Recent zbMATH articles in MSC 35Khttps://zbmath.org/atom/cc/35K2021-06-15T18:09:00+00:00WerkzeugOverlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations.https://zbmath.org/1460.651152021-06-15T18:09:00+00:00"Li, Xiao"https://zbmath.org/authors/?q=ai:li.xiao"Ju, Lili"https://zbmath.org/authors/?q=ai:ju.lili"Hoang, Thi-Thao-Phuong"https://zbmath.org/authors/?q=ai:hoang.thi-thao-phuongSummary: The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen-Cahn equation as a special case. We first study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to verify the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space.Approximate controllability for degenerate heat equation with bilinear control.https://zbmath.org/1460.930162021-06-15T18:09:00+00:00"Li, Lingfei"https://zbmath.org/authors/?q=ai:li.lingfei"Gao, Hang"https://zbmath.org/authors/?q=ai:gao.hangSummary: This paper investigates the nonnegative approximate controllability for the one-dimensional degenerate heat equation governed by bilinear control. Both non-controllability and approximate controllability are studied for the system. If the control is restricted to act on a fixed domain, it is not controllable. If the control is allowed to mobile, it is approximately controllable.On the critical exponent ``instantaneous blow-up'' versus ``local solubility'' in the Cauchy problem for a model equation of Sobolev type.https://zbmath.org/1460.350532021-06-15T18:09:00+00:00"Korpusov, Maxim O."https://zbmath.org/authors/?q=ai:korpusov.maksim-olegovich"Panin, Alexander A."https://zbmath.org/authors/?q=ai:panin.aleksandr-anatolevich"Shishkov, Andrey E."https://zbmath.org/authors/?q=ai:shishkov.andrey-eA new result for boundedness of solutions to a higher-dimensional quasilinear chemotaxis system with a logistic source.https://zbmath.org/1460.353592021-06-15T18:09:00+00:00"Xie, Jianing"https://zbmath.org/authors/?q=ai:xie.jianingA version of the doubly parabolic Keller-Segel system with nonlinear (but nondegenerate) diffusion and logistic source term is studied in bounded domains of \(\mathbb R^N\), \(N\ge 2\). Using a new energy-type inequality global-in-time and uniform boundedness of solutions is studied under suitable assumptions on the diffusion term.
Reviewer: Piotr Biler (Wrocław)Quasi linear parabolic PDE posed on a network with non linear Neumann boundary condition at vertices.https://zbmath.org/1460.351962021-06-15T18:09:00+00:00"Ohavi, Isaac"https://zbmath.org/authors/?q=ai:ohavi.isaacSummary: The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.Global regularity for degenerate/singular parabolic equations involving measure data.https://zbmath.org/1460.350552021-06-15T18:09:00+00:00"Byun, Sun-Sig"https://zbmath.org/authors/?q=ai:byun.sun-sig"Park, Jung-Tae"https://zbmath.org/authors/?q=ai:park.jung-tae"Shin, Pilsoo"https://zbmath.org/authors/?q=ai:shin.pilsooThe authors study the following Cauchy-Dirichlet problem for a degenerate/singular parabolic equation \[ \begin{cases} u_t-\mathrm{div}\, \mathbf{a}(Du,x.t)=\mu & \quad\text{ in } \Omega_T\\
u=0 & \quad\text{ on } \partial_p\Omega_T \end{cases} \] where the nonlinear operator \(\mathbf{a}\) is measurable in \((x,t)\) and satisfies suitable growth and ellipticity conditions. There are obtained global regularity estimates for the spatial gradient os the solutions by introducing the intrinsic fractional maximal function of a given measure and making use of the
Reviewer: Lubomira Softova (Salerno)Patterns versus spatial heterogeneity -- from a variational viewpoint.https://zbmath.org/1460.350332021-06-15T18:09:00+00:00"Takagi, Izumi"https://zbmath.org/authors/?q=ai:takagi.izumiSummary: By a pattern we usually mean a spatially nontrivial structure and hence its antonym is spatial homogeneity. Alan Turing found that, in a reaction-diffusion system of two species, different diffusion rates can destabilize a spatially uniform state, leading to spontaneous formation of a pattern. This chapter proposes to generalize the notion of pattern to that of spatially heterogeneous environments and to build a unified theory of spontaneous emergence of patterns against spatially homogeneous or heterogeneous backgrounds.
For the entire collection see [Zbl 1459.91003].Vanishing phenomena in fast decreasing generalized bistable equations.https://zbmath.org/1460.350382021-06-15T18:09:00+00:00"Li, Qi"https://zbmath.org/authors/?q=ai:li.qi|li.qi.1"Lou, Bendong"https://zbmath.org/authors/?q=ai:lou.bendongSummary: Consider the reaction diffusion equation \(u_t = u_{x x} + f(u)\) with generalized bistable nonlinearity: \(f(0) = f(\theta) = f(1) = 0\) for some \(\theta \in(0, 1)\), \(f(u) \leq 0\) in \((0, \theta)\), \(f(u) > 0\) in \((\theta, 1)\) and \(f(u) < 0\) in \((1, \infty)\). We show that when \(f(u)\) decreases sufficiently fast for \(u \gg 1\), there exists \(\varepsilon_0 > 0\) such that, for any nonnegative initial data \(u_0(x)\) with \(\mathrm{supp}( u_0) \subset [- \varepsilon_0, \varepsilon_0]\) (no matter how large \(\| u_0 \|_{L^\infty}\) is), the solution \(u(x, t)\) to the Cauchy problem with initial data \(u_0(x)\) always vanishes, that is, \(u(x, t) \to 0\) as \(t \to \infty\) in the \(L^\infty(\mathbb{R})\) norm.A subordination principle for subdiffusion equations with memory.https://zbmath.org/1460.352152021-06-15T18:09:00+00:00"Ponce, Rodrigo"https://zbmath.org/authors/?q=ai:ponce.rodrigo-fSummary: We study the existence of mild solutions to subdiffusion equations with memory
\[
\partial_t^\alpha u(t)=Au(t)+\int_0^t\kappa(t-s)Au(s)ds,\quad t\geq0,\tag{\(\ast\)}
\]
with the initial condition \(u(0)=x\), where \(0<\alpha<1\), \(A\) is a closed linear operator defined on a Banach space \(X\), the initial value \(x\) belongs to \(X\) and \(\kappa\) is a suitable kernel in \(L_{\mathrm{loc}}^1(\mathbb{R}_+)\). First, we find a subordination formula for the solution operator of \((\ast)\) and then we study its connection with the existence of mild solution to the first order diffusion equation with memory.Computational nonimaging geometric optics: Monge-Ampère.https://zbmath.org/1460.780022021-06-15T18:09:00+00:00"Awanou, Gerard"https://zbmath.org/authors/?q=ai:awanou.gerardFrom the text:: The goal of computational nonimaging geometric optics is the efficient design of optical lenses and mirrors for the accurate control of light. Light waste in the United States is equivalent to 72.9 million mwh of unnecessary electricity generated at a cost of \$6.9 billion a year and the amount of CO2 generated in that process is equivalent to 9.5 million cars on the roads. Light pollution also has adverse health impacts on wildlife and humans. Other examples where an accurate control of light is required include projection displays, laser weapons, concentrated solar energy, and medical illuminators. Freeform illumination design, i.e., with no a priori symmetry assumption, often leads to numerically solving a nonlinear second order partial differential equation of Monge-Ampère type with nonlocal boundary conditions.Existence of attractors for stochastic diffusion equations with fractional damping and time-varying delay.https://zbmath.org/1460.350462021-06-15T18:09:00+00:00"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyu"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xupingSummary: This paper deals with the well-posedness and existence of attractors of a class of stochastic diffusion equations with fractional damping and time-varying delay on unbounded domains. We first prove the well-posedness and the existence of a continuous non-autonomous cocycle for the equations and the uniform estimates of solutions and the derivative of the solution operators with respect to the time-varying delay. We then show pullback asymptotic compactness of solutions and the existence of random attractors by utilizing the Arzelà-Ascoli theorem and the uniform estimates for the derivative of the solution operator in the fractional Sobolev space \(H^\alpha(\mathbb{R}^n)\), with \(0 < \alpha < 1\).
{\copyright 2021 American Institute of Physics}Global and local optimization in identification of parabolic systems.https://zbmath.org/1460.353942021-06-15T18:09:00+00:00"Krivorotko, Olga"https://zbmath.org/authors/?q=ai:krivorotko.olga-i"Kabanikhin, Sergey"https://zbmath.org/authors/?q=ai:kabanikhin.sergei-i"Zhang, Shuhua"https://zbmath.org/authors/?q=ai:zhang.shuhua.1|zhang.shuhua"Kashtanova, Victoriya"https://zbmath.org/authors/?q=ai:kashtanova.victoriyaSummary: The problem of identification of coefficients and initial conditions for a boundary value problem for parabolic equations that reduces to a minimization problem of a misfit function is investigated. Firstly, the tensor train decomposition approach is presented as a global convergence algorithm. The idea of the proposed method is to extract the tensor structure of the optimized functional and use it for multidimensional optimization problems. Secondly, for the refinement of the unknown parameters, three local optimization approaches are implemented and compared: Nelder-Mead simplex method, gradient method of minimum errors, adaptive gradient method. For gradient methods, the evident formula for the continuous gradient of the misfit function is obtained. The identification problem for the diffusive logistic mathematical model which can be applied to social sciences (online social networks), economy (spatial Solow model) and epidemiology (coronavirus COVID-19, HIV, etc.) is considered. The numerical results for information propagation in online social network are presented and discussed.Convolutions in \((\mu,\nu)\)-pseudo-almost periodic and \((\mu,\nu)\)-pseudo-almost automorphic function spaces and applications to solve integral equations.https://zbmath.org/1460.340502021-06-15T18:09:00+00:00"Békollè, David"https://zbmath.org/authors/?q=ai:bekolle.david"Ezzinbi, Khalil"https://zbmath.org/authors/?q=ai:ezzinbi.khalil"Fatajou, Samir"https://zbmath.org/authors/?q=ai:fatajou.samir"Danga, Duplex Elvis Houpa"https://zbmath.org/authors/?q=ai:danga.duplex-elvis-houpa"Béssémè, Fritz Mbounja"https://zbmath.org/authors/?q=ai:besseme.fritz-mbounjaSummary: In this paper we give sufficient conditions on \(k\in L^1(\mathbb{R})\) and the positive measures \(\mu, \nu\) such that the doubly-measure pseudo-almost periodic (respectively, doubly-measure pseudo-almost automorphic) function spaces are invariant by the convolution product \(\zeta f=k\ast f\). We provide an appropriate example to illustrate our convolution results. As a consequence, we study under Acquistapace-Terreni conditions and exponential dichotomy, the existence and uniqueness of (\(\mu,\nu\))-pseudo-almost periodic (respectively, (\(\mu,\nu\))-pseudo-almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces like neutral systems.Unstructured space-time finite element methods for optimal control of parabolic equations.https://zbmath.org/1460.490022021-06-15T18:09:00+00:00"Langer, Ulrich"https://zbmath.org/authors/?q=ai:langer.ulrich"Steinbach, Olaf"https://zbmath.org/authors/?q=ai:steinbach.olaf"Tröltzsch, Fredi"https://zbmath.org/authors/?q=ai:troltzsch.fredi"Yang, Huidong"https://zbmath.org/authors/?q=ai:yang.huidongExponential convergence of parabolic optimal transport on bounded domains.https://zbmath.org/1460.352172021-06-15T18:09:00+00:00"Abedin, Farhan"https://zbmath.org/authors/?q=ai:abedin.farhan"Kitagawa, Jun"https://zbmath.org/authors/?q=ai:kitagawa.junThe authors study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. They are able to prove an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. They obtain this important exponential convergence and the control of the oscillation in time of solutions to the parabolic equation by deriving a differential Harnack inequality for a special class of functions that solve the linearized problem and by certain techniques specific to mass transport. Additionally, in the course of the proof, they discover an interesting connection with the pseudo-Riemannian framework introduced by Kim and McCann in the context of optimal transport.
