundary value problem (ibvp) for the Korteweg-de Vries equation on the half-line is studied for initial data \(u_0 (x)\) in spatial Sobolev spaces \(H^s (0, \infty)\), \(s > - 3 / 4\), and Robin and Neumann boundary data \(\varphi (t)\) in the temporal Sobolev spaces suggested by the time regularity of the Cauchy problem for the corresponding linear equation. First, linear estimates in Bourgain spaces are derived by utilizing the Fokas solution formula of the ibvp for the forced linear equation. Then, using these and the needed bilinear estimates, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space.Martingale solution of stochastic hybrid Korteweg-de Vries-Burgers equationhttps://zbmath.org/1492.352672022-10-04T19:40:27.024758Z"Karczewska, Anna"https://zbmath.org/authors/?q=ai:karczewska.anna"Szczeciński, Maciej"https://zbmath.org/authors/?q=ai:szczecinski.maciejSummary: In the paper, we consider a stochastic hybrid Korteweg-de Vries-Burgers type equation with multiplicative noise in the form of cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied. The proof of the existence of the solution is based on two approximations of the considered problem and the compactness method. First, we introduce an auxiliary problem corresponding to the equation studied. Then, we prove the existence of a martingale solution to this problem. Finally, we show that the solution of the auxiliary problem converges, in some sense, coincides to the solution of the equation under consideration.Fixed analytic radius lower bound for the dissipative KdV equation on the real linehttps://zbmath.org/1492.352722022-10-04T19:40:27.024758Z"Liu, Ke"https://zbmath.org/authors/?q=ai:liu.ke"Wang, Ming"https://zbmath.org/authors/?q=ai:wang.mingSummary: We study the global analyticity for the dissipative KdV equation with an analytic initial data on the real line. We show that the analytic radius of the solution has a fixed positive lower bound uniformly for all time. This reflects the dissipative effect to some extent, since the best known analytic radius of the KdV equation may decay polynomially as time goes to infinity.Global existence and blow-up phenomena for a periodic modified Camassa-Holm equation (MOCH)https://zbmath.org/1492.352732022-10-04T19:40:27.024758Z"Luo, Zhaonan"https://zbmath.org/authors/?q=ai:luo.zhaonan"Qiao, Zhijun"https://zbmath.org/authors/?q=ai:qiao.zhijun"Yin, Zhaoyang"https://zbmath.org/authors/?q=ai:yin.zhaoyangSummary: In this paper, we study global existence and blow-up for a periodic modified Camassa-Holm equation in nonhomogeneous Sobolev spaces. Also, we provide a key blow-up criteria to investigate norm inflation and ill-posedness problem for the equation in the critical Sobolev space.The non-commutative Korteweg-de Vries hierarchy and combinatorial Pöppe algebrahttps://zbmath.org/1492.352752022-10-04T19:40:27.024758Z"Malham, Simon J. A."https://zbmath.org/authors/?q=ai:malham.simon-j-aSummary: We give a constructive proof, to all orders, that each member of the non-commutative potential Korteweg-de Vries hierarchy is a Fredholm Grassmannian flow and is therefore linearisable. Indeed we prove this for any linear combination of fields from this hierarchy. That each member of the hierarchy is linearisable, and integrable in this sense, means that the time evolving solution can be generated from the solution to the corresponding linear dispersion equation in the hierarchy, combined with solving an associated linear Fredholm equation representing the Marchenko equation. Further, we show that within the class of polynomial partial differential fields, at every order, each member of the non-commutative potential Korteweg-de Vries hierarchy is unique. Indeed, we prove to all orders, that each such member matches the non-commutative Lax hierarchy field, which is therefore a polynomial partial differential field. We achieve this by constructing the abstract combinatorial algebra that underlies the non-commutative potential Korteweg-de Vries hierarchy. This algebra is the non-commutative polynomial algebra over the real line generated by the set of all compositions endowed with the Pöppe product. This product is the abstract representation of the product rule for Hankel operators pioneered by Ch. Pöppe for integrable equations such as the Sine-Gordon and Korteweg-de Vries equations. Integrability of the hierarchy members translates, in the combinatorial algebra, to proving the existence of a `Pöppe polynomial' expansion for basic compositions in terms of `linear signature expansions'. Proving the existence of such Pöppe polynomial expansions boils down to solving a linear algebraic problem for the expansion coefficients, which we solve constructively to all orders.On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersionhttps://zbmath.org/1492.352772022-10-04T19:40:27.024758Z"Morales Paredes, Jorge"https://zbmath.org/authors/?q=ai:morales-paredes.jorge"Méndez, Félix Humberto Soriano"https://zbmath.org/authors/?q=ai:mendez.felix-humberto-sorianoSummary: In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated with a family of equations of ZK-KP-type
\[
\begin{cases}
u_t = u_{xxx}-\mathscr{H}D_x^{\alpha}u_{yy}+uu_x,\\
u(0) = \psi \in Z
\end{cases}
\]
in anisotropic Sobolev spaces, where \(1\leqslant \alpha \leqslant 1\), \(\mathscr{H}\) is the Hilbert transform and \(D_x^{\alpha}\) is the fractional derivative, both with respect to \(x\).The existence of global weak solutions for a generalized Camassa-Holm equationhttps://zbmath.org/1492.352872022-10-04T19:40:27.024758Z"Tu, Xi"https://zbmath.org/authors/?q=ai:tu.xi"Yin, Zhaoyang"https://zbmath.org/authors/?q=ai:yin.zhaoyangSummary: In this paper, we mainly study the Cauchy problem of a generalized Camassa-Holm equation. We prove the existence of global weak solutions for this generalized Camassa-Holm equation by the viscous approximation method.High-speed excited multi-solitons in competitive power nonlinear Schrödinger equationshttps://zbmath.org/1492.352952022-10-04T19:40:27.024758Z"Bai, Mengxue"https://zbmath.org/authors/?q=ai:bai.mengxue"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian.2|zhang.jian.1|zhang.jian.4|zhang.jian.7|zhang.jian.3|zhang.jian|zhang.jian.6|zhang.jian.5Summary: This paper deals with the competitive power nonlinear Schrödinger equation, which originates from the cubic-quintic model in physics. The equation admits infinitely many excited solitons, and the Cauchy problem is globally well-posed in the energy space. In terms of \textit{R. Côte} and \textit{S. Le Coz}'s argument [J. Math. Pures Appl. (9) 96, No. 2, 135--166 (2011; Zbl 1225.35215)], high-speed excited multi-solitons of the equation are constructed, which extend Côte and Le Coz's results from the focusing nonlinear cases to the competitive nonlinear cases combining the focusing nonlinearities and defocusing nonlinearities.Scattering for the non-radial inhomogenous biharmonic NLS equationhttps://zbmath.org/1492.352982022-10-04T19:40:27.024758Z"Campos, Luccas"https://zbmath.org/authors/?q=ai:campos.luccas"Guzmán, Carlos M."https://zbmath.org/authors/?q=ai:guzman.carlos-mSummary: We consider the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in \(H^2(\mathbb{R}^N)\),
\[
iu_t + \Delta^2 u - |x|^{-b}|u|^\alpha u=0,
\]
