Recent zbMATH articles in MSC 35Khttps://zbmath.org/atom/cc/35K2024-11-01T15:51:55.949586ZWerkzeugBoundary behaviour of Neumann harmonic functions with Lebesuge, Hardy and BMO traces in the upper half-spacehttps://zbmath.org/1544.310152024-11-01T15:51:55.949586Z"Chen, Jiahe"https://zbmath.org/authors/?q=ai:chen.jiahe"Li, Bo"https://zbmath.org/authors/?q=ai:li.bo.5"Ma, Bolin"https://zbmath.org/authors/?q=ai:ma.bolin"Wu, Yinhuizi"https://zbmath.org/authors/?q=ai:wu.yinhuizi"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.6Summary: This paper is concerned with the boundary value problem for the elliptic equation of Neumann type on the upper-half space \(\mathbb{R}^n\times \mathbb{R}_+\), \[\begin{cases} -\partial^2_t u(x, t)-\mathrm{div} A\nabla u(x, t)=0, \quad & x\in \mathbb{R}^n, t>0, \\ \partial_{x_{n}} u(x', 0, t)=0 \quad & x'\in\mathbb{R}^{n-1}, t>0, \\ u(x, 0) =u_0(x), \quad & x\in\mathbb{R}^n, \end{cases} \] where the matrix \(A=(a^{ij}(x))_{n\times n}\) is even with respect to the \(n\)-th variable and satisfies the ellipticity condition. By using the reflection method from \textit{W. A. Strauss}'s book [Partial differential equations: an introduction. Hoboken, NJ: John Wiley \& Sons (2008; Zbl 1160.35002)], we derive that the solution \(u\) to the above equation can be represented as the Poisson integral (with an additional perturbation) of the initial value \(u_0\). As applications, the real-variable characterizations of Neumann harmonic functions with Lebesuge, Hardy and BMO traces are established, respectively, via the gluing technology. Finally, the boundary value problem for the parabolic equation is also considered.Quasilinear elliptic and parabolic systems with nondiagonal principal matrices and strong nonlinearities in the gradient. Solvability and regularity problemshttps://zbmath.org/1544.350032024-11-01T15:51:55.949586Z"Arkhipova, A. A."https://zbmath.org/authors/?q=ai:arkhipova.arina-a-arkhipovaSummary: We consider nondiagonal elliptic and parabolic systems of equations with strongly nonlinear terms in the gradient. We review and comment known solvability and regularity results and describe the our last results in this field.Fast-reaction limit of reaction-diffusion systems with nonlinear diffusionhttps://zbmath.org/1544.350252024-11-01T15:51:55.949586Z"Crooks, Elaine"https://zbmath.org/authors/?q=ai:crooks.elaine-c-m"Du, Yini"https://zbmath.org/authors/?q=ai:du.yiniSummary: In this paper, we present an approach to characterizing fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form \(u_{xx}\) in usual reaction-diffusion equation, which represent linear diffusion, by terms of form \(\phi(u)_{xx}\), representing nonlinear diffusion. We prove the convergence in the fast-reaction limit \(k\to\infty\) that is determined by the unique solution of a certain scalar nonlinear diffusion problem.Asymptotic expansion for convection-dominated transport in a thin graph-like junctionhttps://zbmath.org/1544.350282024-11-01T15:51:55.949586Z"Mel'nyk, Taras"https://zbmath.org/authors/?q=ai:melnyk.taras-a"Rohde, Christian"https://zbmath.org/authors/?q=ai:rohde.christianSummary: We consider for a small parameter \(\varepsilon>0\) a parabolic convection-diffusion problem with Péclet number of order \(\mathcal{O}(\varepsilon^{- 1})\) in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order \(\mathcal{O}(\varepsilon)\) connected through a domain (node) of diameter \(\mathcal{O}(\varepsilon)\). Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as \(\varepsilon \to 0\), i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems \((\varepsilon=0)\) are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed.\(L^2\)-estimates of error in homogenization of parabolic equations with correctors taken into accounthttps://zbmath.org/1544.350302024-11-01T15:51:55.949586Z"Pastukhova, S. E."https://zbmath.org/authors/?q=ai:pastukhova.svetlana-evgenievnaSummary: We consider second-order parabolic equations with bounded measurable \(\epsilon \)-periodic coefficients. To solve the Cauchy problem in the layer \(\mathbb{R}^d \times (0, T)\) with the nonhomogeneous equation, we obtain approximations in the norm \({\Vert \bullet \Vert }_{{L}^2\left({\mathbb{R}}^d\times \left(0,T\right)\right)}\) with remainder of order \(\epsilon^2\) as \(\epsilon \rightarrow 0.\)Mixed local and nonlocal parabolic equation: global existence, decay and blow-uphttps://zbmath.org/1544.350332024-11-01T15:51:55.949586Z"Zhao, Yanan"https://zbmath.org/authors/?q=ai:zhao.yanan"Zhang, Binlin"https://zbmath.org/authors/?q=ai:zhang.binlinSummary: In this paper, we use the modified potential well method and the Galerkin method to study the following mixed local and nonlocal parabolic equation:
\[
\begin{cases}
u_t - \Delta u+(-\Delta)^s u = |u|^{p-2}u &\text{in }\Omega\times\mathbb{R}^+,\\
u(x, 0) = u_0(x) &\text{in }\Omega,\\
u(x, t) = 0 &\text{in } \mathbb{R}^N\backslash\Omega\times\mathbb{R}_0^+,
\end{cases}
\]
where \(\Delta\) is the Laplace operator, \((-\Delta)^s\) is the fractional Laplace operator, \(\Omega\subset \mathbb{R}^N\) is a bounded domain with Lipschitz boundary \(\partial\Omega\), \(N > 2s\), \(2 < p \leq 2^\ast\) and \(s\in(0, 1)\). In the cases of low and critical initial energy, we not only prove the existence of global solutions and the decay rate of the \(L^2\) norm for global solutions, but also obtain blow-up of solutions in finite time and the lower and upper bounds of the blow-up time. In the case of high initial energy, we give sufficient conditions for the global existence and blow-up of solutions, and the lower and upper bounds on the blow-up time.Stable spatially inhomogeneous periodic solutions for a diffusive Leslie-Gower predator-prey modelhttps://zbmath.org/1544.350342024-11-01T15:51:55.949586Z"Jiang, Heping"https://zbmath.org/authors/?q=ai:jiang.heping(no abstract)Stability and bifurcation diagram for a shadow Gierer-Meinhardt system in one spatial dimensionhttps://zbmath.org/1544.350352024-11-01T15:51:55.949586Z"Kaneko, Yuki"https://zbmath.org/authors/?q=ai:kaneko.yuki"Miyamoto, Yasuhito"https://zbmath.org/authors/?q=ai:miyamoto.yasuhito"Wakasa, Tohru"https://zbmath.org/authors/?q=ai:wakasa.tohruSummary: We are concerned with a Neumann problem of a shadow system of the Gierer-Meinhardt model in an interval \(I=(0,1)\). A stationary problem is studied, and we consider the diffusion coefficient \(\varepsilon>0\) as a bifurcation parameter. Then a complete bifurcation diagram of the stationary solutions is obtained, and a stability of every stationary solution is determined. In particular, for each \(n\geqslant 1\), two branches of \(n\)-mode solutions emanate from a trivial branch. All 1-mode solutions are stable for small \(\tau>0\), and all \(n\)-mode solutions, \(n\geqslant 2\), are unstable for all \(\tau>0\), where \(\tau>0\) is a time constant. The system is known for having stationary spiky patterns with large amplitude for small \(\varepsilon>0\). Then, asymptotic expansions of maximum and minimum values of a stationary solution as \(\varepsilon\rightarrow 0\) are also obtained.