Reviewer: Vincenzo Vespri (Firenze)On cross-diffusion systems for two populations subject to a common congestion effect.https://zbmath.org/1460.351952021-06-15T18:09:00+00:00"Laborde, Maxime"https://zbmath.org/authors/?q=ai:laborde.maximeLet \(\Omega\) be a convex and relatively compact open subset of \(\mathbb{R}^n\) with smooth boundary and \(|\Omega|>2\) and consider two potentials \(V_i\in W^{1,\infty}(\Omega)\), \(i=1,2\). Existence of global weak solutions is proved for the system \begin{align*} \partial_t \rho_1 - \mathrm{div}\left( \nabla\rho_1 + \rho_1 \left( \nabla V_1 + \nabla p \right) \right)=0\text{ in }(0,\infty)\times \Omega\,, \\ \partial_t \rho_2 - \mathrm{div}\left( \nabla\rho_2 + \rho_2 \left( \nabla V_2 + \nabla p \right) \right)=0\text{ in }(0,\infty)\times \Omega\,, \end{align*} supplemented with constraints \[ p\ge 0\,, \qquad \rho_1+\rho_2 \le 1\,, \] along with no flux boundary conditions and initial conditions \((\rho_{1,0},\rho_{2,0})\in \mathcal{P}^{ac}(\Omega;\mathbb{R}^2)\) (that is, probability measures on \(\Omega\) which are absolutely continuous with respect to the Lebesgue measure) satisfying \(\rho_{1,0},\rho_{2,0}\le 1\) a.e. in \(\Omega\). In addition, when \(m\ge 1\), existence of global weak solutions is also established for the system \begin{align*} \partial_t \rho_1 - \mathrm{div}\left( \nabla\rho_1 + \rho_1 \left( \nabla V_1 + \frac{m}{m-1} \nabla [(\rho_1+\rho_2)^{m-1}] \right) \right)=0\text{ in }(0,\infty)\times \Omega\,, \\ \partial_t \rho_2 - \mathrm{div}\left( \nabla\rho_2 + \rho_2 \left( \nabla V_2 + \frac{m}{m-1} \nabla [(\rho_1+\rho_2)^{m-1}] \right) \right)=0\text{ in }(0,\infty)\times \Omega\,,\end{align*} supplemented with no flux boundary conditions and initial conditions \((\rho_{1,0},\rho_{2,0})\in \mathcal{P}^{ac}(\Omega;\mathbb{R}^2)\) (when \(m=1\), \(m(\rho_1+\rho_2)^{m-1}/(m-1)\) has to be replaced by \(\ln{(\rho_1+\rho_2)}\) as usual). In both cases, the proof relies on the underlying gradient flow structure with respect to the \(2\)-Wasserstein distance and makes use of the flow interchange technique to obtain the compactness estimates needed for the convergence of the variational scheme. The last section is devoted to the particular case \(\nabla V_1=\nabla V_2\), for which \(\rho=\rho_1+\rho_2\) solves a closed equation. Additional regularity on \(\rho\), including \(L^\infty\)-estimates, are derived. Numerical simulations are also provided.
Reviewer: Philippe Laurençot (Toulouse)Exponential attractor for Hindmarsh-Rose equations in neurodynamics.https://zbmath.org/1460.350492021-06-15T18:09:00+00:00"Phan, Chi"https://zbmath.org/authors/?q=ai:phan.chi"You, Yuncheng"https://zbmath.org/authors/?q=ai:you.yunchengSummary: The existence of exponential attractor for the diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain in the study of neurodynamics is proved through uniform estimates and a new theorem on the squeezing property of the abstract reaction-diffusion equation established in this paper. This result on the exponential attractor infers that the global attractor whose existence has been proved in [the authors and \textit{J. Su},``Global attractors for Hindmarsh-Rose equationsin neurodynamics'', Preprint, \url{arXiv:1907.13225}] for the diffusive Hindmarsh-Rose semiflow has a finite fractal dimension.The influence of density in population dynamics with strong and weak allee effect.https://zbmath.org/1460.921702021-06-15T18:09:00+00:00"Keya, Kamrun Nahar"https://zbmath.org/authors/?q=ai:keya.kamrun-nahar"Kamrujjaman, Md."https://zbmath.org/authors/?q=ai:kamrujjaman.md"Islam, Mohammad Shafiqul"https://zbmath.org/authors/?q=ai:islam.mohammad-shafiqulSummary: In this paper, we consider a reaction-diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.Spots, stripes, and spiral waves in models for static and motile cells. GTPase patterns in cells.https://zbmath.org/1460.920252021-06-15T18:09:00+00:00"Liu, Yue"https://zbmath.org/authors/?q=ai:liu.yue"Rens, Elisabeth G."https://zbmath.org/authors/?q=ai:rens.elisabeth-g"Edelstein-Keshet, Leah"https://zbmath.org/authors/?q=ai:edelstein-keshet.leahSummary: The polarization and motility of eukaryotic cells depends on assembly and contraction of the actin cytoskeleton and its regulation by proteins called GTPases. The activity of GTPases causes assembly of filamentous actin (by GTPases Cdc42, Rac), resulting in protrusion of the cell edge. Mathematical models for GTPase dynamics address the spontaneous formation of patterns and nonuniform spatial distributions of such proteins in the cell. Here we revisit the wave-pinning model for GTPase-induced cell polarization, together with a number of extensions proposed in the literature. These include introduction of sources and sinks of active and inactive GTPase (by the group of A. Champneys), and negative feedback from F-actin to GTPase activity. We discuss these extensions singly and in combination, in 1D, and 2D static domains. We then show how the patterns that form (spots, waves, and spirals) interact with cell boundaries to create a variety of interesting and dynamic cell shapes and motion.Regional controllability of a class of time-fractional systems.https://zbmath.org/1460.353802021-06-15T18:09:00+00:00"Tajani, Asmae"https://zbmath.org/authors/?q=ai:tajani.asmae"El Alaoui, Fatima-Zahrae"https://zbmath.org/authors/?q=ai:el-alaoui.fatima-zahrae"Boutoulout, Ali"https://zbmath.org/authors/?q=ai:boutoulout.aliSummary: The main purpose of this paper is to develop the concept of regional controllability for an important class of Caputo time-fractional semi-linear systems using the analytical approach, where the dynamic of the considered system is generates by an analytical semigroup. This approach use the fixed point techniques and semigroup theory. Finally, we present some numerical simulations to approve our theoretical results.
For the entire collection see [Zbl 1459.35003].Hybrid models of chemotaxis with application to leukocyte migration.https://zbmath.org/1460.920312021-06-15T18:09:00+00:00"Lu, Hannah"https://zbmath.org/authors/?q=ai:lu.hannah"Um, Kimoon"https://zbmath.org/authors/?q=ai:um.kimoon"Tartakovsky, Daniel M."https://zbmath.org/authors/?q=ai:tartakovsky.daniel-mA model of the immune response to inflammation processes is proposed and its efficiency is studied numerically. This hybrid type model consists of PDE-chemotaxis part describing diffusion and reaction, discrete stochastic simulation of movement of leukocytes and bacteria, and computation of biochemical transformations.
Reviewer: Piotr Biler (Wrocław)Optimal partial boundary condition for degenerate parabolic equations.https://zbmath.org/1460.351902021-06-15T18:09:00+00:00"Zhan, Huashui"https://zbmath.org/authors/?q=ai:zhan.huashui"Feng, Zhaosheng"https://zbmath.org/authors/?q=ai:feng.zhaoshengSummary: For the stability of the non-Newtonian fluid equation
\[\frac{ \partial u}{ \partial t} - \mathrm{div} \left( a ( x ) | \nabla u |^{p - 2} \nabla u \right) - \sum_{i = 1}^N b_i(x) D_i u + c(x, t) u = f(x, t),\]
where \(a(x) |_{x \in \Omega} > 0\), \(a(x) |_{x \in \partial \Omega} = 0\) and \(b_i(x) \in C^1( \overline{\Omega})\), we know that the degeneracy of \(a(x)\) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the stability of weak solutions.On a non-isothermal Cahn-Hilliard model based on a microforce balance.https://zbmath.org/1460.351992021-06-15T18:09:00+00:00"Marveggio, Alice"https://zbmath.org/authors/?q=ai:marveggio.alice"Schimperna, Giulio"https://zbmath.org/authors/?q=ai:schimperna.giulioThis article presents some new results for the diffuse interface Cahn-Hilliard-type model for non-isothermal phase separation system as derived in [\textit{A. Miranville} and \textit{G. Schimperna}, Discrete Contin. Dyn. Syst., Ser. B 5, No. 3, 753--768 (2005; Zbl 1140.80388)]. The model presented here is based on a balance law for internal microforces proposed by \textit{M. E. Gurtin} [Physica D 92, No. 3--4, 178--192 (1996; Zbl 0885.35121)]. The authors present proof of the global (in time) existence for the initial-boundary value problem associated to two different formulations: the ``entropy formulation'' and the ``weak formulation''.
Reviewer: Joseph Shomberg (Providence)Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth.https://zbmath.org/1460.351922021-06-15T18:09:00+00:00"Zhou, Jun"https://zbmath.org/authors/?q=ai:zhou.jun.3|zhou.jun.1|zhou.jun.2Summary: This paper deals with a fourth order parabolic PDE arising in the theory of epitaxial growth, which was studied in [\textit{C. Escudero} et al., J. Math. Pures Appl. (9) 103, No. 4, 924--957 (2015; Zbl 1406.35114)]. We estimated the lifespan under the blow-up conditions given in [loc. cit.]. Moreover, we extend the blow-up conditions of [loc. cit.] from subcritical initial energy to critical initial energy.Parabolic equations involving Laguerre operators and weighted mixed-norm estimates.https://zbmath.org/1460.350662021-06-15T18:09:00+00:00"Fan, Huiying"https://zbmath.org/authors/?q=ai:fan.huiying"Ma, Tao"https://zbmath.org/authors/?q=ai:ma.taoSummary: In this paper, we study evolution equation \(\partial_tu=-L_\alpha u+f\) and the corresponding Cauchy problem, where \(L_\alpha\) represents the Laguerre operator \(L_\alpha=\frac{1}{2}(-\frac{d^2}{dx^2}+x^2+\frac{1}{x^2}(\alpha^2-\frac{1}{4}))\), for every \(\alpha\geq-\frac{1}{2}\). We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup \(\{e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0}\). In addition, we define the Poisson operator related to the fractional power \((\partial_t+L_\alpha)^s\) and reveal weighted mixed-norm estimates for revelent maximal operators.A two dimensional mathematical model of heat propagation equation and its applications.https://zbmath.org/1460.351862021-06-15T18:09:00+00:00"Talbi, Nassima"https://zbmath.org/authors/?q=ai:talbi.nassima"Dhahbi, Anis Ben"https://zbmath.org/authors/?q=ai:dhahbi.anis-ben"Boulaaras, Salah"https://zbmath.org/authors/?q=ai:boulaaras.salah-mahmoud"Baltache, Hadj"https://zbmath.org/authors/?q=ai:baltache.hadj"Alnegga, Mohammad"https://zbmath.org/authors/?q=ai:alnegga.mohammadSummary: We propose a two-dimensional mathematical model based on Galerkin's spatial method combined with a theta time scheme applied to the heat equations. This model has been applied to a hypothetical example in which the obtained results are compared with the real experimental data. This comparison allow us to predict the soil temperature at different depths as well as at different time periods according to certain conditions imposed on the weather.On a class of degenerate abstract parabolic problems and applications to some eddy current models.https://zbmath.org/1460.353392021-06-15T18:09:00+00:00"Pauly, Dirk"https://zbmath.org/authors/?q=ai:pauly.dirk"Picard, Rainer"https://zbmath.org/authors/?q=ai:picard.rainer-h"Trostorff, Sascha"https://zbmath.org/authors/?q=ai:trostorff.sascha"Waurick, Marcus"https://zbmath.org/authors/?q=ai:waurick.marcusThe authors develop a framework for parabolic problems which can be degenerate in certain spatial regions. The approach used by the authors is related to evolution equations in Hilbert spaces, and involves only minimal assumptions on the boundary. This framework is used to analyze the structure of the degenerate eddy current problem. This eddy current problem is then justified as a limiting model of Maxwell's equations.
Reviewer: Eric Stachura (Marietta)Local existence and nonexistence for fractional in time weakly coupled reaction-diffusion systems.https://zbmath.org/1460.353792021-06-15T18:09:00+00:00"Suzuki, Masamitsu"https://zbmath.org/authors/?q=ai:suzuki.masamitsuSummary: We study a fractional in time weakly coupled reaction-diffusion system in a bounded domain with the Dirichlet boundary condition. The domain is imbedded in an \(N\)-dimensional space and it has \(C^2\) boundary, and fractional derivatives are meant in a generalized Caputo sense. The system can be referred to as a standard reaction-diffusion system in two components with polynomial growth. We obtain integrability conditions on the initial state functions which determine the existence/nonexistence of a local in time mild solution.Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains.https://zbmath.org/1460.350502021-06-15T18:09:00+00:00"Shi, Lin"https://zbmath.org/authors/?q=ai:shi.lin"Wang, Xuemin"https://zbmath.org/authors/?q=ai:wang.xuemin"Li, Dingshi"https://zbmath.org/authors/?q=ai:li.dingshiSummary: This article is concerned with the limiting behavior of dynamics of a class of non-autonomous stochastic partial differential equations driven by colored noise on unbounded thin domains. We first prove the existence of tempered pullback random attractors for the equations defined on \((n+1)\)-dimensional unbounded thin domains. Then, we show the upper semicontinuity of these attractors when the \((n+1)\)-dimensional unbounded thin domains collapse onto the \(n\)-dimensional space \(\mathbb{R}^n\). Here, the tail estimates are utilized to deal with the non-compactness of Sobolev embeddings on unbounded domains.On the well-posedness of a nonlinear pseudo-parabolic equation.https://zbmath.org/1460.352122021-06-15T18:09:00+00:00"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Au, Vo Van"https://zbmath.org/authors/?q=ai:au.vo-van"Tri, Vo Viet"https://zbmath.org/authors/?q=ai:tri.vo-viet"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper we consider the Cauchy problem for the pseudo-parabolic equation:
\[
\dfrac{\partial}{\partial t}(u+\mu(-\Delta)^{s_1}u)+(-\Delta)^{s_2}u=f(u),\quad x\in\Omega,\,t>0.