when \(b > 0\) and \(N \ge 5\). We first obtain a small data global result in \(H^2\), which, in the five-dimensional case, improves a previous result from the second author and \textit{A. Pastor} [Nonlinear Anal., Real World Appl. 56, Article ID 103174, 35 p. (2020; Zbl 1451.35185)]. In the sequel, we show the main result, scattering below the mass-energy threshold in the intercritical case, that is, \(\frac{8-2b}{N} < \alpha < \frac{8-2b}{N-4}\), without assuming radiality of the initial data. The proof combines the decay of the nonlinearity with Virial-Morawetz-type estimates to avoid the radial assumption, allowing for a much simpler proof than the Kenig-Merle roadmap.The nonlinear Schrödinger-Airy equation in weighted Sobolev spaceshttps://zbmath.org/1492.353002022-10-04T19:40:27.024758Z"Castro, A. J."https://zbmath.org/authors/?q=ai:castro.alejandro-j"Jabbarkhanov, K."https://zbmath.org/authors/?q=ai:jabbarkhanov.k"Zhapsarbayeva, L."https://zbmath.org/authors/?q=ai:zhapsarbaeva.lyailya-kurmantaevnaSummary: We study the persistence property of the solution for the nonlinear Schrödinger-Airy equation with initial data in the weighted Sobolev space \(H^{1 / 4} (\mathbb{R}) \cap L^2 (|x|^{2m} dx)\), \(0 < m \leq 1/8\), via the contraction principle.The unconditional uniqueness for the energy-supercritical NLShttps://zbmath.org/1492.353012022-10-04T19:40:27.024758Z"Chen, Xuwen"https://zbmath.org/authors/?q=ai:chen.xuwen"Shen, Shunlin"https://zbmath.org/authors/?q=ai:shen.shunlin"Zhang, Zhifei"https://zbmath.org/authors/?q=ai:zhang.zhifei.1|zhang.zhifeiSummary: We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the \(\mathbb{R}^d\) and \(\mathbb{T}^d\) energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [\textit{X. Chen} and \textit{J. Holmer}, Invent. Math. 217, No. 2, 433--547 (2019; Zbl 1422.35148); Forum Math. Pi 10, Paper No. e3, 49 p. (2022; Zbl 1483.35201)]], the unconditional uniqueness problems for \(H^1\)-critical and \(H^1\)-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [\textit{F. Merle} et al., Invent. Math. 227, No. 1, 247--413 (2022; Zbl 1487.35353)] are the only possible \(C([0,T);\dot{H}^{s_c})\) solutions if exist in these domains.Well-posedness and blow-up of Virial type for some fractional inhomogeneous Choquard equationshttps://zbmath.org/1492.353022022-10-04T19:40:27.024758Z"Chergui, L."https://zbmath.org/authors/?q=ai:chergui.lassaadSummary: In the subcritical energy case, local well-posedness is established in the radial energy space for a class of fractional inhomogeneous Choquard equations. The best constant of a Gagliardo-Nirenberg type inequality is obtained. Moreover, a sharp threshold of global existence versus blow-up dichotomy is obtained for mass super-critical and energy subcritical solutions.Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speedhttps://zbmath.org/1492.353042022-10-04T19:40:27.024758Z"Dinh, Van Duong"https://zbmath.org/authors/?q=ai:dinh.van-duongSummary: We study the existence and stability of standing waves associated with the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose-Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, mass-critical, and mass-supercritical nonlinearities. Our result extends a recent work of \textit{J. Bellazzini} et al. [Commun. Math. Phys. 353, No. 1, 229--251 (2017; Zbl 1367.35150)], where the existence and stability of standing waves for the supercritical NLS with a partial confinement were established.Global dynamics in nonconservative nonlinear Schrödinger equationshttps://zbmath.org/1492.353092022-10-04T19:40:27.024758Z"Jaquette, Jonathan"https://zbmath.org/authors/?q=ai:jaquette.jonathan"Lessard, Jean-Philippe"https://zbmath.org/authors/?q=ai:lessard.jean-philippe"Takayasu, Akitoshi"https://zbmath.org/authors/?q=ai:takayasu.akitoshiAuthors' abstract: In this paper, we study the global dynamics of a class of nonlinear Schrödinger equations using perturbative and non-perturbative methods. We prove the semi-global existence of solutions for initial conditions close to constant. That is, solutions will exist for all positive time or all negative time. The existence of an open set of initial data which limits to zero in both forward and backward time is also demonstrated. This result in turn forces the non-existence of any real-analytic conserved quantities. For the quadratic case, we prove the existence of two (infinite) families of nontrivial unstable equilibria and prove the existence of heteroclinic orbits limiting to the nontrivial equilibria in backward time and to zero in forward time. By a time reversal argument, we also obtain heteroclinic orbits limiting to the nontrivial equilibria in forward time and to zero in backward time. The proofs for the quadratic equation are computer-assisted and rely on three separate ingredients: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of the flow (starting from the unstable manifold) and a proof that the solution enters a validated stable set (hence showing convergence to zero).
Reviewer: Anthony D. Osborne (Keele)The defocusing energy-supercritical nonlinear Schrödinger equation in high dimensionshttps://zbmath.org/1492.353122022-10-04T19:40:27.024758Z"Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.6|li.jing.13|li.jing|li.jing.12|li.jing.3|li.jing.10|li.jing.1|li.jing.5|li.jing.11|li.jing.2|li.jing.7|li.jing.4"Li, Kuijie"https://zbmath.org/authors/?q=ai:li.kuijieSharp condition for inhomogeneous nonlinear Schrödinger equations by cross-invariant manifoldshttps://zbmath.org/1492.353132022-10-04T19:40:27.024758Z"Lin, Qiang"https://zbmath.org/authors/?q=ai:lin.qiang"Yang, Chao"https://zbmath.org/authors/?q=ai:yang.chao|yang.chao.1|yang.chao.3|yang.chao.2Summary: In this paper, we study a class of cross constrained variational problem for the inhomogeneous nonlinear Schrödinger equation in \({\mathbb{R}}^N\). By constructing cross-invariant manifolds, we derive a sharp condition for blow-up phenomenon and global well-posedness of solutions.Novel optical soliton waves in metamaterials with parabolic law of nonlinearity via the IEFM and ISEMhttps://zbmath.org/1492.353142022-10-04T19:40:27.024758Z"Li, Xiaoyan"https://zbmath.org/authors/?q=ai:li.xiaoyan"Manafian, Jalil"https://zbmath.org/authors/?q=ai:manafian-heris.jalil"Abotaleb, Mostafa"https://zbmath.org/authors/?q=ai:abotaleb.mostafa"Ilhan, Onur Alp"https://zbmath.org/authors/?q=ai:ilhan.onur-alp"Oudah, Atheer Y."https://zbmath.org/authors/?q=ai:oudah.atheer-y"Prakaash, A. S."https://zbmath.org/authors/?q=ai:prakaash.a-s(no abstract)Normalized ground states for general pseudo-relativistic Schrödinger equationshttps://zbmath.org/1492.353152022-10-04T19:40:27.024758Z"Luo, Haijun"https://zbmath.org/authors/?q=ai:luo.haijun"Wu, Dan"https://zbmath.org/authors/?q=ai:wu.danSummary: In this paper, we consider the pseudo-relativistic type Schrödinger equations with general nonlinearities. By studying the related constrained minimization problems, we obtain the existence of ground states via applying the concentration-compactness principle. Then some properties of the ground states have been discussed, including regularity, symmetry and etc. Furthermore, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near it for all time.Behaviour of solutions to the 1D focusing stochastic \(L^2\)-critical and supercritical nonlinear Schrödinger equation with space-time