{{\copyright} 2024 IOP Publishing Ltd \& London Mathematical Society}On a robust stability criterion of the radially symmetric heat equationhttps://zbmath.org/1544.350392024-11-01T15:51:55.949586Z"Temoltzi-Ávila, R."https://zbmath.org/authors/?q=ai:temoltzi-avila.raulSummary: This paper establishes a robust stability criterion in the radially symmetric heat equation that admits heat sources belonging to a set of bounded functions. The robust stability criterion is determined by extending a definition of stability under constant-acting perturbations that was originally established for systems of ordinary differential equations. It is assumed that heat sources admit a Fourier series representation whose coefficients are bounded and piecewise continuous functions. The robust stability criterion obtained is useful to conclude that the solution of the heat equation, as well as its first partial derivatives with respect to the radial axis and with respect to time, are bounded by a constant whose value is initially established. The results obtained are illustrated numerically.Strong nonlocal-to-local convergence of the Cahn-Hilliard equation and its operatorhttps://zbmath.org/1544.350412024-11-01T15:51:55.949586Z"Abels, Helmut"https://zbmath.org/authors/?q=ai:abels.helmut"Hurm, Christoph"https://zbmath.org/authors/?q=ai:hurm.christophSummary: We prove convergence of a sequence of weak solutions of the nonlocal Cahn-Hilliard equation to the strong solution of the corresponding local Cahn-Hilliard equation. The analysis is done in the case of sufficiently smooth bounded domains with Neumann boundary condition and a \(W^{1, 1}\)-kernel. The proof is based on the relative entropy method. Additionally, we prove the strong \(L^2\)-convergence of the nonlocal operator to the negative Laplacian together with a rate of convergence.Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction-repulsion chemotaxis models with logisticshttps://zbmath.org/1544.350432024-11-01T15:51:55.949586Z"Columbu, Alessandro"https://zbmath.org/authors/?q=ai:columbu.alessandro"Díaz Fuentes, Rafael"https://zbmath.org/authors/?q=ai:diaz-fuentes.rafael"Frassu, Silvia"https://zbmath.org/authors/?q=ai:frassu.silviaSummary: The following fully nonlinear attraction-repulsion and zero-flux chemotaxis model is studied:
\[
\begin{cases}
u_t = \nabla \cdot((u + 1)^{m_1 - 1} \nabla u - \chi u (u + 1)^{m_2 - 1} \nabla v \\
\qquad+ \xi u(u + 1)^{m_3 - 1} \nabla w) + \lambda u - \mu u^r & \text{in }\Omega\times(0, T_{max}), \\
\tau v_t = \Delta v - \phi(t, v) + f(u) & \text{in }\Omega\times(0, T_{max}), \\
\tau w_t = \Delta w - \psi(t, w) + g(u) & \text{in }\Omega\times(0, T_{max}).
\end{cases}\tag{\(\diamondsuit\)}
\]
Herein, \(\Omega\) is a bounded and smooth domain of \(\mathbb{R}^n\), for \(n\in\mathbb{N}\), \(\chi\), \(\xi\), \(\lambda\), \(\mu\), \(r\) proper positive numbers, \(m_1, m_2, m_3\in\mathbb{R}\), and \(f(u)\) and \(g(u)\) regular functions that generalize the prototypes \(f(u) \simeq u^k\) and \(g(u) \simeq u^l\), for some \(k, l > 0\) and all \(u \geq 0\). Moreover, \(\tau\in\{0, 1\}\), and \(T_{max}\in(0, \infty]\) is the maximal interval of existence of solutions to the model. Once suitable initial data \(u_0(x)\), \(\tau v_0(x)\), \(\tau w_0(x)\) are fixed, we are interested in deriving sufficient conditions implying globality (i.e., \(T_{max} = \infty\)) and boundedness (i.e., \(\|u(\cdot, t)\|_{L^\infty(\Omega)} + \|v(\cdot, t) \|_{L^\infty(\Omega)} + \|w(\cdot, t)\|_{L^\infty(\Omega)} \leq C\) for all \(t\in(0, \infty)\)) of solutions to problem (\(\diamondsuit\)). This is achieved in the following scenarios:
\(\vartriangleright\) For \(\phi(t, v)\) proportional to \(v\) and \(\psi(t, w)\) to \(w\), whenever \(\tau = 0\) and provided one of the following conditions
\[
(\mathrm{I})\;m_2 + k < m_3 + l, \quad (\mathrm{II})\;m_2 + k < r, \quad (\mathrm{III})\; m_2 + k < m_1 + \frac{2}{n}
\]
is accomplished or \(\tau = 1\) in conjunction with one of these restrictions
\[
\begin{alignedat}{1}