\]
Here, the orders \(s_1,s_2\) satisfy \(0<s_1\neq s_2 <1\) (order of diffusion-type terms). We establish the local well-posedness of the solutions to the Cauchy problem when the source \(f\) is globally Lipschitz. In the case when the source term \(f\) satisfies a locally Lipschitz condition, the existence in large time, blow-up in finite time and continuous dependence on the initial data of the solutions are given.Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production.https://zbmath.org/1460.351972021-06-15T18:09:00+00:00"Pan, Xu"https://zbmath.org/authors/?q=ai:pan.xu"Wang, Liangchen"https://zbmath.org/authors/?q=ai:wang.liangchenSummary: This paper deals with the chemotaxis system with nonlinear signal secretion
\[
\begin{cases}
u_t=\nabla\cdot(D(u)\nabla u-S(u)\nabla v), & x\in\Omega,\quad t>0,\\ v_t=\Delta v-v+g(u), & x\in \Omega,\quad t>0,
\end{cases}
\]
under homogeneous Neumann boundary conditions in a bounded domain \(\Omega\subset\mathbb{R}^n\) \((n\ge 2)\). The diffusion function \(D(s)\in C^2([0,\infty))\) and the chemotactic sensitivity function \(S(s)\in C^2([0,\infty))\) are given by \(D(s)\geq C_d(1+s)^{-\alpha}\) and \(0<S(s)\leq C_ss(1+s)^{\beta-1}\) for all \(s\geq 0\) with \(C_d,C_s>0\) and \(\alpha,\beta\in\mathbb{R}\). The nonlinear signal secretion function \(g(s)\in C^1([0,\infty))\) is supposed to satisfy \(g(s)\leq C_g s^{\gamma}\) for all \(s\geq 0\) with \(C_g,\gamma>0\). Global boundedness of solution is established under the specific conditions:
\[
0<\gamma\leq 1\text{ and }\alpha+\beta<\min\left\lbrace 1+\frac{1}{n},1+\frac{2}{n}-\gamma \right\rbrace.
\]
The purpose of this work is to remove the upper bound of the diffusion condition assumed in [\textit{X. Tao} et al., J. Math. Anal. Appl. 474, No. 1, 733--747 (2019; Zbl 07056518)], and we also give the necessary constraint \(\alpha+\beta<1+\frac{1}{n}\), which is ignored in [loc. cit., Theorem 1.1].Vanishing viscosity limit to the 3D Burgers equation in Gevrey class.https://zbmath.org/1460.350172021-06-15T18:09:00+00:00"Selmi, Ridha"https://zbmath.org/authors/?q=ai:selmi.ridha"Chaabani, Abdelkerim"https://zbmath.org/authors/?q=ai:chaabani.abdelkerimSummary: We consider the Cauhcy problem to the 3D diffusive periodic Burgers equation. We prove that a unique solution exists on time interval independent of the viscosity and tends, as the viscosity vanishes, to the solution of the limiting equation, the inviscid periodic three-dimensional Burgers equation, in Gevrey-Sobolev spaces. Compared to Navier-Stokes equations, the main difficulties come from the lack of the divergence-free condition which is essential to handle the nonlinear term. Our alternative tool will be to use a change of functions to estimate nonlinearities. Fourier analysis and compactness methods are widely used.Some recent developments in the theory and applications of reaction-diffusion waves.https://zbmath.org/1460.350032021-06-15T18:09:00+00:00"Volpert, Vitaly"https://zbmath.org/authors/?q=ai:volpert.vitaly-aSummary: Some recent developments in the theory and applications of travelling wave solutions of parabolic equations are discussed. These results continue the works by Aizik Volpert on index and solvability conditions of elliptic problems, topological degree, spectral properties and bifurcations, wave existence and stability.Homogenization of Richards' equations in multiscale porous media with soft inclusions.https://zbmath.org/1460.350192021-06-15T18:09:00+00:00"Jäger, Willi"https://zbmath.org/authors/?q=ai:jager.willi"Woukeng, Jean Louis"https://zbmath.org/authors/?q=ai:woukeng.jean-louisApplying the multiscale-sigma-convergence method the authors perform the homogenization of Richards' type equations posed in a deterministic multiscale porous medium filled with soft inclusions. They also provide an approximation scheme for the effective coefficients, which is useful for an eventual numerical computation.
Reviewer: Adrian Muntean (Karlstad)Global generalized solutions for a class of chemotaxis-consumption systems with generalized logistic source.https://zbmath.org/1460.353542021-06-15T18:09:00+00:00"Lyu, Wenbin"https://zbmath.org/authors/?q=ai:lyu.wenbinThe author studies a model of chemotaxis with chemoattractant consumption, a general smooth chemotactic sensitivity tensor function and a general logistic type reaction term. Under suitable assumptions the initial boundary value problem in bounded domains of \(\mathbb{R}^n\) is shown to be well posed globally in time so that finite time blowup cannot occur for those systems.
Reviewer: Piotr Biler (Wrocław)A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation.https://zbmath.org/1460.350262021-06-15T18:09:00+00:00"Li, Yanan"https://zbmath.org/authors/?q=ai:li.yanan"Carvalho, Alexandre N."https://zbmath.org/authors/?q=ai:nolasco-de-carvalho.alexandre"Luna, Tito L. M."https://zbmath.org/authors/?q=ai:luna.tito-l-m"Moreira, Estefani M."https://zbmath.org/authors/?q=ai:moreira.estefani-mSummary: In this paper we study the asymptotic behaviour of solutions for a non-local non-autonomous scalar quasilinear parabolic problem in one space dimension. Our aim is to give a fairly complete description of the forward asymptotic behaviour of solutions for models with Kirchhoff type diffusion. In the autonomous case we use the gradient structure, symmetry properties and comparison results to obtain a sequence of bifurcations of equilibria, analogous to what is seen in the local diffusivity case. We provide conditions so that the autonomous problem admits at most one positive equilibrium and analyse the existence of sign changing equilibria. Also using symmetry and the comparison results (developed here) we construct what is called non-autonomous equilibria to describe part of the asymptotics of the associated non-autonomous non-local parabolic problem.Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation.https://zbmath.org/1460.352182021-06-15T18:09:00+00:00"Gong, Shuyu"https://zbmath.org/authors/?q=ai:gong.shuyu"Zhou, Ziwei"https://zbmath.org/authors/?q=ai:zhou.ziwei"Bao, Jiguang"https://zbmath.org/authors/?q=ai:bao.jiguangSummary: In this paper, we use the Perron method to prove the existence and uniqueness of the exterior problem for a kind of parabolic Monge-Ampère equation \(-u_t+\log\det D^2u=f(x)\) with prescribed asymptotic behavior at infinity, where \(f\) is asymptotically close to a radial function at infinity. We generalize the results of both the elliptic exterior problems and the parabolic interior problems for the Monge-Ampère equations.The effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries.https://zbmath.org/1460.354002021-06-15T18:09:00+00:00"Zhao, Meng"https://zbmath.org/authors/?q=ai:zhao.meng"Li, Wantong"https://zbmath.org/authors/?q=ai:li.wan-tong"Du, Yihong"https://zbmath.org/authors/?q=ai:du.yihongSummary: In this paper, we examine an epidemic model which is described by a system of two equations with nonlocal diffusion on the equation for the infectious agents \(u\), while no dispersal is assumed in the other equation for the infective humans \(v\). The underlying spatial region \([g(t),h(t)]\) (i.e., the infected region) is assumed to change with time, governed by a set of free boundary conditions. In the recent work [\textit{M. Zhao} et al., J. Differ. Equations 269, No. 4, 3347--3386 (2020; Zbl 1442.35486)], such a model was considered where the growth rate of \(u\) due to the contribution from \(v\) is given by \(cv\) for some positive constant \(c\). Here this term is replaced by a nonlocal reaction function of \(v\) in the form \(c\int_{g(t)}^{h(t)}K(x-y)v(t,y)dy\) with a suitable kernel function \(K\), to represent the nonlocal effect of \(v\) on the growth of \(u\). We first show that this problem has a unique solution for all \(t>0\), and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy, which indicates that the long-time dynamics of the model is not vastly altered by this change of the term \(cv\). We also obtain sharp criteria for spreading and vanishing, which reveal that changes do occur in these criteria from the earlier model in [loc. cit] where the term \(cv\) was used; in particular, small nonlocal dispersal rate of \(u\) alone no longer guarantees successful spreading of the disease as in the model of [loc. cit.].Motion of interfaces for a damped hyperbolic Allen-Cahn equation.https://zbmath.org/1460.350132021-06-15T18:09:00+00:00"Folino, Raffaele"https://zbmath.org/authors/?q=ai:folino.raffaele"Lattanzio, Corrado"https://zbmath.org/authors/?q=ai:lattanzio.corrado"Mascia, Corrado"https://zbmath.org/authors/?q=ai:mascia.corradoSummary: This paper concerns with the motion of the interface for a damped hyperbolic Allen-Cahn equation, in a bounded domain of \(\mathbb{R}^n\), for \(n=2\) or \(n=3\). In particular, we focus the attention on radially symmetric solutions and extend to the hyperbolic framework some well-known results of the classic parabolic case: it is shown that, under appropriate assumptions on the initial data and on the boundary conditions, the interface moves by mean curvature as the diffusion coefficient goes to \(0\).Passivity and synchronization of coupled reaction-diffusion complex-valued memristive neural networks.https://zbmath.org/1460.351942021-06-15T18:09:00+00:00"Huang, Yanli"https://zbmath.org/authors/?q=ai:huang.yanli"Hou, Jie"https://zbmath.org/authors/?q=ai:hou.jie"Yang, Erfu"https://zbmath.org/authors/?q=ai:yang.erfuSummary: This paper considers two types of coupled reaction-diffusion complex-valued memristive neural networks (CRDCVMNNs). The nodes of the first type CRDCVMNN are coupled through their state and the second one is coupled by spatial diffusion coupling term. For the former, some novel criteria for the passivity and synchronization are derived by constructing an appropriate controller and utilizing some inequality techniques as well as Lyapunov functional method. For the latter, we establish some sufficient conditions which guarantee that this type of CRDCVMNNs can realize passivity and synchronization. Finally, the effectiveness and correctness of the acquired theoretical results are verified by two numerical examples.Large time behavior of ODE type solutions to parabolic \(p\)-Laplacian type equations.https://zbmath.org/1460.350352021-06-15T18:09:00+00:00"Eom, Junyong"https://zbmath.org/authors/?q=ai:eom.junyong"Sato, Ryuichi"https://zbmath.org/authors/?q=ai:sato.ryuichiSummary: Let \(u\) be a solution to the Cauchy problem for a nonlinear diffusion equation \[
\begin{cases}
\partial_t u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)+u^\alpha & \text{
in }\mathbb{R}^N\times(0, \infty), \\
u(x, 0)=\lambda+\varphi(x) & \text{ in }\mathbb{R}^N,
\end{cases}
\]
where \(N\geq 1\), \(2N/(N+1)<p\neq 2\), \(\alpha\in (-\infty, 1)\), \(\lambda>0\) and \(\varphi\in BC(\mathbb{R}^N)\cap L^1(\mathbb{R}^N)\) with \(\varphi\geq 0\) in \(\mathbb{R}^N\). Then the solution \(u\) behaves like a positive solution to ODE \(\zeta'=\zeta^\alpha\) in \((0,\infty)\). In this paper we show that the large time behavior of the solution \(u\) is described by a rescaled Barenblatt solution.Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems.https://zbmath.org/1460.651162021-06-15T18:09:00+00:00"Wu, Shu-Lin"https://zbmath.org/authors/?q=ai:wu.shulin"Zhou, Tao"https://zbmath.org/authors/?q=ai:zhou.taoPDEs constrained optimisation (PDECO) encompasses analysis, discretization, and the development of dedicated optimization methods for minimization problems constrained by partial differential equations. Several challenges must be addressed to solve these problems, consisting of ill-conditioning, nonlinearities, computational requirements, and software implementation. PDECO is deeply rooted in engineering and science and will continue to advance as a result of the ever-growing demand for analysis to support decision-making processes. To this end, based on the diagonalization technique established recently, the paper introduces a modified approach in PinT algorithms for the parabolic type PDE-constrained optimization problems with distributed control (in literature often referred to as tracking type problems). Solving parabolic PDE-constrained optimization problems requires to take into account all the discrete time points simultaneously, which means that the computation procedure is often time-consuming. This motivates the need for algorithms and computational methods that are efficient, accurate, and applicable in the context of complicated dynamics. The main reason behind developing parallel algorithms was to reduce the computation time of an algorithm. Thus, evaluating the execution time of an algorithm is extremely important in analyzing its efficiency. Selecting a proper designing technique for a parallel algorithm is the most difficult and important task. Motivated by this challenge, authors devise a novel modification, which dramatically reduces the condition number of the matrix representing the time-discretization.