white noisehttps://zbmath.org/1492.353162022-10-04T19:40:27.024758Z"Millet, Annie"https://zbmath.org/authors/?q=ai:millet.annie"Roudenko, Svetlana"https://zbmath.org/authors/?q=ai:roudenko.svetlana"Yang, Kai"https://zbmath.org/authors/?q=ai:yang.kaiSummary: We study the focusing stochastic nonlinear Schrödinger equation in 1D in the \(L^2\)-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the \(L^2\)-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions. In particular, we show that the noise may induce blow up, thus ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow up happens, it has the same dynamics as in the deterministic setting; however, there is a (random) shift of the blow-up centre, which can be described as a random variable normally distributed.Scattering threshold for a coupled focusing nonlinear Schrödinger systemhttps://zbmath.org/1492.353182022-10-04T19:40:27.024758Z"Saanouni, T."https://zbmath.org/authors/?q=ai:saanouni.tarekSummary: It is the purpose of this note, to obtain a scattering versus finite time blow-up dichotomy for a mass super-critical and energy sub-critical coupled Schrödinger system.Normalized ground state traveling solitary waves for the half-wave equations with combined nonlinearitieshttps://zbmath.org/1492.353232022-10-04T19:40:27.024758Z"Zhang, Guoqing"https://zbmath.org/authors/?q=ai:zhang.guoqing"Li, Yawen"https://zbmath.org/authors/?q=ai:li.yawenSummary: In this paper, we consider the half-wave equations with combined power nonlinearities
\[
i\partial_t u=\sqrt{-\Delta}u-\mu |u|^{q-1}u-|u|^{p-1}u,\; (t,x) \in \mathbb{R}\times \mathbb{R}^d,
\]
where \(d\geq 2\), \(\mu \in \mathbb{R}\) and \(1<q<p< 1+\frac{2}{d-1}\). We study traveling solitary waves of the form
\[
u(x,t)=e^{i\omega t}Q(x-vt),
\]
with frequency \(w \in \mathbb{R}\), and velocity \(v\in \mathbb{R}^d\). As \(|v| \geqslant 1\), we establish a general nonexistence of traveling solitary waves by using Riesz transforms and a virial-type identity. As \(0< |v| < 1\), under different assumptions on \(q< p\), we prove several existence results for traveling solitary waves. In particular, we consider cases when
\[
1< q< 1+\frac{2}{d}< p< 1+\frac{2}{d-1},
\]
i.e., the two nonlinearities have different character with respect to the \(L^2\)-critical exponent. Note that such traveling solitary waves \(Q\) is not radially symmetric in \(x \in \mathbb{R}^d\), and we need to overcome the lack of compactness and obtain the existence of mountain-pass-type solution and saddle-type solution. In addition, based on the existence and properties of traveling solitary waves, we also prove that small data scattering fails to hold for the nonlinear half-wave equations.Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticityhttps://zbmath.org/1492.353312022-10-04T19:40:27.024758Z"Aouadi, Moncef"https://zbmath.org/authors/?q=ai:aouadi.moncefThe paper presents a nonlocal linearized model for porous elastic materials. The configuration is described by the deformation and the volume fraction field (related to the presence of voids in the bulk). The second gradients of both variables feature in the constitutive relations. The PDE system is composed of two hyperbolic equations with two parameters related to nonlocal terms. The authors consider the case of dimension one and study well-posedness and exponential decay of the system. The approach is based on nonlinear semigroups and monotone operators.
Reviewer: Giuliano Lazzaroni (Firenze)Global existence and propagation of moments for a Vlasov-Poisson equation with a point chargehttps://zbmath.org/1492.353422022-10-04T19:40:27.024758Z"Miot, Evelyne"https://zbmath.org/authors/?q=ai:miot.evelyneSummary: Le but de ce texte est de présenter des résultats, en collaboration avec \textit{L. Desvillettes} and \textit{C. Saffirio} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 32, No. 2, 373--400 (2015; Zbl 1323.35178)], à propos de l'existence globale d'une solution pour un système de Vlasov-Poisson avec une charge ponctuelle en dimension trois.Rigorous analysis and dynamics of Hibler's sea ice modelhttps://zbmath.org/1492.353482022-10-04T19:40:27.024758Z"Brandt, Felix"https://zbmath.org/authors/?q=ai:brandt.felix"Disser, Karoline"https://zbmath.org/authors/?q=ai:disser.karoline"Haller-Dintelmann, Robert"https://zbmath.org/authors/?q=ai:haller-dintelmann.robert"Hieber, Matthias"https://zbmath.org/authors/?q=ai:hieber.matthiasSummary: This article develops for the first time a rigorous analysis of Hibler's model of sea ice dynamics. Identifying Hibler's ice stress as a quasilinear second-order operator and regarding Hibler's model as a quasilinear evolution equation, it is shown that a regularized version of Hibler's coupled sea ice model, i.e., the model coupling velocity, thickness and compactness of sea ice, is locally strongly well-posed within the \(L_q\)-setting and also globally strongly well-posed for initial data close to constant equilibria.Second order local minimal-time mean field gameshttps://zbmath.org/1492.353522022-10-04T19:40:27.024758Z"Ducasse, Romain"https://zbmath.org/authors/?q=ai:ducasse.romain"Mazanti, Guilherme"https://zbmath.org/authors/?q=ai:mazanti.guilherme"Santambrogio, Filippo"https://zbmath.org/authors/?q=ai:santambrogio.filippoSummary: The paper considers a forward-backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain \(\Omega\) in minimal expected time. Agents are constrained by a bound on the drift depending on the density of other agents at their location. Existence for a finite time horizon \(T\) is proven via a fixed point argument, but the natural setting for this problem is in infinite time horizon. Estimates are needed to treat the limit \(T\rightarrow \infty\), and the asymptotic behavior of the solution obtained in this way is also studied. This passes through classical parabolic arguments and specific computations for MFGs. Both the Fokker-Planck equation on the density of agents and the Hamilton-Jacobi-Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach \(\partial\Omega\). The initial datum for the density is given, and the long-time limit of the value function is characterized as the solution of a stationary problem.On classical solutions to the mean field game system of controlshttps://zbmath.org/1492.353532022-10-04T19:40:27.024758Z"Kobeissi, Ziad"https://zbmath.org/authors/?q=ai:kobeissi.ziadSummary: We consider a class of mean field games in which the optimal strategy of a representative agent depends on the statistical distribution of both the states and controls. We prove some existence results for the forward-backward system of PDEs in a regime never considered so far, where agents may somehow favor a velocity close to the average one. The main step of the proof consists of obtaining \textit{a priori} estimates on the gradient of the value function by Bernstein's method. Uniqueness is also proved under more restrictive assumptions. Finally, we discuss some examples to which the previously mentioned results apply.Mean field games with monotonous interactions through the law of states and controls of the agentshttps://zbmath.org/1492.353542022-10-04T19:40:27.024758Z"Kobeissi, Ziad"https://zbmath.org/authors/?q=ai:kobeissi.ziadSummary: We consider a class of Mean Field Games in which the agents may interact through the