&(\mathrm{i})\;\max[m_2 + k, m_3 + l] < r, \qquad(\mathrm{ii})\;\max[m_2 + k, m_3 + l] < m_1 + \frac{2}{n},\\
&(\mathrm{iii})\; m_2 + k < r \text{ and }m_3 + l < m_1 + \frac{2}{n},\qquad(\mathrm{iv})\;m_2 + k < m_1 + \frac{2}{n}\text{ and }m_3 + l < r;
\end{alignedat}
\]
\(\vartriangleright\) For \(\phi(t, v) = \frac{1}{|\Omega|}\int_\Omega f(u)\) and \(\psi(t, w) = \frac{1}{|\Omega|}\int_\Omega g(u)\), whenever \(\tau = 0\) if moreover one among (I), (II), (III) is fulfilled. Our research partially improves and extends some results derived in
[\textit{Z. Jiao} et al., Nonlinear Anal., Real World Appl. 77, Article ID 104023, 15 p. (2024; Zbl 1537.35104);
\textit{G. Ren} and \textit{B. Liu}, Z. Angew. Math. Phys. 73, No. 2, Paper No. 58, 25 p. (2022; Zbl 1485.35060);
\textit{Y. Chiyo} and \textit{T. Yokota}, Z. Angew. Math. Phys. 73, No. 2, Paper No. 61, 27 p. (2022; Zbl 1485.35072);
\textit{A. Columbu} et al., Stud. Appl. Math. 151, No. 4, 1349--1379 (2023; Zbl 1537.35344)].Asymptotic profiles for inhomogeneous heat equations with memoryhttps://zbmath.org/1544.350442024-11-01T15:51:55.949586Z"Cortázar, Carmen"https://zbmath.org/authors/?q=ai:cortazar.carmen"Quirós, Fernando"https://zbmath.org/authors/?q=ai:quiros-gracian.fernando"Wolanski, Noemí"https://zbmath.org/authors/?q=ai:wolanski.noemi-iSummary: We study the large-time behavior in all \(L^{p}\) norms of solutions to an inhomogeneous nonlocal heat equation in \(\mathbb{R}^{N}\) involving a Caputo \(\alpha\)-time derivative and a power \(\beta\) of the Laplacian when the dimension is large, \(N > 4 \beta\). The asymptotic profiles depend strongly on the space-time scale and on the time behavior of the spatial \(L^{1}\) norm of the forcing term.Persistence of a competition model of plankton allelopathy in time-space periodic environmenthttps://zbmath.org/1544.350452024-11-01T15:51:55.949586Z"Du, Li-Jun"https://zbmath.org/authors/?q=ai:du.lijun"Zhang, Li"https://zbmath.org/authors/?q=ai:zhang.li.28|zhang.li.1|zhang.li.6|zhang.li.3|zhang.li.7|zhang.li.66|zhang.li.40|zhang.li.72|zhang.li.59|zhang.li.10|zhang.li.15|zhang.li.8|zhang.li.89|zhang.li.14|zhang.li.57|zhang.li.13|zhang.li.11|zhang.li.56|zhang.li.27|zhang.li.5|zhang.li.9|zhang.li.20|zhang.li.58|zhang.li.35"Cao, Qian"https://zbmath.org/authors/?q=ai:cao.qianSummary: This work is devoted to the study of a competition model of plankton allelopathy imposed in time-space periodic environment. We prove that the system admits positive periodic solutions under certain conditions. We further obtain some sufficient conditions for the uniqueness and global stability of the positive periodic solution, which shows that the model is persistent. The main tools for our arguments are comparison theorems based on the maximum principle, sub- and supersolutions method, and an iteration method, which also permit the treatment of some more general reaction-diffusion models in periodic environment.Global boundedness and stability of a predator-prey model with alarm-taxishttps://zbmath.org/1544.350492024-11-01T15:51:55.949586Z"Li, Songzhi"https://zbmath.org/authors/?q=ai:li.songzhi"Wang, Kaiqiang"https://zbmath.org/authors/?q=ai:wang.kaiqiangSummary: This paper deals with the global boundedness and stability of classical solutions to an important alarm-taxis ecosystem that is significant in understanding the behaviors of prey and predators. Specifically, it studies the case where prey attracts the secondary predators when threatened by the primary predators. The secondary consumers pursue the signal generated by the interaction between the prey and direct consumers. However, obtaining the necessary gradient estimates for global existence seems difficult in the critical case due to the strong coupled structure. Therefore, a new approach is developed to estimate the gradient of prey and primary predators, which takes advantage of slightly higher damping power. Subsequently, the boundedness of classical solutions in two-dimension with Neumann boundary conditions