The approach can be outlined as follows. The authors are interested in minimizing the tracking-type functionals with distributed control, subject to a time-dependent PDE with second-order spatial operator. They consider two types of conditions: Time-periodic and initial-value. Following an optimize-then-discretize approach and then by applying the first-order optimality condition, the KKT system in matrix form is obtained. ``First optimize then discretize approach''. As the name suggests, requires first writing the optimality conditions at the continuous level and then discretizing them. This in turn is reduced to saddle point system (SAS), by eliminating control variable. Then, after space and time discretizations, the discrete version of SAS becomes the computation model for the purpose of numerical simulations. For the time-periodic condition, the matrix representing the time-discretization can be diagonalized directly and this yields direct PinT computation for all the time discrete points. A direct diagonalization results in large condition number of the eigenvector matrix contributing to the unboundedness of roundoff error. To address this shortcoming, paper's main contribution, as claimed by the authors, is a novel modification, which significantly reduces the condition number. The modification involves a dual strategy by performing parametrization and scaling, to show eventually that the condition number is of moderate quantity and is robust with respect to the problem/discretization parameters.
For the initial-value condition, the matrix representing the time-discretization, however cannot be diagonalized with explicit formulas. Motivated by recently published work, the discrete KKT system is solved, rather than digonalizing directly using command in MATLAB to which FFT is inapplicable. Authors are of the opinion that this is not the first effort toward PinT computation for time-dependent PDE-constrained optimization problems and several relevant work can be found in the literature. But, the algorithms studied in these previous work are completely different from the algorithm proposed in this paper, because for the proposed algorithm the key intelligence lies in diagonalizing a class of KKT discrete system, which never appears in these previous work. Use of uniform step-size for time discretization without needing a decomposition of the time interval is another distinguishing feature of the algorithm.
Several numerical experiments are carried out and the results (numerical, graphical and contour plots) are showcased to verify and validate the claims made by the authors. Noteworthy among them are: illustrating how the condition number of the eigenvector matrix impacts the accuracy of the numerical solution, comparison between the numerical solutions computed by diagonalization technique and directly solving by the inv command in Matlab, Comparisons of condition number obtained by the direct diagonalization and the modified diagonalization based on parametrization and scaling, In the concluding section numerical results to illustrate the advantages of the proposed algorithm are shown by considering the 2D optimal control problem with time-periodic condition or initial-value condition. For this problem, the computation time of solving via two strategies: the PinT computation by the diagonalization technique and the sequential-in-time computation by two different methods proposed in earlier published works. In terms of time complexity (execution time), it is observed that, after 6 minutes approximately 92\% of the whole computation by the diagonalization technique is finished, while by the sequential-in-time method only 9\% is finished, an enormous saving in computation effort and time.
Considering the initial-value condition for the optimal control problem, the residuals of the GMRES and BiCGStab methods using the two preconditioners introduced in the paper, are plotted against iteration numbers, for fixed mesh size and number of time discrete points. Finally, the computation time (in minutes) of the the GMRES and BiCGStab methods, for fixed mesh size and differing time discrete points are plotted. The computation time (in minutes) of the Krylov subspace solvers using the two preconditioners are compared. The classical preconditioner is used in the sequential-in-time pattern and the new one is used in the PinT pattern. It is clear that, the new preconditioner significantly reduces the computation time.
Key observations:
\begin{itemize}
\item[1)] The diagonalization technique in the algorithmic style is provided for clear and better understanding.
\item[2)] For fixed dimension, increasing the number of time points does not impact the convergence rate of the Krylov subspace solvers.
\item[3)] For both the GMRES and BiCGStab methods, the convergence rates are robust with respect to the regularization parameter.
\item[4)] Clustering of the eigenvalues and singular values of the preconditioned matrix is justified theoretically and empirically.
\item[5)] Due to potentiality of PinT computation, the new preconditioner significantly reduces the computation time, compared to the classical preconditioner.
\end{itemize}
Reviewer: Chandrasekhar Salimath (Bengaluru)Null-controllability of perturbed porous medium gas flow.https://zbmath.org/1460.930152021-06-15T18:09:00+00:00"Geshkovski, Borjan"https://zbmath.org/authors/?q=ai:geshkovski.borjanSummary: In this work, we investigate the null-controllability of a nonlinear degenerate parabolic equation, which is the equation satisfied by a perturbation around the self-similar solution of the porous medium equation in Lagrangian-like coordinates. We prove a local null-controllability result for a regularized version of the nonlinear problem, in which singular terms have been removed from the nonlinearity. We use spectral techniques and the source-term method to deal with the linearized problem and the conclusion follows by virtue of a Banach fixed-point argument. The spectral techniques are also used to prove a null-controllability result for the linearized thin-film equation, a degenerate fourth order analog of the problem under consideration.Extended mean-field games.https://zbmath.org/1460.350202021-06-15T18:09:00+00:00"Lions, Pierre-Louis"https://zbmath.org/authors/?q=ai:lions.pierre-louis"Souganidis, Panagiotis E."https://zbmath.org/authors/?q=ai:souganidis.panagiotis-eSummary: We introduce a new class of coupled forward-backward in time systems consisting of a forward Hamilton-Jacobi and a backward quasilinear transport equation, which we call extended mean-field games system. This new class of equations strictly contains the classical mean-field games system with no common noise and its homogenization limit, and optimal transportation-type control problems. We also identify a new and meaningful ``monotonicity''-type condition that yields well-posedeness. The same condition yields uniqueness in the Hilbertian setting for the master equation without common noise as well as the hyperbolic system describing finite-state mean-field games.Well-posedness for the backward problems in time for general time-fractional diffusion equation.https://zbmath.org/1460.353742021-06-15T18:09:00+00:00"Floridia, Giuseppe"https://zbmath.org/authors/?q=ai:floridia.giuseppe"Li, Zhiyuan"https://zbmath.org/authors/?q=ai:li.zhiyuan"Yamamoto, Masahiro"https://zbmath.org/authors/?q=ai:yamamoto.masahiroSummary: In this article, we consider an evolution partial differential equation with Caputo time-derivative with the zero Dirichlet boundary condition: \( \partial^{\alpha}_tu + Au = F\) where \(0 < \alpha < 1\) and the principal part \(-A\), is a non-symmetric elliptic operator of the second order. Given a source \(F\), we prove the well-posedness for the backward problem in time and our result generalizes the existing results assuming that \(-A\) is symmetric. The key is a perturbation argument and the completeness of the generalized eigenfunctions of the elliptic operator \(A\).Existence of solutions for a parabolic equation in bounded domain.https://zbmath.org/1460.351892021-06-15T18:09:00+00:00"Ghanmi, A."https://zbmath.org/authors/?q=ai:ghanmi.allal|ghanmi.ahmed|ghanmi.abdejabbar"Kenzizi, T."https://zbmath.org/authors/?q=ai:kenzizi.tarekSummary: In this paper, we study the existence of nonnegative solutions for the parabolic problem
\[
\begin{cases}\Delta u+Vu+2\frac{\nabla\varphi}{\varphi}. \nabla u-\frac{\partial u}{\partial t}=0, \quad\text{in }D\times(0,+\infty),\\ \varphi(x)u(x,t)=0,(x,t)\in\partial D\times(0,+\infty),\\ u(x,0)=u_0(x),\quad x\in D,\end{cases}
\]
where \(D\) is a smooth domain of \(\mathbb{R}^n\), \((n\geq 3)\) the function \(u_0\) is such that \(\varphi u_0\in L^2(D)\) and \(\varphi\) is the normalized positive eigenfunction corresponding to the first eigenvalue of \(-\Delta\) and \(V\) belongs to a class of time-dependant potentials which can be written as a nonlinear combination of derivatives of a function.The obstacle problem for singular doubly nonlinear equations of porous medium type.https://zbmath.org/1460.352132021-06-15T18:09:00+00:00"Schätzler, Leah"https://zbmath.org/authors/?q=ai:schatzler.leahSummary: In this paper we prove the existence of variational solutions to the obstacle problem associated with doubly nonlinear equations \(\partial_t (|u|^{m-1}u) - \text{div}(D_\xi f(Du)) = 0\) with \(m > 1\) and a convex function \(f\) satisfying a standard \(p\)-growth condition for an exponent \(p \in (1,\infty)\) in a bounded space-time cylinder \(\Omega_T := \Omega \times (0,T)\). The obstacle function \(\psi\) and the boundary values \(g\) are time dependent. The proof relies on a nonlinear version of the method of minimizing movements.Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: fronts, pulses, and wave trains.https://zbmath.org/1460.352022021-06-15T18:09:00+00:00"Zemskov, Evgeny P."https://zbmath.org/authors/?q=ai:zemskov.evgeny-p"Tsyganov, Mikhail A."https://zbmath.org/authors/?q=ai:tsyganov.mikhail-a"Kassner, Klaus"https://zbmath.org/authors/?q=ai:kassner.klaus"Horsthemke, Werner"https://zbmath.org/authors/?q=ai:horsthemke.wernerSummary: We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [the first author et al., ``Wavy fronts and speed bifurcation in excitable systems with cross diffusion'', Phys. Rev. E (3) 77, No. 3, Article ID 036219, 6 p. (2008; \url{doi:10.1103/PhysRevE.77.036219}); ``Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion'', ibid. 95, No. 1, Article ID 012203, 9 p. (2017; \url{10.1103/PhysRevE.95.012203})] for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.
{\copyright 2021 American Institute of Physics}Conditional estimates in three-dimensional chemotaxis-Stokes systems and application to a Keller-Segel-fluid model accounting for gradient-dependent flux limitation.https://zbmath.org/1460.353582021-06-15T18:09:00+00:00"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelThe goal of this paper is to study effects of the Stokes flow on the fully parabolic chemotaxis system with suitable flux limitation in the equation for the evolution of the density of population. Results on the absence of blowup of solutions obtained in the three-dimensional case are similar to those for the chemotaxis system without fluid. General estimates derived for fluid motion and taxis gradients have also an independent interest for study of global-in-time existence of bounded solutions in related problems.
Reviewer: Piotr Biler (Wrocław)Small time equivalents for the density of a planar quadratric diffusion.https://zbmath.org/1460.580202021-06-15T18:09:00+00:00"Franchi, Jacques"https://zbmath.org/authors/?q=ai:franchi.jacquesThis paper presents asymptotic estimates for the joint probability density function \(p_\varepsilon (w,y)\) of the random vector \(\left( B_\varepsilon , \int_0^\varepsilon B^2_s ds \right)\) as \(\varepsilon\) tends to zero, where \((B_s)_{s\in {\mathbb R}_+}\) is a standard Brownian motion. Exact equivalents are given for \(p_\varepsilon (0, y)\) and \(p_\varepsilon (w, y)\), and for \(p_\varepsilon (0,\varepsilon y)\), \(p_\varepsilon (w,\varepsilon y)\) in the scaled regime \((w,\varepsilon y)\), \(w \in {\mathbb R}^*\), \(y > 0\). The proofs rely on expressions of the joint Fourier transform of quadratic Langevin-type diffusion processes, and on their inversion via a careful analysis of contour and oscillatory integrals by saddle-point methods.