statistical distribution of their states and controls. It is supposed that the Hamiltonian behaves like a power of its arguments as they tend to infinity, with an exponent larger than one. A monotonicity assumption is also made. Existence and uniqueness are proved using a priori estimates which stem from the monotonicity assumptions and Leray-Schauder theorem. Applications of the results are given.Global Fujita-Kato's type solutions and long-time behavior for the multidimensional chemotaxis modelhttps://zbmath.org/1492.353582022-10-04T19:40:27.024758Z"Chen, Qiong Lei"https://zbmath.org/authors/?q=ai:chen.qionglei"Hao, Xiao Nan"https://zbmath.org/authors/?q=ai:hao.xiaonan"Li, Jing Yue"https://zbmath.org/authors/?q=ai:li.jingyueSummary: We establish the global well-posedness for the multidimensional Chemotaxis model with some classes of large initial data, especially the case when the rate of variation of ln \(v_0 (v_0\) is the chemical concentration) contains high oscillation and the initial density near the equilibrium is allowed to have large oscillation in 3D. Besides, we show the optimal time-decay rates of the strong solutions under an additional perturbation assumption, which include specially the situations of \(d = 2, 3\) and improve the previous time-decay rates. Our method mainly relies on the introduce of the effective velocity and the application of the localization in Fourier spaces.On kinetic and macroscopic models for the stripe formation in engineered bacterial populationshttps://zbmath.org/1492.353662022-10-04T19:40:27.024758Z"Jiang, Ning"https://zbmath.org/authors/?q=ai:jiang.ning"Liang, Jiangyan"https://zbmath.org/authors/?q=ai:liang.jiangyan"Luo, Yi-Long"https://zbmath.org/authors/?q=ai:luo.yi-long"Tang, Min"https://zbmath.org/authors/?q=ai:tang.min.1|tang.min"Zhang, Yaming"https://zbmath.org/authors/?q=ai:zhang.yamingSummary: We study the well-posedness of the biological models with AHL-dependent cell mobility on engineered \textit{Escherichia coli} populations. For the kinetic model proposed by \textit{X. Xue}, \textit{C. Xue} and \textit{M. Tang} recently [``The role of intracellular signaling in the stripe formation in engineered \textit{Escherichia coli} populations'', PLoS Comput. Biol. 14, No. 6, e1006178, 23 p. (2018; \url{doi:10.1371/journal.pcbi.1006178})], the local existence for large initial data is proved first. Furthermore, the positivity and local conservation laws for density and nutrient concentration with initial assumptions (1.24) and (1.26) are justified. Based on these properties, it can be extended globally in time near the equilibrium \((0, 0, 0)\). Considering the asymptotic behaviors of faster response CheZ turnover rate, one formally derives an anisotropic diffusion engineered Escherichia coli populations model (in short, AD-EECP) for which we find a key extra a priori estimate to overcome the difficulties coming from the nonlinearity of the diffusion structure. The local well-posedness and the positivity and local conservation laws for density and nutrient of the AD-EECP are justified. Furthermore, the global existence around the stable steady-state is obtained.Superprocesses for the population of rabbits on grasslandhttps://zbmath.org/1492.353682022-10-04T19:40:27.024758Z"Ji, Lina"https://zbmath.org/authors/?q=ai:ji.lina"Xiong, Jie"https://zbmath.org/authors/?q=ai:xiong.jie.1|xiong.jieSummary: Motivated by the control of rabbits on grassland, a model of a population with branching dynamics in a random environment is constructed. The system is described as the solution to a conditional martingale problem given the random environment which satisfies a stochastic partial differential equation (SPDE). The weak uniqueness of the solution to the system is established by characterizing its conditional log-Laplace transform through the solution to a related nonlinear SPDE.Infection spreading in cell culture as a reaction-diffusion wavehttps://zbmath.org/1492.353732022-10-04T19:40:27.024758Z"Mahiout, Latifa Ait"https://zbmath.org/authors/?q=ai:mahiout.latifa-ait"Bessonov, Nikolai"https://zbmath.org/authors/?q=ai:bessonov.nikolai"Kazmierczak, Bogdan"https://zbmath.org/authors/?q=ai:kazmierczak.bogdan"Sadaka, Georges"https://zbmath.org/authors/?q=ai:sadaka.georges"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: Infection spreading in cell culture occurs due to virus replication in infected cells and its random motion in the extracellular space. Multiplicity of infection experiments in cell cultures are conventionally used for the characterization of viral infection by the number of viral plaques and the rate of their growth. We describe this process with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells, virus, and interferon. Time delay corresponds to the duration of viral replication inside infected cells. We show that infection propagates in cell culture as a reaction-diffusion wave, we determine the wave speed and prove its existence. Next, we carry out numerical simulations and identify three stages of infection progression: infection decay during time delay due to virus replication, explosive growth of viral load when infected cells begin to reproduce it, and finally, wave-like infection progression in cell culture characterized by a constant or slowly growing total viral load. The modelling results are in agreement with the experimental data for the coronavirus infection in a culture of epithelial cells and for some other experiments. The presence of interferon produced by infected cells decreases the viral load but does not change the speed of infection progression in cell culture. In the 2D modelling, the total viral load grows faster than in the 1D case due to the increase of plaque perimeter.Cardiac electro-mechanical activity in a deforming human cardiac tissue: modeling, existence-uniqueness, finite element computation and application to multiple ischemic diseasehttps://zbmath.org/1492.353752022-10-04T19:40:27.024758Z"Pargaei, Meena"https://zbmath.org/authors/?q=ai:pargaei.meena"Kumar, B. V. Rathish"https://zbmath.org/authors/?q=ai:kumar.b-v-rathish|kumar.bayya-venkatesulu-rathish"Pavarino, Luca F."https://zbmath.org/authors/?q=ai:pavarino.luca-franco"Scacchi, Simone"https://zbmath.org/authors/?q=ai:scacchi.simoneSummary: In this study, the cardiac electro-mechanical model in a deforming domain is taken with the addition of mechanical feedback and stretch-activated channel current coupled with the ten Tusscher human ventricular cell level model that results in a coupled PDE-ODE system. The existence and uniqueness of such a coupled system in a deforming domain is proved. At first, the existence of a solution is proved in the deformed domain. The local existence of the solution is proved using the regularization and the Faedo-Galerkin technique. Then, the global existence is proved using the energy estimates in appropriate Banach spaces, Gronwall lemma, and the compactness procedure. The existence of the solution in an undeformed domain is proved using the lower semi-continuity of the norms. Uniqueness is proved using Young's inequality, Gronwall lemma, and the Cauchy-Schwartz inequality. For the application purpose, this model is applied to understand the electro-mechanical activity in ischemic cardiac tissue. It also takes care of the development of