can be established by energy estimates and semigroup theory. Moreover, by constructing Lyapunov functional, it is proved that the coexistence homogeneous steady states are asymptotically stable, and the convergence rate is exponential under certain assumptions on the system coefficients.On a fourth order equation describing single-component film modelshttps://zbmath.org/1544.350502024-11-01T15:51:55.949586Z"Magliocca, Martina"https://zbmath.org/authors/?q=ai:magliocca.martinaSummary: We study existence results for a fourth order problem describing single-component film models assuming initial data in Wiener spaces.A critical exponent in a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effectshttps://zbmath.org/1544.350532024-11-01T15:51:55.949586Z"Stinner, Christian"https://zbmath.org/authors/?q=ai:stinner.christian"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelIn this paper, the authors study a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects endowed with homogeneous Neumann boundary conditions in a smoothly bounded domain \(\mathbb{R}^{n}\), \(n\geq3\) with sufficiently regular functions \(D(u)\) and \(S(u)\). When \(\frac{S(u)}{D(u)}\cong Cu^{\alpha}\) with \(C>0\), they establish some criticality of the value \(\alpha=\frac{2}{n}\) for \(n\geq3\), without any additional assumption on the behavior of \(D(s)\) as \(s\rightarrow\infty\), in particular without requiring any algebraic lower bound for \(D\). Moreover, applied to the Keller-Segel system with volume-filling effect for probability distribution functions of the type \(Q(s)=\exp(-s^{\beta})\), \(s\geq0\), for global solvability the exponent \(\beta=\frac{n-2}{n}\) is seen to be critical.
These results are undoubtedly novel and intriguing, and they extend previous understanding in a seamless manner. The derivation is presented and organized exceptionally well, building upon existing ideas from the literature and appropriately citing the relevant references. Moreover, the work introduces significant and nontrivial new ideas. Overall, in my opinion, this is an exemplary piece of research.
Reviewer: Pan Zheng (Chongqing)Instability of homogeneous steady states in chemotaxis systems with flux limitationhttps://zbmath.org/1544.350582024-11-01T15:51:55.949586Z"Mao, Xuan"https://zbmath.org/authors/?q=ai:mao.xuan"Li, Yuxiang"https://zbmath.org/authors/?q=ai:li.yuxiang.2|li.yuxiang.1|li.yuxiangThe authors consider the parabolic-elliptic chemotaxis model with flux limitation
\[
\begin{cases} u_t = \Delta u - \nabla \cdot (u (1 + |\nabla v|^2)^\frac{\alpha-2}{2}) \nabla v), \\
0 = \Delta v - \mu + u, \quad \mu = \frac{1}{|\Omega|} \int_\Omega u_0, \end{cases}
\]
complemented with Neumann boundary and initial conditions, in smooth, bounded domains \(\Omega \subset \mathbb R^n\), \(n \ge 3\). In [\textit{M. Winkler}, Indiana Univ. Math. J. 71, No. 4, 1437--1465 (2022; Zbl 1501.35094)] it has been shown that all solutions are global in time if \(\alpha < \frac{n-1}{n}\) but when \(\alpha > \frac{n-1}{n}\), for each \(\mu > 0\) one can find initial data \(u_0\) with \(\frac{1}{|\Omega|} \int_\Omega u_0 = \mu\) such that the corresponding solution blows up in finite time.
The authors show that a critical mass phenomenon detected in [\textit{M. Winkler}, Math. Ann. 373, No. 3--4, 1237--1282 (2019; Zbl 1416.35049)] for \(\alpha = 2\) also holds in the regime \(\alpha \in [\frac{n}{n-1}, 2)\). That is, their main result states that if \(\mu\) is large enough and \(\Omega\) is a ball, \textit{all} radially symmetric initial data strictly more concentrated than the average mass lead to finite-time blow-up, while in contrast for sufficiently small \(\mu\), the homogeneous steady state \((\mu, \mu)\) is locally asymptotically stable with respect to the \(L^\infty\) topology.
The blow-up proof relies on an intricate comparison argument for the transformed quantity \(\int_0^{\xi^\frac1n} \rho^{n-1} u(\rho, t) \,\mathrm d\rho\), while the stability result is obtained by semigroup arguments.