Reviewer: Nicolas Privault (Singapore)Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains.https://zbmath.org/1460.370722021-06-15T18:09:00+00:00"Li, Dingshi"https://zbmath.org/authors/?q=ai:li.dingshi"Wang, Xuemin"https://zbmath.org/authors/?q=ai:wang.xueminSummary: This paper deals with the limiting dynamical behavior of non-autonomous stochastic reaction-diffusion equations on thin domains. Firstly, we prove the existence and uniqueness of the regular random attractor. Then we prove the upper semicontinuity of the regular random attractors for the equations on a family of \((n+1)\)-dimensional thin domains collapses onto an \(n\)-dimensional domain.Solvability of a coupled quasilinear reaction-diffusion system.https://zbmath.org/1460.351932021-06-15T18:09:00+00:00"Ambrazevičius, A."https://zbmath.org/authors/?q=ai:ambrazevicius.algirdas"Skakauskas, V."https://zbmath.org/authors/?q=ai:skakauskas.vladasSummary: The aim of this paper is to investigate the existence, uniqueness, and long-time behaviour of the classical solutions to a coupled system of four quasilinear parabolic equations. Two of them are solved in a domain and the other two are determined on the domain boundary. Such coupled systems arise in modelling of surface reactions between two reactants.Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model.https://zbmath.org/1460.350452021-06-15T18:09:00+00:00"Cantin, Guillaume"https://zbmath.org/authors/?q=ai:cantin.guillaume"Aziz-Alaoui, M. A."https://zbmath.org/authors/?q=ai:aziz-alaoui.m-aSummary: The asymptotic behavior of dissipative evolution problems, determined by complex networks of reaction-diffusion systems, is investigated with an original approach. We establish a novel estimation of the fractal dimension of exponential attractors for a wide class of continuous dynamical systems, clarifying the effect of the topology of the network on the large time dynamics of the generated semi-flow. We explore various remarkable topologies (chains, cycles, star and complete graphs) and discover that the size of the network does not necessarily enlarge the dimension of attractors. Additionally, we prove a synchronization theorem in the case of symmetric topologies. We apply our method to a complex network of competing species systems modeling an heterogeneous biological ecosystem and propose a series of numerical simulations which underpin our theoretical statements.Geometric properties of the Gelfand problem through a parabolic approach.https://zbmath.org/1460.352002021-06-15T18:09:00+00:00"Kim, Sunghoon"https://zbmath.org/authors/?q=ai:kim.sunghoon"Lee, Ki-Ahm"https://zbmath.org/authors/?q=ai:lee.ki-ahmSummary: We consider the asymptotic profiles of the nonlinear parabolic flows
\[
(e^u)_t=\Delta u+\lambda e^u
\]
to show the geometric properties of minimal solutions of the following elliptic nonlinear eigenvalue problems known as the Gelfand problem:
\[
\begin{aligned} & \Delta\varphi+\lambda e^\varphi=0,\quad \varphi>0\text{ in }\Omega\\ & \varphi=0\text{ on }\Omega\end{aligned}
\]
posed in a strictly convex domain \(\Omega\subset\mathbb R^n\). In this work, we show that there is a strictly increasing function \(f(s)\) such that \(f^{-1}(\varphi(x))\) is convex for \(0<\lambda\leqslant\lambda^\ast\), i.e., we prove that level set of \(\varphi\) is convex. Moreover, we also present the boundary condition of \(\varphi\) which guarantees the \(f\)-convexity of solution \(\varphi\).Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity.https://zbmath.org/1460.350472021-06-15T18:09:00+00:00"Lee, Jihoon"https://zbmath.org/authors/?q=ai:lee.jihoon"Toi, Vu Manh"https://zbmath.org/authors/?q=ai:toi.vu-manhSummary: We study the long-time behavior of the solutions of the partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type with exponential growth nonlinearity. More precisely, we prove the existence of weak solutions, the regularity of the global attractor and the exponential stability of stationary solutions of the systems.Application of the heat equation to the calculation of temperature rises from pulsed microwave exposure.https://zbmath.org/1460.920772021-06-15T18:09:00+00:00"Laurence, Jocelyn A."https://zbmath.org/authors/?q=ai:laurence.jocelyn-a"McKenzie, David R."https://zbmath.org/authors/?q=ai:mckenzie.david-r"Foster, Kenneth R."https://zbmath.org/authors/?q=ai:foster.kenneth-rSummary: The calculation of temperature rises in tissue is important in experiments in which the effects of exposure to pulsed microwave fields are being studied. In a previous paper [the first author et al., ``Biological effects of electromagnetic fields -- mechanisms for the effects of pulsed microwave radiation on protein conformation'', J. Theor. Biology 206, No. 2, 291--298 (2000; \url{doi:10.1006/jtbi.2000.2123})], we presented results on the temperature rise induced in tissue by pulsed microwave exposures.
This Letter corrects an error in that paper. In addition, we extend the discussion in that paper to consider the possible magnitude of temperature fluctuations induced on microscopic distance scales in tissue due to non-uniform energy absorption, assuming maximum ``worst case'' exposure conditions allowed by international exposure guidelines.
We will assume a distribution of energy eposited in the tissue medium according to the formula for the rate of energy deposition per unit volume:
\[
\dot q= A \mathrm e^{-x/1}\mathrm e^{-t/t_0}, \tag{1}
\]
where \(x\) is the distance from the surface of the medium, \(t_0\) is the characteristic time of the radiation pulse and \(l\) is the characteristic attenuation distance. Eq. (1) describes an exponential decay of the energy deposited with distance into the medium and a pulse shape which has an initial sudden rise followed by an exponential decrease. A is a scaling parameter determining the magnitude of this pulse.On competition models under Allee effect: asymptotic behavior and traveling waves.https://zbmath.org/1460.350292021-06-15T18:09:00+00:00"Feng, Wei"https://zbmath.org/authors/?q=ai:feng.wei"Freeze, Michael"https://zbmath.org/authors/?q=ai:freeze.michael"Lu, Xin"https://zbmath.org/authors/?q=ai:lu.xinSummary: In this article, we study a reaction-diffusion model on infinite spatial domain for two competing biological species (\(u\) and \(v\)). Under one-side Allee effect on \(u\)-species, the model demonstrates complexity on its coexistence and \(u\)-dominance steady states. The conditions for persistence, permanence and competitive exclusion of the species are obtained through analysis on asymptotic behavior of the solutions and stability of the steady states, including the attraction regions and convergent rates depending on the biological parameters. When the Allee effect constant \(K\) is large relative to other biological parameters, the asymptotic stability of the \(v\)-dominance state \((0,1)\) indicates the competitive exclusion of the \(u\)-species. Applying upper-lower solution method, we further prove that for a family of wave speeds with specific minimum wave speed determined by several biological parameters (including the magnitude of the \(u\)-dominance states), there exist traveling wave solutions flowing from the \(u\)-dominance states to the \(v\)-dominance state. The asymptotic rates of the traveling waves at \(\xi\rightarrow\mp\infty\) are also explicitly calculated. Finally, numerical simulations are presented to illustrate the theoretical results and population dynamics of coexistence or dominance-shifting.On the heat content for the Poisson kernel over the unit ball in the Euclidean space.https://zbmath.org/1460.351882021-06-15T18:09:00+00:00"Valverde, Luis Acuña"https://zbmath.org/authors/?q=ai:acuna-valverde.luisSummary: This paper studies, by employing analytical tools, the small-time behavior of the heat content for the Poisson kernel over the unit ball in \(\mathbb{R}^d, d \geqslant 2\) by working with its related set covariance function. As a result, we obtain a representation for the third term in the expansion of the heat content over the unit ball and provide the explicit form of this term in the particular cases \(d = 2\) and \(d = 3\).Spreading speed of the periodic Lotka-Volterra competition model.https://zbmath.org/1460.352012021-06-15T18:09:00+00:00"Liu, Xiaolin"https://zbmath.org/authors/?q=ai:liu.xiaolin"Ouyang, Zigen"https://zbmath.org/authors/?q=ai:ouyang.zigen"Huang, Zhe"https://zbmath.org/authors/?q=ai:huang.zhe"Ou, Chunhua"https://zbmath.org/authors/?q=ai:ou.chunhuaThis paper is concerned with the minimal speed (spreading speed) selection mechanism of traveling waves to a periodic diffusive Lotka-Volterra model with monostable nonlinearity. For the Lotka-Volterra competition model with constant coefficients, the minimal speed determinacy of traveling waves has been investigated by \textit{A. Alhasanat} and \textit{C. Ou} [J. Differ. Equations 266, No. 11, 7357--7378 (2019; Zbl 1408.35067)]. The main method is the upper-lower solution technique.
Taking into account the influence of seasonal changes, the authors studied a diffusive Lotka-Volterra model with the periodic coefficients. The method of upper and lower solutions pair is also applied to establish the existence of traveling waves and the minimal speed determinacy. Comparing with the model with constant coefficients, it is not easy to find upper (or lower) solutions for both equations simultaneously. By constructing new pairs of upper and lower solutions that are totally different from the classical ones, the authors established new results on both the linear and nonlinear speed selection mechanism. In addition, they showed that the significant nature of the nonlinear selection is to find a lower solution pair with the density of the first species decaying in a faster rate. This provides a practical way to find a bound estimate for the spreading speed as well as when the nonlinear selection is realized.
Reviewer: Guobao Zhang (Lanzhou)Semilinear Caputo time-fractional pseudo-parabolic equations.https://zbmath.org/1460.353812021-06-15T18:09:00+00:00"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Au, Vo Van"https://zbmath.org/authors/?q=ai:au.vo-van"Xu, Runzhang"https://zbmath.org/authors/?q=ai:xu.runzhangSummary: This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory including existence, uniqueness and regularity of the local solution, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem with the source term satisfying the globally Lipschitz condition, we prove the global existence theorem.A unique continuation property for a class of parabolic differential inequalities in a bounded domain.https://zbmath.org/1460.350622021-06-15T18:09:00+00:00"Zheng, Guojie"https://zbmath.org/authors/?q=ai:zheng.guojie"Xu, Dihong"https://zbmath.org/authors/?q=ai:xu.dihong"Wang, Taige"https://zbmath.org/authors/?q=ai:wang.taigeSummary: This article is concerned with a strong unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain \(\Omega\) prescribed with some regularity and growth conditions. Our results show that the value of the solutions can be determined uniquely by its value on an arbitrary open subset \(\omega\) in \(\Omega\) at any given positive time \(T\). We also derive the quantitative nature of this unique continuation, that is, the estimate of a \(L^2(\Omega)\) norm of the initial data, which is majorized by that of solution on the bounded open subset \(\omega\) at terminal moment \(t=T\).Existence results for impulsive partial functional fractional differential equation with state dependent delay.https://zbmath.org/1460.353842021-06-15T18:09:00+00:00"Abada, Nadjet"https://zbmath.org/authors/?q=ai:abada.nadjet"Chahdane, Helima"https://zbmath.org/authors/?q=ai:chahdane.helima"Hammouche, Hadda"https://zbmath.org/authors/?q=ai:hammouche.haddaSummary: In this paper, we study the existence of mild solutions of impulsive fractional semilinear differential equation with state dependent delay of order \(0<\alpha<1\). We shall rely on fixed point theorem for the sum of completely continuous and contraction operators due to Burton and Kirk. An example is given to illustrate the theory.
For the entire collection see [Zbl 1459.35003].Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food.https://zbmath.org/1460.350272021-06-15T18:09:00+00:00"Xing, Chao"https://zbmath.org/authors/?q=ai:xing.chao"Pan, Jiaojiao"https://zbmath.org/authors/?q=ai:pan.jiaojiao"Luo, Hong"https://zbmath.org/authors/?q=ai:luo.hongSummary: The article aims to investigate the dynamic transitions of a toxin-producing phytoplankton zooplankton model with additional food in a 2D-rectangular domain. The investigation is based on the dynamic transition theory for dissipative dynamical systems. Firstly, we verify the principle of exchange of stability by analysing the corresponding linear eigenvalue problem. Secondly, by using the technique of center manifold reduction, we determine the types of transitions. Our results imply that the model may bifurcate two new steady state solutions, which are either attractors or saddle points. In addition, the model may also bifurcate a new periodic solution as the control parameter passes critical value. Finally, some numerical results are given to illustrate our conclusions.New primal-dual weak Galerkin finite element methods for convection-diffusion problems.https://zbmath.org/1460.651442021-06-15T18:09:00+00:00"Cao, Waixiang"https://zbmath.org/authors/?q=ai:cao.waixiang"Wang, Chunmei"https://zbmath.org/authors/?q=ai:wang.chunmeiSummary: This article devises a new primal-dual weak Galerkin finite element method for the convection-diffusion equation. Optimal order error estimates are established for the primal-dual weak Galerkin approximations in various discrete norms and the standard \(L^2\) norms. A series of numerical experiments are conducted and reported to verify the theoretical findings.The research of a Stefan problem with unknown pressure in a liquid phase.https://zbmath.org/1460.353982021-06-15T18:09:00+00:00"Tilepiev, Murat"https://zbmath.org/authors/?q=ai:tilepiev.murat"Beisebay, Perizat"https://zbmath.org/authors/?q=ai:beisebay.perizat"Aruova, Aliya"https://zbmath.org/authors/?q=ai:aruova.aliya"Akzhigitov, Erbulat"https://zbmath.org/authors/?q=ai:akzhigitov.erbulatThe authors consider a one-dimensional mathematical model describing the melting process of crude oil based on heavy paraffins. The melted oil is located in the porous medium, the pressure in this phase is unknown. The temperature of melting depends on the pressure. The model leads to a free boundary problem for three parabolic equations, where the unknowns are the temperature in the solid and liquid phases and the pressure in the liquid phase. This is a Stefan type problem with the melting point that depends on pressure. The unique solvability of the problem in Hölder function spaces is proved for the small values of time. The result is obtained by fixed point theorem on the base of the investigation of the correspoding linear problem and estimates of nonlinear terms.