active tension, conductive, convective, and ionic feedback. The Second Piola-Kirchoff stress tensor arising in Lagrangian mapping between reference and moving frames is taken as a combination of active, passive, and volumetric components. We investigated the effect of varying strength of hyperkalemia and hypoxia, in the ischemic subregions of human cardiac tissue with local multiple ischemic subregions, on the electro-mechanical activity of healthy and ischemic zones. This system is solved numerically using the \(Q_1\) finite element method in space and the implicit-explicit Euler method in time. Discontinuities arising with the modeled multiple ischemic regions are treated to the desired order of accuracy by a simple regularization technique using the interpolating polynomials. We examined the cardiac electro-mechanical activity for several cases in multiple hyperkalemic and hypoxic human cardiac tissue. We concluded that local multiple ischemic subregions severely affect the cardiac electro-mechanical activity more, in terms of action potential \((v)\) and mechanical parameters, intracellular calcium ion concentration \([\mathrm{Ca}^{2+}]_i\), active tension \((T_A)\), stretch \((\lambda)\) and stretch rate \((\frac{d\lambda}{dt})\), of a healthy cell in its vicinity, compared to a single Hyperkalemic or Hypoxic subregion. The four moderate hypoxically generated ischemic subregions affect the waveform of the stretch along the fiber and the stretch rate more than a single severe ischemic subregion.Analysis of the McKendrick-von Foerster equation with weighted white noise by means of stochastic functional differential equationshttps://zbmath.org/1492.353772022-10-04T19:40:27.024758Z"Ponosov, A."https://zbmath.org/authors/?q=ai:ponosov.arkadii-vladimirovichSummary: The existence and uniqueness of solutions of the boundary value problem for the recently introduced McKendrick-Von Foerster equation with continuous stochastic noise is proven. Motivated by the applications in mathematical biology, the boundary conditions are assumed to depend on the aggregated age variables, which makes the problem both non-local and nonlinear. The techniques used in the paper are based on a special transformation method converting the stochastic McKendrick-Von Foerster equation into a pair of finite dimensional systems of stochastic functional differential equations.The synergistic interplay of amyloid beta and tau proteins in Alzheimer's disease: a compartmental mathematical modelhttps://zbmath.org/1492.353812022-10-04T19:40:27.024758Z"Tesi, Maria Carla"https://zbmath.org/authors/?q=ai:tesi.maria-carlaSummary: The purpose of this Note is to present and discuss some mathematical results concerning a compartmental model for the synergistic interplay of Amyloid beta and tau proteins in the onset and progression of Alzheimer's disease. We model the possible mechanisms of interaction between the two proteins by a system of Smoluchowski equations for the Amyloid beta concentration, an evolution equation for the dynamics of misfolded tau and a kinetic-type transport equation for a function taking into accout the degree of malfunctioning of neurons. We provide a well-posedness results for our system of equations. This work extends results obtained in collaboration with \textit{M. Bertsch}, \textit{B. Franchi} and \textit{A. Tosin} [J. Phys. A, Math. Theor. 50, No. 41, Article ID 414003, 22 p. (2017; Zbl 1376.92021); SIAM J. Math. Anal. 50, No. 3, 2362--2388 (2018; Zbl 1397.35313)].
For the entire collection see [Zbl 1487.35004].A reaction-advection-diffusion model of cholera epidemics with seasonality and human behavior changehttps://zbmath.org/1492.353832022-10-04T19:40:27.024758Z"Wang, Xueying"https://zbmath.org/authors/?q=ai:wang.xueying"Wu, Ruiwen"https://zbmath.org/authors/?q=ai:wu.ruiwen"Zhao, Xiao-Qiang"https://zbmath.org/authors/?q=ai:zhao.xiao-qiangSummary: Cholera is a water- and food-borne infectious disease caused by \textit{V. cholerae}. To investigate multiple effects of human behavior change, seasonality and spatial heterogeneity on cholera spread, we propose a reaction-advection-diffusion model that incorporates human hosts and aquatic reservoir of \textit{V. cholerae}. We derive the basic reproduction number \(\mathcal{R}_0\) for this system and then establish a threshold type result on its global dynamics in terms of \(\mathcal{R}_0\). Further, we show that the bacterial loss at the downstream end of the river due to water flux can reduce the disease risk, and describe the asymptotic behavior of \(\mathcal{R}_0\) for small and large diffusion in a special case (where the diffusion rates of infected human and the pathogen are constant). We also study the transmission dynamics at the early stage of cholera outbreak numerically, and find that human behavior change may lower the infection level and delay the disease peak. Moreover, the relative rate of bacterial loss, together with convection rate, plays an important role in identifying the severely infected areas. Meanwhile spatial heterogeneity may dilute or amplify cholera infection, which in turn would increase the complexity of disease spread.Analysis of a time-delayed free boundary problem for solid tumor growth with angiogenesis and direct influence of inhibitorshttps://zbmath.org/1492.353842022-10-04T19:40:27.024758Z"Xu, Shihe"https://zbmath.org/authors/?q=ai:xu.shihe"Su, Dan"https://zbmath.org/authors/?q=ai:su.danSummary: In this paper we consider a free boundary problem for tumor growth under direct effect of inhibitors with angiogenesis and time delays in proliferation. The existence and uniqueness of the steady state solution is studied. The asymptotic behavior of steady state solution is proved, and the condition under which the tumor will tend to disappear is given. Finally, we also discuss the effects of the concentration of external inhibitors, the concentration of external nutrients, and the consumption rate of nutrients and inhibitors on the growth of tumors. The results show that under certain conditions the tumor will eventually disappear or will tend to a steady state. The increase of inhibitor concentration (or consumption rate) will lead to the reduction of the radius of the tumor, and the increase of nutrient concentration (or consumption rate) will lead to the increase of the radius of the tumor.Role of seasonality and spatial heterogeneous in the transmission dynamics of avian influenzahttps://zbmath.org/1492.353872022-10-04T19:40:27.024758Z"Zheng, Tingting"https://zbmath.org/authors/?q=ai:zheng.tingting"Nie, Linfei"https://zbmath.org/authors/?q=ai:nie.linfei"Zhu, Huaiping"https://zbmath.org/authors/?q=ai:zhu.huaiping"Luo, Yantao"https://zbmath.org/authors/?q=ai:luo.yantao"Teng, Zhidong"https://zbmath.org/authors/?q=ai:teng.zhi-dongSummary: In this paper, we propose a time-periodic reaction-diffusion model with environment transmission and spatial heterogeneity to describe the transmission of avian influenza virus between birds, poultry and human population. Firstly, we study the well-posedness of solutions, including the existence and boundedness of global solutions and the existence of the global attractor of solution semiflow. Then, the basic reproduction number \(\mathcal{R}_0\) of the model is defined, and we prove \(\mathcal{R}_0\) is a threshold determining the global dynamics of the model. Furthermore, we analyze the dynamic behavior of the autonomous reaction-diffusion model, and obtain the existence and global attractivity of endemic equilibrium. Finally, we give some numerical examples and simulations to illustrate the main theoretical results and investigate the impacts of some model parameters on \(\mathcal{R}_0\).Complexity analysis of stochastic gradient methods for PDE-constrained optimal control problems with uncertain parametershttps://zbmath.org/1492.353942022-10-04T19:40:27.024758Z"Martin, Matthieu"https://zbmath.org/authors/?q=ai:martin.matthieu"Krumscheid, Sebastian"https://zbmath.org/authors/?q=ai:krumscheid.sebastian"Nobile, Fabio"https://zbmath.org/authors/?q=ai:nobile.fabioSummary: We consider the numerical approximation of an optimal control problem for an elliptic Partial Differential Equation (PDE) with random coefficients. Specifically, the control function is a deterministic, distributed forcing term that minimizes the expected squared \(L^2\) misfit between the state (\textit{i.e.