Reviewer: Elena Frolova (Sankt-Peterburg)High-order Wong-Zakai approximations for non-autonomous stochastic \(p\)-Laplacian equations on \(\mathbb{R}^N\).https://zbmath.org/1460.354042021-06-15T18:09:00+00:00"Zhao, Wenqiang"https://zbmath.org/authors/?q=ai:zhao.wenqiang"Zhang, Yijin"https://zbmath.org/authors/?q=ai:zhang.yijinSummary: In this paper, we investigate the approximations of stochastic \(p\)-Laplacian equation with additive white noise by a family of piecewise deterministic partial differential equations driven by a stationary stochastic process. We firstly obtain the tempered pullback attractors for the random \(p\)-Laplacian equation with a general diffusion. We secondly prove the convergence of solutions and the upper semi-continuity of pullback attractors of the Wong-Zakai approximation equations in a Hilbert space for the additive case. Thirdly, by a truncation technique, the uniform compactness of pullback attractor with respect to the quantity of approximations is derived in the space of \(q\)-times integrable functions, where the upper semi-continuity of the attractors of the approximation equations is well established.Spatial modeling and dynamics of organic matter biodegradation in the absence or presence of bacterivorous grazing.https://zbmath.org/1460.922312021-06-15T18:09:00+00:00"Chang, Xiaoyuan"https://zbmath.org/authors/?q=ai:chang.xiaoyuan"Shi, Junping"https://zbmath.org/authors/?q=ai:shi.junping"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.4Summary: Biodegradation is a pivotal natural process for elemental recycling and preservation of an ecosystem. Mechanistic modeling of biodegradation has to keep track of chemical elements via stoichiometric theory, under which we propose and analyze a spatial movement model in the absence or presence of bacterivorous grazing. Sensitivity analysis shows that the organic matter degradation rate is most sensitive to the grazer's death rate when the grazer is present and most sensitive to the bacterial death rate when the grazer is absent. Therefore, these two death rates are chosen as the primary parameters in the conditions of most mathematical theorems. The existence, stability and persistence of solutions are proven by applying linear stability analysis, local and global bifurcation theory, and the abstract persistence theory. Through numerical simulations, we obtain the transient and asymptotic dynamics and explore the effects of all parameters on the organic matter decomposition. Grazers either facilitate biodegradation or has no impact on biodegradation, which resolves the ``decomposition-facilitation paradox'' in the spatial context.Second order monotone difference schemes with approximation on non-uniform grids for two-dimensional quasilinear parabolic convection-diffusion equations.https://zbmath.org/1460.651332021-06-15T18:09:00+00:00"Hieu, Le M."https://zbmath.org/authors/?q=ai:hieu.le-minh"Thanh, Dang N. H."https://zbmath.org/authors/?q=ai:thanh.dang-n-h"Surya Prasath, V. B."https://zbmath.org/authors/?q=ai:prasath.v-b-suryaSummary: The present communication is devoted to the construction of monotone difference schemes of the second order of local approximation on non-uniform grids in space for 2D quasi-linear parabolic convection-diffusion equation. With the help of difference maximum principle, two-sided estimates of the difference solution are established and an important a priori estimate in a uniform norm \(C\) is proved. It is interesting to note that the maximal and minimal values of the difference solution do not depend on the diffusion and convection coefficients.A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDEs.https://zbmath.org/1460.350222021-06-15T18:09:00+00:00"Sandrić, Nikola"https://zbmath.org/authors/?q=ai:sandric.nikola"Valentić, Ivana"https://zbmath.org/authors/?q=ai:valentic.ivanaUsing probabilistic techniques, the authors study the periodic homogenization of a class of degenerate elliptic and degenerate parabolic equations (with non-oscillaating Dirichlet boundary conditions).
Reviewer: Adrian Muntean (Karlstad)The inverse first passage time problem for killed Brownian motion.https://zbmath.org/1460.352032021-06-15T18:09:00+00:00"Ettinger, Boris"https://zbmath.org/authors/?q=ai:ettinger.boris"Hening, Alexandru"https://zbmath.org/authors/?q=ai:hening.alexandru"Wong, Tak Kwong"https://zbmath.org/authors/?q=ai:wong.tak-kwongSummary: The classical inverse first passage time problem asks whether, for a Brownian motion \((B_t)_{t \geq 0}\) and a positive random variable \(\xi\), there exists a barrier \(b : \mathbb{R}_+ \to \mathbb{R}\) such that \(\mathbb{P} \{B_s > b(s), 0 \leq s \leq t\} = \mathbb{P} \{\xi > t\}\), for all \(t \geq 0\). We study a variant of the inverse first passage time problem for killed Brownian motion. We show that if \(\lambda > 0\) is a killing rate parameter and \(1_{(-\infty, 0]}\) is the indicator of the set \((-\infty, 0]\) then, under certain compatibility assumptions, there exists a unique continuous function \(b : \mathbb{R}_+ \to \mathbb{R}\) such that \(\mathbb{E} [-\lambda \int_0^t 1_{(-\infty, 0]} (B_s - b(s))ds] = \mathbb{P} \{\zeta > t\}\) holds for all \(t \geq 0\). This is a significant improvement of a result of the first author et al. [ibid. 24, No. 1, 1--33 (2014; Zbl 1328.60188)].
The main difficulty arises because \(1_{(-\infty, 0]}\) is discontinuous. We associate a semilinear parabolic partial differential equation (PDE) coupled with an integral constraint to this version of the inverse first passage time problem. We prove the existence and uniqueness of weak solutions to this constrained PDE system. In addition, we use the recent Feynman-Kac representation results of \textit{K. Glau} [Finance Stoch. 20, No. 4, 1021--1059 (2016; Zbl 1355.60060)] to prove that the weak solutions give the correct probabilistic interpretation.A boundary value problem of sand transport equations: an existence and homogenization results.https://zbmath.org/1460.350242021-06-15T18:09:00+00:00"Thiam, B. K."https://zbmath.org/authors/?q=ai:thiam.b-k"Baldé, M. A. M. T."https://zbmath.org/authors/?q=ai:balde.mouhamadou-aliou-m-t"Faye, I."https://zbmath.org/authors/?q=ai:faye.ibrahima"Seck, D."https://zbmath.org/authors/?q=ai:seck.diarafSummary: In this paper, we consider degenerate parabolic sand transport equations in a non periodic domain with Neumann boundary condition. We give existence and uniqueness results for the models which are also homogenized. Finally some corrector results are given.
For the entire collection see [Zbl 1458.00035].Global solvability results for parabolic equations with \(p\)-Laplacian type diffusion.https://zbmath.org/1460.352062021-06-15T18:09:00+00:00"Chagas, J. Q."https://zbmath.org/authors/?q=ai:chagas.j-q"Guidolin, P. L."https://zbmath.org/authors/?q=ai:guidolin.p-l"Zingano, P. R."https://zbmath.org/authors/?q=ai:zingano.paulo-r-aSummary: We give conditions that assure global existence of bounded weak solutions to the Cauchy problem of general conservative 2nd-order parabolic equations with \(p\)-Laplacian type diffusion \((p>2)\) and initial data \(u_0\in L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)\). Related results of interest are also given.Steady states of thin film droplets on chemically heterogeneous substrates.https://zbmath.org/1460.350252021-06-15T18:09:00+00:00"Liu, Weifan"https://zbmath.org/authors/?q=ai:liu.weifan"Witelski, Thomas P."https://zbmath.org/authors/?q=ai:witelski.thomas-pSummary: We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. We show how the analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast of the substrate pattern on the linear stability of droplets and the time evolution for dewetting on small domains. Results are also applied to describe 2D droplets on hydrophilic square patches and striped regions used in microfluidic applications.Null controllability of a system of degenerate nonlinear coupled equations derived from population dynamics.https://zbmath.org/1460.352112021-06-15T18:09:00+00:00"Birba, Mamadou"https://zbmath.org/authors/?q=ai:birba.mamadou"Traoré, Oumar"https://zbmath.org/authors/?q=ai:traore.oumarSummary: In this paper, we study the null controllability property of a nonlinear coupled model with degenerate diffusion term. Firstly, we establish a Carleman type inequality for the adjoint system of an intermediate model. From this inequality, we derive our observability inequality. Next, by a fixed point argument, we prove the null controllability result with an internal control acting on a small subset of the domain.
For the entire collection see [Zbl 1458.00035].Locating small inclusions in diffuse optical tomography by a direct imaging method.https://zbmath.org/1460.353912021-06-15T18:09:00+00:00"Jiang, Yu"https://zbmath.org/authors/?q=ai:jiang.yu.3"Nakamura, Gen"https://zbmath.org/authors/?q=ai:nakamura.gen"Wang, Haibing"https://zbmath.org/authors/?q=ai:wang.haibingSummary: Optical tomography is a typical non-invasive medical imaging technique, which aims to reconstruct geometric and physical properties of tissues by passing near infrared light through tissues for obtaining the intensity measurements. Other than optical properties of tissues, we are interested in finding locations of small inclusions inside the object from boundary measurements, based on the time-dependent diffusion model. First, we analyze the asymptotic behavior of the boundary measurements weighted by the fundamental solution of a backward diffusion equation as the diameters of inclusions go to zero. Then, we derive an efficient algorithm for locating small inclusions by finite boundary measurements. This algorithm is direct, simple and easy to be implemented numerically, since it only involves matrix operations and has no iteration process. Finally, some numerical results are presented to illustrate the feasibility and robustness of the algorithm. A new observation of the algorithm is that we can take the source points and test points independently and increase the resolution of numerical results by taking more test points.Non-uniqueness results for entropy two-phase solutions of forward-backward parabolic problems with unstable phase.https://zbmath.org/1460.352082021-06-15T18:09:00+00:00"Terracina, Andrea"https://zbmath.org/authors/?q=ai:terracina.andreaSummary: This paper studies the well-posedness of the entropy formulation given by \textit{P. I. Plotnikov} in [Differ. Equations 30, No. 4, 614--622 (1994; Zbl 0824.35100); translation from Differ. Uravn. 30, No. 4, 665--674 (1994)] for forward-backward parabolic problem obtained as singular limit of a proper pseudoparabolic approximation. It was proved in \textit{C. Mascia} et al. [Arch. Ration. Mech. Anal. 194, No. 3, 887--925 (2009; Zbl 1183.35163)] that such a formulation gives uniqueness when the solution takes values in the stable phases. Here we consider the situation in which unstable phase is taken in account, proving that, in general, uniqueness does not hold.Global synchronization of coupled reaction-diffusion neural networks with general couplings via an iterative approach.https://zbmath.org/1460.351982021-06-15T18:09:00+00:00"Tseng, Jui-Pin"https://zbmath.org/authors/?q=ai:tseng.jui-pinSummary: We establish a framework to investigate the global synchronization of coupled reaction-diffusion neural networks with time delays. The coupled networks under consideration can incorporate both the internal delays in each individual network and the transmission delays across different networks. The coupling scheme for the coupled networks is rather general, and its performance is not adversely affected by the restrictions commonly imposed by existing relevant investigations. Based on the proposed iterative approach, the problem of global synchronization is transformed into that of solving the corresponding homogeneous linear system of algebraic equations. The synchronization criterion is subsequently derived and can be verified with simple computations. Three numerical examples are presented to illustrate the effectiveness of the synchronization theory presented in this paper.Stable asymmetric spike equilibria for the Gierer-Meinhardt model with a precursor field.https://zbmath.org/1460.350142021-06-15T18:09:00+00:00"Kolokolnikov, Theodore"https://zbmath.org/authors/?q=ai:kolokolnikov.theodore"Paquin-Lefebvre, Frédéric"https://zbmath.org/authors/?q=ai:paquin-lefebvre.frederic"Ward, Michael J."https://zbmath.org/authors/?q=ai:ward.michael-jSummary: Precursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1D Gierer-Meinhardt (GM) model, we show that a non-constant precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric and two-spike patterns, corresponding to localized peaks in the activator of different heights. These stable, asymmetric patterns co-exist in the same parameter space as symmetric two-spike patterns. This is in contrast to a constant precursor case, for which asymmetric spike patterns are known to be unstable. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach, we analyse the spectrum of the linearization of the GM model around the two-spike steady state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis, a new class of vector-valued non-local eigenvalue problem is derived and analysed.Pattern formation in a slowly flattening spherical cap: delayed bifurcation.https://zbmath.org/1460.350322021-06-15T18:09:00+00:00"Charette, Laurent"https://zbmath.org/authors/?q=ai:charette.laurent"Macdonald, Colin B."https://zbmath.org/authors/?q=ai:macdonald.colin-b"Nagata, Wayne"https://zbmath.org/authors/?q=ai:nagata.wayneSummary: This article describes a reduction of a non-autonomous Brusselator reaction-diffusion system of partial differential equations on a spherical cap with time-dependent curvature using the method of centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case. The evolving domain functions and quasi-patternless solutions are derived as well as a method to obtain this non-autonomous normal form. The coefficients of such a normal form are computed and the reduction solutions are compared to numerical solutions.Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups.https://zbmath.org/1460.810612021-06-15T18:09:00+00:00"Dang, Nguyen Viet"https://zbmath.org/authors/?q=ai:dang.nguyen-viet"Zhang, Bin"https://zbmath.org/authors/?q=ai:zhang.bin.2|zhang.bin.4|zhang.bin.1|zhang.bin.3Summary: Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes on Riemannian manifolds. Our method uses complex powers of elliptic operators but involves several complex parameters in the spirit of \textit{analytic renormalization} by Speer, to build mathematical foundations for the renormalization of perturbative interacting quantum field theories. Our main result shows that spectrally regularized Feynman amplitudes admit analytic continuation as meromorphic germs with linear poles in the sense of the works of Guo-Paycha and the second author. We also give an explicit determination of the affine hyperplanes supporting the poles. Our proof relies on suitable resolution of singularities of products of heat kernels to make them smooth.
As an application of the analytic continuation result, we use a universal projection from meromorphic germs with linear poles on holomorphic germs to construct renormalization maps which subtract singularities of Feynman amplitudes of Euclidean fields. Our renormalization maps are shown to satisfy consistency conditions previously introduced in the work of Nikolov-Todorov-Stora in the case of flat space-times.On the parabolic equation for portfolio problems.https://zbmath.org/1460.912572021-06-15T18:09:00+00:00"Zawisza, Dariusz"https://zbmath.org/authors/?q=ai:zawisza.dariuszSummary: We consider a semilinear equation linked to the finite horizon consumption-investment problem under stochastic factor framework, prove it admits a classical solution and provide all obligatory estimates to successfully apply a verification reasoning. The paper covers the standard time additive utility, as well as the recursive utility framework. We extend existing results by considering more general factor dynamics including a nontrivial diffusion part and a stochastic correlation between assets and factors. In addition, this is the first paper which compromise many other optimization problems in finance, for example those related to the indifference pricing or the quadratic hedging problem. The extension of the result to the stochastic differential utility and robust portfolio optimization is provided as well. The essence of our paper lays in using improved stochastic methods to prove gradient estimates for suitable HJB equations with restricted control space.
For the entire collection see [Zbl 1460.93006].Existence and upper semicontinuity of pullback attractors for non-autonomous \(p\)-Laplacian parabolic problems.https://zbmath.org/1460.352072021-06-15T18:09:00+00:00"Simsen, Jacson"https://zbmath.org/authors/?q=ai:simsen.jacson"Nascimento, Marcelo J. D."https://zbmath.org/authors/?q=ai:nascimento.marcelo-jose-dias"Simsen, Mariza S."https://zbmath.org/authors/?q=ai:simsen.mariza-stefanelloSummary: We study the asymptotic behavior of parabolic \(p\)-Laplacian problems of the form
\[
\frac{\partial u_\lambda}{\partial t}(t)-\mathrm{div}(D_\lambda(t)|\nabla u_\lambda(t)|^{p-2}\nabla u_\lambda(t))+|u_\lambda(t)|^{p-2}u_\lambda(t)=B(t,u_\lambda(t))
\]
in a bounded smooth domain \(\varOmega\) in \(\mathbb R^n\), where \(n\geqslant 1\), \(p>2\), \(D_\lambda\in L^\infty ([\tau,T]\times\varOmega)\) with \(0<\beta\leqslant D_\lambda(t,x)\leqslant M\) a.e. in \([\tau,T]\times\varOmega\), \(\lambda\in [0,\infty)\) and for each \(\lambda\in[0,\infty)\) we have \(|D_\lambda(s,x)-D_\lambda(t,x)|\leqslant C_\lambda|s-t|^{\theta_\lambda}\) for all \(x\in\varOmega\), \(s,t\in[\tau,T]\) for some positive constants \(\theta_\lambda\) and \(C_\lambda\). Moreover, \(D_\lambda\to D_{\lambda_1}\) in \(L^\infty([\tau,T]\times\varOmega)\) as \(\lambda\to\lambda_1\). We prove that for each \(\lambda\in[0,\infty)\) the evolution process of this problem has a pullback attractor and we show that the family of pullback attractors behaves upper semicontinuously at \(\lambda_1\).Existence and non-existence of caloric morphisms with Bateman space-mapping for radial metrics.https://zbmath.org/1460.350962021-06-15T18:09:00+00:00"Shimomura, Katsunori"https://zbmath.org/authors/?q=ai:shimomura.katsunoriSummary: In semi-euclidean spaces, conformal mappings are consists of similarities, inversions, and Bateman mapping [\textit{K. Shimomura}, Math. J. Ibaraki Univ. 45, 7--13 (2013; Zbl 1277.31015)]. In this note, we shall discuss problems whether there exist caloric morphisms with Bateman space mapping for radial semi-euclidean metrics. It is based on the similar arguments as were used in [\textit{K. Shimomura}, Math. J. Ibaraki Univ. 35, 35--53 (2003; Zbl 1056.31007); ibid. Univ. 37, 81--103 (2005; Zbl 1125.31004); ibid. 43, 13--41 (2011; Zbl 1239.31005)].Blow-up analyses in parabolic equations with anisotropic nonstandard damping source.https://zbmath.org/1460.350542021-06-15T18:09:00+00:00"Liu, Bingchen"https://zbmath.org/authors/?q=ai:liu.bingchen"Xin, Qingna"https://zbmath.org/authors/?q=ai:xin.qingna"Dong, Mengzhen"https://zbmath.org/authors/?q=ai:dong.mengzhenSummary: In this paper, we consider the nonlinear parabolic problems with anisotropic nonstandard growth conditions and damping terms. After obtaining well-posedness of solutions, we give blow-up criteria of solutions through constructing different control functions and generalizing eigenfunction method, respectively. We also classify global solutions in all scope of the variable exponents. Moreover, the sharp blow-up rates, blow-up time and blow-up set are determined, which seem to be rarely studied for parabolic problems with anisotropic nonstandard damping sources. It is interesting that the asymptotic estimates of blow-up solutions rely not only on maxima and minima of the anisotropic exponents, but also on the geometry properties of the spatial domain and the scope of variable coefficients.The diffusive logistic equation on periodically evolving domains.https://zbmath.org/1460.352042021-06-15T18:09:00+00:00"Jiang, Dan-Hua"https://zbmath.org/authors/?q=ai:jiang.danhua"Wang, Zhi-Cheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.1|wang.zhi-cheng.2Summary: A diffusive logistic equation on \(n\)-dimensional periodically and isotropically evolving domains is investigated. We first derive the model and present the eigenvalue problem on evolving domains. Then we prove that the species persists if the diffusion rate \(d\) is below the critical value \(\underline{D}_0\), while the species become extinct if it is above the critical value \(\overline{D}_0\). Finally, we analyze the effect of domain evolution rate on the persistence of a species. Precisely, it depends on the average value \(\overline{\rho^{-2}}\), where \(\rho(t)\) is the domain evolution rate, and \(\overline{\rho^{-2}}=\frac{1}{T}\int_0^T\frac{1}{\rho^2(t)}\,dt\). If \(\overline{\rho^{-2}}>1\), the periodical domain evolution has a negative effect on the persistence of a species. If \(\overline{\rho^{-2}}<1\), the periodical domain evolution has a positive effect on the persistence of a species. If \(\overline{\rho^{-2}}=1\), the periodical domain evolution has no effect on the persistence of a species. Numerical simulations are also presented to illustrate the analytical results.A doubly degenerate diffusion equation in multiplicative noise removal models.https://zbmath.org/1460.352092021-06-15T18:09:00+00:00"Zhou, Zhenyu"https://zbmath.org/authors/?q=ai:zhou.zhenyu"Guo, Zhichang"https://zbmath.org/authors/?q=ai:guo.zhichang"Wu, Boying"https://zbmath.org/authors/?q=ai:wu.boyingSummary: In this paper, we consider the Neumann problem of a doubly degenerate diffusion equation which is proposed to deal with multiplicative noise. We prove the existence and the extremum principle of weak solutions for the problem.Book review of: L. R. Evangelista and E. Kaminski Lenzi, Fractional diffusion equations and anomalous diffusion.https://zbmath.org/1460.000172021-06-15T18:09:00+00:00"Magin, Richard L."https://zbmath.org/authors/?q=ai:magin.richard-lReview of [Zbl 1457.35001]Convergence rate from hyperbolic systems of balance laws to parabolic systems.https://zbmath.org/1460.350152021-06-15T18:09:00+00:00"Li, Yachun"https://zbmath.org/authors/?q=ai:li.yachun"Peng, Yue-Jun"https://zbmath.org/authors/?q=ai:peng.yuejun"Zhao, Liang"https://zbmath.org/authors/?q=ai:zhao.liang.2|zhao.liang.4|zhao.liang.1|zhao.liang.5|zhao.liang.3|zhao.liangSummary: It is proved recently that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish error estimates between the smooth solutions of the hyperbolic systems of balance laws and those of the parabolic limit systems in one space dimension. The proof of the error estimates uses a stream function technique together with energy estimates. As applications of the results, we give five examples arising from physical models.Finite-time blow-up for inhomogeneous parabolic equations with nonlinear memory.https://zbmath.org/1460.350512021-06-15T18:09:00+00:00"Alqahtani, Awatif"https://zbmath.org/authors/?q=ai:alqahtani.awatif"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: In this paper, we study the effect of an inhomogeneity \(w=w(x)\) on the finite-time blow-up of solutions to the nonlinear heat equation
\[
\partial_tu-\Delta u=\frac{1}{\Gamma(1-\gamma)}\int_0^t(t-s)^{-\gamma}|u(s)|^p\mathrm{ d}s+w(x),
\]
\[
(t,x)\in(0,T)\times\mathbb{R}^N,
\]
where \(N\geq 1\), \(0<\gamma<1\) and \(p>1\). It is well known that in the homogeneous case \(w\equiv 0\), the Fujita critical exponent is given by
\[
p_\ast=\max\left\{\frac{1}{\gamma}+\frac{4-2\gamma}{(N-2+2\gamma)^+}\right\}.