} solution to the PDE) and a target function, subject to a regularization for well posedness. For the numerical treatment of this risk-averse Optimal Control Problem (OCP) we consider a Finite Element discretization of the underlying PDE, a Monte Carlo sampling method, and gradient-type iterations to obtain the approximate optimal control. We provide full error and complexity analyses of the proposed numerical schemes. In particular we investigate the complexity of a conjugate gradient method applied to the fully discretized OCP (so called Sample Average Approximation), in which the Finite Element discretization and Monte Carlo sample are chosen in advance and kept fixed over the iterations. This is compared with a \textit{Stochastic Gradient} method on a fixed or varying Finite Element discretization, in which the expectation in the computation of the steepest descent direction is approximated by Monte Carlo estimators, independent across iterations, with small sample sizes. We show in particular that the second strategy results in an improved computational complexity. The theoretical error estimates and complexity results are confirmed by numerical experiments.The generalized porous medium equation on graphs: existence and uniqueness of solutions with \(\ell^1\) datahttps://zbmath.org/1492.353992022-10-04T19:40:27.024758Z"Bianchi, Davide"https://zbmath.org/authors/?q=ai:bianchi.davide"Setti, Alberto G."https://zbmath.org/authors/?q=ai:setti.alberto-g"Wojciechowski, Radosław K."https://zbmath.org/authors/?q=ai:wojciechowski.radoslaw-krzysztofSummary: We study solutions of the generalized porous medium equation on infinite graphs. For nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild solutions on any graph. For changing sign integrable data, we show existence and uniqueness under extra assumptions such as local finiteness or a uniform lower bound on the node measure.Study on the Biswas-Arshed equation with the beta time derivativehttps://zbmath.org/1492.354062022-10-04T19:40:27.024758Z"Akbulut, Arzu"https://zbmath.org/authors/?q=ai:akbulut.arzu"Islam, S. M. Rayhanul"https://zbmath.org/authors/?q=ai:islam.s-m-rayhanul(no abstract)A complete study of the lack of compactness and existence results of a fractional Nirenberg equation via a flatness hypothesis. IIhttps://zbmath.org/1492.354072022-10-04T19:40:27.024758Z"Alghanemi, Azeb"https://zbmath.org/authors/?q=ai:alghanemi.azeb"Abdelhedi, Wael"https://zbmath.org/authors/?q=ai:abdelhedi.wael"Chtioui, Hichem"https://zbmath.org/authors/?q=ai:chtioui.hichemSummary: This is a sequel to [the second author et al., Anal. PDE 9, No. 6, 1285--1315 (2016; Zbl 1366.35211)] where the prescribed \(\sigma \)-curvature problem on the standard sphere was studied under the hypothesis that the flatness order at critical points of the prescribed function lies in \((1, n - 2 \sigma]\). We provide a complete description of the lack of compactness of the problem when the flatness order varies in \((1, n)\) and we establish an existence theorem based on an Euler-Hopf type formula. As a product, we extend the existence results of [loc. cit.; \textit{T. Jin} et al., J. Eur. Math. Soc. (JEMS) 16, No. 6, 1111--1171 (2014; Zbl 1300.53041); Int. Math. Res. Not. 2015, No. 6, 1555--1589 (2015; Zbl 1319.53031)] and deliver a new one.Concentration phenomena for fractional magnetic NLS equationshttps://zbmath.org/1492.354082022-10-04T19:40:27.024758Z"Ambrosio, Vincenzo"https://zbmath.org/authors/?q=ai:ambrosio.vincenzoIn this paper the author studies existence, multiplicity of solutions for a semiclassical fractional magnetic Schrödinger equation in the whole space in presence of a suitable subcritical nonlinearity. Under suitable assumptions on the external potential and the nonlineariy, he proves that the number of solutions is estimated below by the Ljusternick-Schnirelmann category of the set of the minima of the external potential in the semiclassical limit. Penalization technique, generalized Nehari manifold method and Ljusternik-Schnirelman theory are used. As usual in this type of problems, also concentration is showed.
Reviewer: Gaetano Siciliano (São Paulo)Ground state solutions for a fractional system involving critical non-linearitieshttps://zbmath.org/1492.354102022-10-04T19:40:27.024758Z"Guo, Zhenyu"https://zbmath.org/authors/?q=ai:guo.zhenyu|guo.zhenyu-v"Deng, Yan"https://zbmath.org/authors/?q=ai:deng.yanSummary: The aim of this paper is to study a fractional system involving critical non-linearities. Using the Mountain Pass Theorem, the existence of ground state solutions for our problem is obtained in two cases.Ground state solutions for fractional Schrödinger-Choquard-Kirchhoff type equations with critical growthhttps://zbmath.org/1492.354112022-10-04T19:40:27.024758Z"Huang, Ling"https://zbmath.org/authors/?q=ai:huang.ling"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.4|wang.li.1|wang.li.6|wang.li.3|wang.li.5|wang.li|wang.li.2"Feng, Shenghao"https://zbmath.org/authors/?q=ai:feng.shenghaoSummary: In this paper, we investigate the existence of ground state solutions for fractional Schrödinger-Choquard-Kirchhoff type equations with critical growth
\[
\begin{cases}
\left(a+b \displaystyle\int_{\mathbb{R}^N} |(-\Delta)^{\frac{s}{2}} u|^2 \mathrm{d}x \right) (-\Delta)^s u+V(x)u \\
\quad\quad = \lambda f(x,u)+ [|x|^{-\mu} \ast |u|^{2_{\mu,s}^\ast}]|u|^{2_{\mu,s}^\ast-2} u, \quad x \in \mathbb{R}^N, \\
u \in H^s(\mathbb{R}^N),
\end{cases}
\] where \(a, b>0\) are constants, \(\lambda>0\) is a parameter, \(0<s<1\), \((-\Delta)^s\) denotes the fractional Laplacian of order \(s\), \(N>2s\), \(0<\mu<2s\) and \(2_{\mu,s}^\ast = \frac{2N-\mu}{N-2s}\). When \(V\) and \(f\) are asymptotically periodic in \(x\), we prove that the equation has a ground state solution for large \(\lambda\) by Nehari method.The Calderón problem for the fractional wave equation: uniqueness and optimal stabilityhttps://zbmath.org/1492.354272022-10-04T19:40:27.024758Z"Kow, Pu-Zhao"https://zbmath.org/authors/?q=ai:kow.pu-zhao"Lin, Yi-Hsuan"https://zbmath.org/authors/?q=ai:lin.yi-hsuan"Wang, Jenn-Nan"https://zbmath.org/authors/?q=ai:wang.jenn-nanAuthors' abstract: We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension \(n\in \mathbb{N}\).