\]
In the case \(\int_{\mathbb{R}^N}w(x)dx>0\) and \(u(0,\cdot)\geq 0\), we prove that the critical exponent is equal to \(\infty\), which means that for all \(p>1\), we have a finite-time blow-up.Blowup and global existence of a free boundary problem with weak spatial source.https://zbmath.org/1460.353992021-06-15T18:09:00+00:00"Wang, Jia-Bing"https://zbmath.org/authors/?q=ai:wang.jiabing"Wang, Jie"https://zbmath.org/authors/?q=ai:wang.jie.1|wang.jie.2|wang.jie.4|wang.jie.3|wang.jie"Cao, Jia-Feng"https://zbmath.org/authors/?q=ai:cao.jia-fengSummary: This paper deals with the blowup and global existence of the solutions for a nonlinear reaction-diffusion equation with free boundaries and very weak spatial sources. For such a problem, we first derive some sufficient conditions to finite time blowup of the solution. Secondly, the existence of global vanishing solutions is proven for a family of sufficiently small initial values. Finally, a sharp threshold trichotomy result is obtained in term of the size of the initial value, by which the blowup solution, the global vanishing solution, and, in particular, the global transition solution are distinguished.A note on the slow convergence of solutions to conservation laws with mean curvature diffusions.https://zbmath.org/1460.350422021-06-15T18:09:00+00:00"Strani, Marta"https://zbmath.org/authors/?q=ai:strani.martaSummary: We study the asymptotic behaviour of solutions to a scalar conservation law with a mean curvature's type diffusion, focusing our attention to the stability/metastability properties of the steady state. In particular, we show the existence of a unique steady state that slowly converges to its asymptotic configuration, with a speed rate which is exponentially small with respect to the viscosity parameter \(\varepsilon\); the rigorous results are also validated by numerical simulations.Stochastic PDEs via convex minimization.https://zbmath.org/1460.354032021-06-15T18:09:00+00:00"Scarpa, Luca"https://zbmath.org/authors/?q=ai:scarpa.luca"Stefanelli, Ulisse"https://zbmath.org/authors/?q=ai:stefanelli.ulisseSummary: We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approach via convex optimization to the approximation of nonlinear stochastic partial differential equations.Inverse source problem for the abstract fractional differential equation.https://zbmath.org/1460.353932021-06-15T18:09:00+00:00"Kostin, Andrey B."https://zbmath.org/authors/?q=ai:kostin.andrey-b"Piskarev, Sergey I."https://zbmath.org/authors/?q=ai:piskarev.sergey-iSummary: In a Banach space, the inverse source problem for a fractional differential equation with Caputo-Dzhrbashyan derivative is considered. The initial and observation conditions are given by elements from \(D(A)\), and the operator function on the right side is sufficiently smooth. Two types of the observation operator are considered: integral and at the final point. Under the assumptions that operator \(A\) is a generator of positive and compact semigroup the uniqueness, existence and stability of the solution are proved.Recovering a time-dependent potential function in a multi-term time fractional diffusion equation by using a nonlinear condition.https://zbmath.org/1460.353902021-06-15T18:09:00+00:00"Jiang, Su Zhen"https://zbmath.org/authors/?q=ai:jiang.suzhen"Wu, Yu Jiang"https://zbmath.org/authors/?q=ai:wu.yujiangSummary: In the present paper, we devote our effort to a nonlinear inverse problem for recovering a time-dependent potential term in a multi-term time fractional diffusion equation from an additional measurement in the form of an integral over the space domain. First we study the existence, uniqueness, regularity and stability of the solution for the direct problem by using the fixed point theorem. And we obtain the uniqueness of the inverse time-dependent potential term problem. Numerically, we use the Levenberg-Marquardt method to find the approximate potential function. Four different examples are presented to show the feasibility and efficiency of the proposed method.Conditional stability in a backward Cahn-Hilliard equation via a Carleman estimate. Estimates for linear Cahn-Hilliard equations.https://zbmath.org/1460.351912021-06-15T18:09:00+00:00"Shang, Yunxia"https://zbmath.org/authors/?q=ai:shang.yunxia"Li, Shumin"https://zbmath.org/authors/?q=ai:li.shuminSummary: We consider a Cahn-Hilliard equation in a bounded domain \(\Omega\) in \(\mathbb{R}^n\) over a time interval \((0,T)\) and discuss the backward problem in time of determining intermediate data \(u(x,\theta)\), \(\theta\in(0,T)\), \(x\in\Omega\) from the measurement of the final data \(u(x,T)\), \(x\in\Omega\). Under suitable a priori boundness assumptions on the solutions \(u(x,t)\), we prove a conditional stability estimate for the semilinear Cahn-Hilliard equation
\[
\Vert u(\,\cdot\,,\theta)\Vert_{L^2(\Omega)}\leq C\Vert u(\,\cdot\,,T)\Vert_{H^2(\Omega)}^{\kappa_0},
\]
and a conditional stability estimate for the linear Cahn-Hilliard equation
\[
\Vert u(\,\cdot\,,\theta)\Vert_{H^{\beta}(\Omega)}\leq C\Vert u(\,\cdot\,,T)\Vert_{H^2(\Omega)}^{\kappa_1},
\]
where \(\theta\in(0,T)\), \(\beta\in(0,4)\) and \(\kappa_0,\kappa_1\in(0,1)\). The proof is based on a Carleman estimate with the weight function \(\text{e}^{2s\text{e}^{\lambda t}}\) with large parameters \(s,\lambda\in\mathbb{R}^+\).A robust nonstandard finite difference scheme for pricing real estate index options.https://zbmath.org/1460.353502021-06-15T18:09:00+00:00"Dube, Mbakisi"https://zbmath.org/authors/?q=ai:dube.mbakisi"Patidar, Kailash C."https://zbmath.org/authors/?q=ai:patidar.kailash-cSummary: Real estate assets can be used to store capital, generate income through rentals and can act as collateral for debt instruments. Common risk management mechanisms for real estate investments use portfolio diversification techniques, but these techniques require large amounts of capital to work effectively. Real estate index derivatives offer an alternative mechanism for managing risks associated with real estate investments. They also increase market liquidity by providing a path for individuals who do not own real estate assets to participate in the real estate market. We consider the problem of pricing real estate index derivative contracts. The market is incomplete, so it is completed by futures derivatives on the same real estate index. A dynamic hedging strategy is employed leading to a parabolic partial differential equation with coefficients which are dependent on time and the real estate index. We then construct a nonstandard finite difference method to price European and American real estate index options. The scheme utilizes complete cubic spline interpolants of the option prices at time-dependent backtrack points on the spatial grid. Bounds for the global error are theoretically established. Numerical experiments are carried out to illustrate the accuracy of the scheme.Stability analysis of anti-periodic solutions of the time-varying delayed hematopoiesis model with discontinuous harvesting terms.https://zbmath.org/1460.352052021-06-15T18:09:00+00:00"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchao"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Fu, Xiangying"https://zbmath.org/authors/?q=ai:fu.xiangyingSummary: This paper is concerned with a time-varying delayed hematopoiesis model with discontinuous harvesting terms. The harvesting terms considered in our hematopoiesis model are discontinuous which are totally different from the previous continuous, Lipschitz continuous or even smooth ones. By means of functional differential inclusions theory, inequality technique and the non-smooth analysis theory with Lyapunov-like approach, some new sufficient criteria are given to ascertain the existence and globally exponential stability of the anti-periodic solution for our proposed hematopoiesis model. Some previously known works are significantly extended and complemented. Moreover, simulation results of two topical numerical examples are also delineated to demonstrate the effectiveness of the theoretical results.Verified computations for solutions to semilinear parabolic equations using the evolution operator.https://zbmath.org/1460.352162021-06-15T18:09:00+00:00"Takayasu, Akitoshi"https://zbmath.org/authors/?q=ai:takayasu.akitoshi"Mizuguchi, Makoto"https://zbmath.org/authors/?q=ai:mizuguchi.makoto"Kubo, Takayuki"https://zbmath.org/authors/?q=ai:kubo.takayuki"Oishi, Shin'ichi"https://zbmath.org/authors/?q=ai:oishi.shinichiSummary: This article presents a theorem for guaranteeing existence of a solution for an initial-boundary value problem of semilinear parabolic equations. The sufficient condition of our main theorem is derived by a fixed-point formulation using the evolution operator. We note that the sufficient condition can be checked by verified numerical computations.
For the entire collection see [Zbl 1334.68018].Asymptotic behavior of solutions to the logarithmic diffusion equation with a linear source.https://zbmath.org/1460.350412021-06-15T18:09:00+00:00"Shimojo, Masahiko"https://zbmath.org/authors/?q=ai:shimojo.masahiko"Takáč, Peter"https://zbmath.org/authors/?q=ai:takac.peter"Yanagida, Eiji"https://zbmath.org/authors/?q=ai:yanagida.eijiSummary: We investigate the behavior of positive solutions to the Cauchy problem
\[
\begin{cases} \partial_t u =\partial_x^2(\log u)+u, & x\in\mathbb{R}, t>0, \\ \mathop{\lim}\limits_{x\rightarrow-\infty} \partial_x(\log u)=\alpha, \quad \mathop{\lim}\limits_{x\rightarrow+\infty}\partial_x(\log u)=\beta, \quad & t>0, \\ u(x,0)=u_0(x), & x\in\mathbb{R}, \end{cases}
\]
where \(\alpha,\beta \) are given positive constants and \(u_0(x)\) is a positive initial value. In the case of mass conservation, i.e., \(\int_{-\infty}^{\infty} u(x,t) dx \equiv \alpha + \beta \) for \(t\geq 0\), we show by an intersection number argument that the solution approaches a traveling wave as \(t\rightarrow \infty \). We then study the behavior in the case of mass expansion or extinction by using a transformation of variables. When the total mass is smaller, we show that extinction of the solution occurs in finite time and a rescaled solution converges to the traveling wave, whereas when the total mass is larger, the solution grows exponentially and a rescaled solution converges to a certain profile. Our results also include some log-concavity properties of solutions.Bespoke Turing systems.https://zbmath.org/1460.920282021-06-15T18:09:00+00:00"Woolley, Thomas E."https://zbmath.org/authors/?q=ai:woolley.thomas-e"Krause, Andrew L."https://zbmath.org/authors/?q=ai:krause.andrew-l"Gaffney, Eamonn A."https://zbmath.org/authors/?q=ai:gaffney.eamonn-aSummary: Reaction-diffusion systems are an intensively studied form of partial differential equation, frequently used to produce spatially heterogeneous patterned states from homogeneous symmetry breaking via the Turing instability. Although there are many prototypical ``Turing systems'' available, determining their parameters, functional forms, and general appropriateness for a given application is often difficult. Here, we consider the reverse problem. Namely, suppose we know the parameter region associated with the reaction kinetics in which patterning is required -- we present a constructive framework for identifying systems that will exhibit the Turing instability within this region, whilst in addition often allowing selection of desired patterning features, such as spots, or stripes. In particular, we show how to build a system of two populations governed by polynomial morphogen kinetics such that the: patterning parameter domain (in any spatial dimension), morphogen phases (in any spatial dimension), and even type of resulting pattern (in up to two spatial dimensions) can all be determined. Finally, by employing spatial and temporal heterogeneity, we demonstrate that mixed mode patterns (spots, stripes, and complex prepatterns) are also possible, allowing one to build arbitrarily complicated patterning landscapes. Such a framework can be employed pedagogically, or in a variety of contemporary applications in designing synthetic chemical and biological patterning systems. We also discuss the implications that this freedom of design has on using reaction-diffusion systems in biological modelling and suggest that stronger constraints are needed when linking theory and experiment, as many simple patterns can be easily generated given freedom to choose reaction kinetics.Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation.https://zbmath.org/1460.352102021-06-15T18:09:00+00:00"Ansini, Nadia"https://zbmath.org/authors/?q=ai:ansini.nadia"Fagioli, Simone"https://zbmath.org/authors/?q=ai:fagioli.simoneSummary: We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density \(u\). In case of \textit{fast-decay} mobilities, namely mobilities functions under an Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process. We show that the rescaled density \(\rho\) is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density \(\rho\) allow us to motivate the aforementioned change of variable and to state the results in terms of the original density \(u\) without prescribing any boundary conditions.Non-local convolution type parabolic equations with fractional and regular time derivative.https://zbmath.org/1460.351872021-06-15T18:09:00+00:00"Piatnitski, Andrey"https://zbmath.org/authors/?q=ai:piatnitski.andrey-l"Zhizhina, Elena"https://zbmath.org/authors/?q=ai:zhizhina.elena-anatolevnaSummary: This note deals with the fundamental solutions of parabolic equations for convolution type non-local operators. Our goal is to compare the large time asymptotics of these fundamental solutions with that of the classical Gaussian heat kernel. A similar problem is considered for evolution equations with a fractional time derivative.
For the entire collection see [Zbl 1445.00026].The obstacle problem for degenerate doubly nonlinear equations of porous medium type.https://zbmath.org/1460.352142021-06-15T18:09:00+00:00"Schätzler, Leah"https://zbmath.org/authors/?q=ai:schatzler.leahSummary: We prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation
\[
\begin{aligned} \partial_t b(u) - {{\,\text{div}\,}}(Df(Du)) = 0, \end{aligned}
\]
where the nonlinearity \(b:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}\) is increasing, piecewise \(C^1\) and satisfies a polynomial growth condition. The prototype is \(b(u):=u^m\) with \(m\in(0,1)\). Further, \(f:\mathbb{R}^n\rightarrow\mathbb{R}_{\ge 0}\) is convex and fulfills a standard \(p\)-growth condition. The proof relies on a nonlinear version of the method of minimizing movements.Well-posedness of a family of degenerate parabolic mixed equations.https://zbmath.org/1460.780262021-06-15T18:09:00+00:00"Acevedo, Ramiro"https://zbmath.org/authors/?q=ai:acevedo.ramiro"Gómez, Christian"https://zbmath.org/authors/?q=ai:gomez.christian"López-Rodríguez, Bibiana"https://zbmath.org/authors/?q=ai:lopez-rodriguez.bibianaSummary: In this study, we present an abstract framework for analyzing a family of linear degenerate parabolic mixed equations. We combine the theory of degenerate parabolic equations with the classical Babuška-Brezzi theory for linear mixed stationary equations to deduce sufficient conditions to prove the well-posedness of the problem. Finally, we illustrate the application of the abstract framework based on examples from physical science applications, including fluid dynamics models and electromagnetic problems.Long time existence of solutions to an elastic flow of networks.https://zbmath.org/1460.350362021-06-15T18:09:00+00:00"Garcke, Harald"https://zbmath.org/authors/?q=ai:garcke.harald"Menzel, Julia"https://zbmath.org/authors/?q=ai:menzel.julia-h"Pluda, Alessandra"https://zbmath.org/authors/?q=ai:pluda.alessandraSummary: The \(L^2\)-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods.Space-fractional diffusion equation with variable coefficients: well-posedness and Fourier pseudospectral approximation.https://zbmath.org/1460.353762021-06-15T18:09:00+00:00"Li, Xue-Yang"https://zbmath.org/authors/?q=ai:li.xueyang"Xiao, Ai-Guo"https://zbmath.org/authors/?q=ai:xiao.aiguoSummary: Multi-dimensional space-fractional diffusion equation with variable coefficients and fractional gradient is a difficult problem in theory and computation. As far as we know, there rarely exist well-posedness results and efficient numerical approaches for such equation. In this paper, we focus on this subject. First, we apply the commutator estimation method to prove the coercivity of the non-positive bilinear form for such equation in both continuous sense and discrete sense, and this is key for the later discussion. Then, we prove the well-posedness of the analytical solution and give the global error estimation of the numerical solution obtained by Crank-Nicolson Fourier pseudospectral scheme. Last, the numerical experiments are used to verify the main results of the theoretical analysis, and a model for the plume of solute through groundwater is exhibited to show the application of space-fractional diffusion theory.Asymptotic stability in a chemotaxis-competition system with indirect signal production.https://zbmath.org/1460.353602021-06-15T18:09:00+00:00"Zheng, Pan"https://zbmath.org/authors/?q=ai:zheng.panA system of chemotaxis and competition with three parabolic equations combining the models of Keller-Segel and Lotka-Volterra type is studied in bounded domains with the homogeneous Neumann boundary conditions. Exponential convergence to a coexistence steady state is shown under some assumptions on the parameters of the problem, as well as a result on an algebraic decay to an extinction steady state in another parameter range.
Reviewer: Piotr Biler (Wrocław)