Reviewer: Giovanni S. Alberti (Genova)A quantitative rigidity result for a two-dimensional Frenkel-Kontorova modelhttps://zbmath.org/1492.370682022-10-04T19:40:27.024758Z"Dipierro, Serena"https://zbmath.org/authors/?q=ai:dipierro.serena"Poggesi, Giorgio"https://zbmath.org/authors/?q=ai:poggesi.giorgio"Valdinoci, Enrico"https://zbmath.org/authors/?q=ai:valdinoci.enricoSummary: We consider a Frenkel-Kontorova system of harmonic oscillators in a two-dimensional Euclidean lattice and we obtain a quantitative estimate on the angular function of the equilibria. The proof relies on a PDE method related to a classical conjecture by \textit{E. De Giorgi} [in: Proceedings of the Intern. Meeting on Recent Methods in Nonlinear Analysis, Rome 1978, 131--188 (1979; Zbl 0405.49001)], also in view of an elegant technique based on complex variables that was introduced by \textit{A. Farina} [Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 6, No. 3, 685--692 (2003; Zbl 1115.35045)].
In the discrete setting, a careful analysis of the reminders is needed to exploit these types of methodologies inspired by continuum models.On finding exact solutions to coupled systems of partial differential equations by the NDMhttps://zbmath.org/1492.440012022-10-04T19:40:27.024758Z"Rawashdeh, Mahmoud Saleh"https://zbmath.org/authors/?q=ai:rawashdeh.mahmoud-saleh"Maitama, Shehu"https://zbmath.org/authors/?q=ai:maitama.shehuSummary: In this article, we develop a new method called the Natural Decomposition Method (NDM). We use the NDM to find exact solutions for coupled systems of linear and nonlinear partial differential equations. The proposed method is a combination of the Natural Transform Method (NTM) and Adomian Decomposition method (ADM). Besides, the NDM avoid round-off errors which leads to solutions in closed form. The new method always lead to an exact or approximate solution in the form of rapidly convergence series. Hence, the Natural Decomposition Method is elegant refinement of the existing methods and can easily be used to solve a wide class of Linear and Nonlinear Partial Differential Equations.A note on the Sobolev trace inequalityhttps://zbmath.org/1492.460332022-10-04T19:40:27.024758Z"Ho, Pak Tung"https://zbmath.org/authors/?q=ai:ho.pak-tungSummary: Consider the classical Sobolev trace inequality
\[
\|\nabla\varphi \|_{L^2(\mathbb{R}^n_+)}\geq K\|\varphi \|_{L^{\frac{2(n-1)}{n-2}}(\partial \mathbb{R}^n_+)}
\]
for all \(\varphi\in W^{1,2}_0(\mathbb{R}^n_+)\), where \(K\) is the best constant. Here, \(W^{1,2}_0(\mathbb{R}^n_+)\) is the space obtained by taking the completion in the norm \(\|\nabla\varphi \|_{L^2(\mathbb{R}^n_+)}\) of the set of all smooth functions with support contained in the closure of \(\mathbb{R}^n_+\), and \(n\geq 3\). Let \(\mathcal{M}\) be the set of functions for which we have equality in the Sobolev trace inequality above. In this note, we show that there is a positive constant \(\alpha\) such that
\[
\|\nabla \varphi \|_{L^2(\mathbb{R}^n_+)}^2-K^2 \|\varphi \|_{L^{\frac{2(n-1)}{n-2}}(\partial \mathbb{R}^n_+)}^2\geq\alpha d(\varphi ,\mathcal{M})^2
\]
for all \(\varphi \in W^{1,2}_0(\mathbb{R}^n_+)\), where \(d\) is the distance in the Sobolev space \(W^{1,2}_0(\mathbb{R}^n_+)\).Adams inequalities with exact growth condition for Riesz-like potentials on \(\mathbb{R}^n\)https://zbmath.org/1492.460382022-10-04T19:40:27.024758Z"Qin, Liuyu"https://zbmath.org/authors/?q=ai:qin.liuyuSummary: We derive sharp Adams inequalities with exact growth condition for the Riesz potential and for more general Riesz-like potentials on \(\mathbb{R}^n\). We also obtain Moser-Trudinger inequalities with exact growth condition for the fractional Laplacian, and for general homogeneous elliptic differential operators with constant coefficients.Multipopulation minimal-time mean field gameshttps://zbmath.org/1492.490372022-10-04T19:40:27.024758Z"Arjmand, Saeed Sadeghi"https://zbmath.org/authors/?q=ai:arjmand.saeed-sadeghi"Mazanti, Guilherme"https://zbmath.org/authors/?q=ai:mazanti.guilhermeRelative growth rate and contact Banach-Mazur distancehttps://zbmath.org/1492.530962022-10-04T19:40:27.024758Z"Rosen, Daniel"https://zbmath.org/authors/?q=ai:rosen.daniel"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.8Summary: In this paper, we define non-linear versions of Banach-Mazur distance in the contact geometry set-up, called contact Banach-Mazur distances and denoted by \(d_\mathrm{CBM}\). Explicitly, we consider the following two set-ups, either on a contact manifold \(W\times S^1\) where \(W\) is a Liouville manifold, or a closed Liouville-fillable contact manifold \(M\). The inputs of \(d_\mathrm{CBM}\) are different in these two cases. In the former case the inputs are (contact) star-shaped domains of \(W\times S^1\) which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact \(1\)-forms of \(M\) inducing the same contact structure. In particular, the contact Banach-Mazur distance \(d_\mathrm{CBM}\) defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg-Polterovich. The main results are the large-scale geometric properties in terms of \(d_\mathrm{CBM}\). In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou-Kashiwara-Schapira's sheaf quantization.The Itô-Tanaka trick: a non-semimartingale approachhttps://zbmath.org/1492.601532022-10-04T19:40:27.024758Z"Coutin, Laure"https://zbmath.org/authors/?q=ai:coutin.laure"Duboscq, Romain"https://zbmath.org/authors/?q=ai:duboscq.romain"Réveillac, Anthony"https://zbmath.org/authors/?q=ai:reveillac.anthonySummary: In this paper we provide an Itô-Tanaka trick formula in a non semimartingale context, filling a gap in the theory of regularisation by noise. In a classical Brownian framework, the Itô-Tanaka trick links the time average of a functionfalong the solution to a Brownian SDE, with the solution of a Fokker-Planck PDE. Our main contribution is to provide such a link in a nonsemimartingale framework, where the solution to the non-available PDE is replaced by a well-chosen random field. This allows us to improve well-posedness results for fractional SDEs with a singular drift coefficient.Pointwise a posteriori error analysis of a finite element method for the Signorini problemhttps://zbmath.org/1492.653162022-10-04T19:40:27.024758Z"Khandelwal, Rohit"https://zbmath.org/authors/?q=ai:khandelwal.rohit"Porwal, Kamana"https://zbmath.org/authors/?q=ai:porwal.kamanaSummary: In this article, we develop a posteriori error control of conforming finite element method in maximum norm for the one-body contact problem. The reliability and the efficiency of the error estimator is discussed. The upper and lower barriers of the exact solution \(\boldsymbol{u}\) have been constructed by rectifying the discrete solution \(\boldsymbol{u_h}\) properly and they are crucially used in obtaining the reliability estimates. Other key ingredients of the analysis are the sign property of the quasi-discrete contact force density as well as bounds on the Green's matrix of the divergence type operator. Numerical experiments are presented for a two dimensional contact problems that exhibit reliability and efficiency of the error estimator confirming theoretical findings.A pressure projection stabilized mixed finite element method for a Stokes hemivariational inequalityhttps://zbmath.org/1492.653202022-10-04T19:40:27.024758Z"Ling, Min"https://zbmath.org/authors/?q=ai:ling.min"Han, Weimin"https://zbmath.org/authors/?q=ai:han.weimin"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: This paper is devoted to the development and analysis of a pressure projection stabilized mixed finite element method, with continuous piecewise linear approximations of velocities and pressures, for solving a hemivariational inequality of the stationary Stokes equations with a nonlinear non-monotone slip boundary condition. We present an existence result for an abstract mixed hemivariational inequality and apply it for a unique solvability analysis of the numerical method for the Stokes hemivariational inequality. An optimal order error estimate is derived for the numerical solution under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence order.A Nitsche hybrid multiscale method with non-matching gridshttps://zbmath.org/1492.653252022-10-04T19:40:27.024758Z"Ming, Pingbing"https://zbmath.org/authors/?q=ai:ming.pingbing"Song, Siqi"https://zbmath.org/authors/?q=ai:song.siqiSummary: We propose a Nitsche method for multiscale partial differential equations, which retrieves the macroscopic information and the local microscopic information at one stroke. We prove the convergence of the method for second order elliptic problem with bounded and measurable coefficients. The rate of convergence may be derived for coefficients with further structures such as periodicity and ergodicity. Extensive numerical results confirm the theoretical predictions.Analysis of finite element methods for surface vector-Laplace eigenproblemshttps://zbmath.org/1492.653292022-10-04T19:40:27.024758Z"Reusken, Arnold"https://zbmath.org/authors/?q=ai:reusken.arnoldSummary: In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential \textit{nonconformities} in the discretization of the variational formulation of the vector-Laplace eigenproblem. We present a general abstract framework applicable to such nonconforming discretizations of eigenproblems. Error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters. Sharpness of these bounds is discussed. Results of a numerical experiment illustrate certain convergence properties of such finite element discretizations of the surface vector-Laplace eigenproblem.Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D/3D MHD equationshttps://zbmath.org/1492.653322022-10-04T19:40:27.024758Z"Xu, Jiali"https://zbmath.org/authors/?q=ai:xu.jiali"Su, Haiyan"https://zbmath.org/authors/?q=ai:su.haiyan"Li, Zhilin"https://zbmath.org/authors/?q=ai:li.zhilinSummary: The main purpose of this paper is to analyze nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix-Raviart-type finite element is used to approximate the velocity and the conforming piecewise linear element \(P_1\) is used for the pressure. Since the finite element method for the velocity field and the pressure is unstable, a simple locally stabilization term is added to satisfy the weak inf-sup condition. Then, the well-posedness and the optimal error estimates of the continuous and discrete problems are analyzed with the nonlinear terms being iteratively updated. Three effective iterative methods are proposed and their stability and convergence analyses are carried out. Finally, the theoretical analysis presented in this paper is verified by numerical experiments.Enhanced existence time of solutions to evolution equations of Whitham typehttps://zbmath.org/1492.760202022-10-04T19:40:27.024758Z"Ehrnström, Mats"https://zbmath.org/authors/?q=ai:ehrnstrom.mats"Wang, Yuexun"https://zbmath.org/authors/?q=ai:wang.yuexunSummary: We show that Whitham type equations \(u_t + u u_x -\mathcal{L} u_x = 0 \), where \(L\) is a general Fourier multiplier operator of order \(\alpha \in [-1, 1], \alpha\neq 0 \), allow for small solutions to be extended beyond their ordinary existence time. The result is valid for a range of quadratic dispersive equations with inhomogenous symbols in the dispersive regime given by the parameter \(\alpha \).Wave diffraction from the PEC finite wedgehttps://zbmath.org/1492.780092022-10-04T19:40:27.024758Z"Kuryliak, Dozyslav B."https://zbmath.org/authors/?q=ai:kuryliak.dozyslav-bSummary: The aim of this paper is to discuss the problem of wave diffraction from the finite wedge on a rigorous level using the method of analytical regularization. We apply the Kontorovich-Lebedev integrals that are considered in principal value sense and the eigenfunctions series for this purpose. The problem is reduced to a couple of the independent infinite systems of linear algebraic equations (ISLAE) of the first kind. The convolution type operators are singled out from them and the inverse operators are represented in analytical form. These two couples of operators are called the regularizing operators. They are used to reduce the initial ISLAE of the first kind to the ISLAE of the second kind. The numerical examples of wave scattering from the wedge are analysed.Dynamics of diverse optical solitary wave solutions to the Biswas-Arshed equation in nonlinear opticshttps://zbmath.org/1492.780182022-10-04T19:40:27.024758Z"Bilal, Muhammad"https://zbmath.org/authors/?q=ai:bilal.muhammad"Ur-Rehman, Shafqat"https://zbmath.org/authors/?q=ai:rehman.shafqat-ur"Ahmad, Jamshad"https://zbmath.org/authors/?q=ai:ahmad.jamshadSummary: The solitary wave solutions gained well-reputed significance because of their peculiar characteristics. Solitary waves are spatially localized waves and are found in a variety of natural systems from mathematical physics and engineering phenomena. This manuscript deals the investigation of optical pulses to the Biswas-Arshed equation with third order dispersion and self-steepening coefficients in nonlinear optics. Various optical pulses are recovered in single and combo shapes like bright, dark, singular, bright-dark, and dark-singular solitons by the virtue of extended sinh-Gordon equation expansion method and \((\frac{G'}{G^2})\)-expansion function method. Besides, the singular periodic wave solutions are also derived. The constraints conditions to ensure the existence criteria of reported optical solutions are also listed. In addition, by selecting different parametric values, the physical representation of some achieved solutions is plotted in 3D graphs with the help of Mathematica. The reported results show that the proposed methods are effective, concise, straightforward, powerful, and they can be used to tackle some more complex nonlinear systems.The viable \(f(G)\) gravity models via reconstruction from the observationshttps://zbmath.org/1492.830452022-10-04T19:40:27.024758Z"Lee, Seokcheon"https://zbmath.org/authors/?q=ai:lee.seokcheon"Tumurtushaa, Gansukh"https://zbmath.org/authors/?q=ai:tumurtushaa.gansukh(no abstract)Resonant instability of axionic dark matter clumpshttps://zbmath.org/1492.830502022-10-04T19:40:27.024758Z"Wang, Zihang"https://zbmath.org/authors/?q=ai:wang.zihang"Shao, Lijing"https://zbmath.org/authors/?q=ai:shao.lijing"Li, Li-Xin"https://zbmath.org/authors/?q=ai:li.lixin(no abstract)Opening the reheating box in multifield inflationhttps://zbmath.org/1492.831272022-10-04T19:40:27.024758Z"Martin, Jérôme"https://zbmath.org/authors/?q=ai:martin.jerome"Pinol, Lucas"https://zbmath.org/authors/?q=ai:pinol.lucas(no abstract)Anisotropic instability in a higher order gravity theoryhttps://zbmath.org/1492.831292022-10-04T19:40:27.024758Z"Pookkillath, Masroor C."https://zbmath.org/authors/?q=ai:pookkillath.masroor-c"De Felice, Antonio"https://zbmath.org/authors/?q=ai:de-felice.antonio"Starobinsky, Alexei A."https://zbmath.org/authors/?q=ai:starobinsky.aleksei-aleksandrovich(no abstract)