Reviewer: Mario Fuest (Hannover)Blowup for semilinear parabolic equation with logarithmic nonlinearityhttps://zbmath.org/1544.350602024-11-01T15:51:55.949586Z"Wang, Xingchang"https://zbmath.org/authors/?q=ai:wang.xingchang"Wang, Yitian"https://zbmath.org/authors/?q=ai:wang.yitianSummary: In this paper, we proposed a new method to prove the blowup at \(+\infty\) for the solution of the initial boundary value problem for a class of semilinear parabolic equations with logarithmic nonlinearity at sub-critical initial energy level by contradiction. Moreover, following the idea of contradiction, we obtained a sufficient criterion for the blowup at \(+\infty\) of the solution with arbitrarily positive initial energy by defining a new auxiliary function, in which the location of the initial data concerning the unstable set is not required.Gradient bounds for strongly singular or degenerate parabolic systemshttps://zbmath.org/1544.350612024-11-01T15:51:55.949586Z"Ambrosio, Pasquale"https://zbmath.org/authors/?q=ai:ambrosio.pasquale"Bäuerlein, Fabian"https://zbmath.org/authors/?q=ai:bauerlein.fabianSummary: We consider weak solutions \(u : \Omega_T \to \mathbb{R}^N\) to parabolic systems of the type
\[
u_t - \operatorname{div} A(x, t, Du) = f \qquad \text{in } \Omega_T = \Omega \times(0, T),
\]
where \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\) for \(n \geq 2\), \(T > 0\) and the datum \(f\) belongs to a suitable Orlicz space. The main novelty here is that the partial map \(\xi \mapsto A(x, t, \xi)\) satisfies standard \(p\)-growth and ellipticity conditions for \(p > 1\) only outside the unit ball \(\{|\xi| < 1\}\). For \(p > \frac{2 n}{n + 2}\) we establish that any weak solution
\[
u \in C^0((0, T); L^2(\Omega, \mathbb{R}^N)) \cap L^p(0, T; W^{1, p}(\Omega, \mathbb{R}^N))
\]
admits a locally bounded spatial gradient \(Du\). Moreover, assuming that \(u\) is essentially bounded, we recover the same result in the case \(1 < p \leq \frac{2n}{n + 2}\) and \(f = 0\). Finally, we also prove the uniqueness of weak solutions to a Cauchy-Dirichlet problem associated with the parabolic system above. We emphasize that our results include both the degenerate case \(p \geq 2\) and the singular case \(1 < p < 2\).Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operatorshttps://zbmath.org/1544.350632024-11-01T15:51:55.949586Z"Guliyev, V. S."https://zbmath.org/authors/?q=ai:guliyev.vagif-sabirSummary: We consider a class of hypoelliptic operators of the following type
\[
\mathcal{L} = \sum_{i, j=1}^{p_0} a_{ij} \partial_{x_i x_j}^2 + \sum \limits_{i, j=1}^N b_{ij} x_i \partial_{x_j}-\partial_t,
\]
where \((a_{ij})\), \((b_{ij})\) are constant matrices and \((a_{ij})\) is symmetric positive definite on \(\mathbb{R}^{p_0}\) (\(p_0 \leq N\)). We obtain generalized Hölder estimates for \(\mathcal{L}\) on \(\mathbb{R}^{N+1}\) by establishing several estimates of singular integrals in generalized Morrey spaces.Optimal global second-order regularity and improved integrability for parabolic equations with variable growthhttps://zbmath.org/1544.350652024-11-01T15:51:55.949586Z"Arora, Rakesh"https://zbmath.org/authors/?q=ai:arora.rakesh"Shmarev, Sergey"https://zbmath.org/authors/?q=ai:shmarev.sergey-iSummary: We consider the homogeneous Dirichlet problem for the parabolic equation
\[
u_t - \operatorname{div} (|\nabla u|^{p(x, t)-2} \nabla u)=f(x, t) + F(x, t, u, \nabla u)
\]
in the cylinder \(Q_T := \Omega \times (0, T)\), where \(\Omega \subset \mathbb{R}^N\), \(N \geq 2\), is a \(C^2\)-smooth or convex bounded domain. It is assumed that \(p \in C^{0,1}(\overline{Q}_T)\) is a given function and that the nonlinear source \(F(x, t, s, \xi)\) has a proper power growth with respect to \(s\) and \(\xi\). It is shown that if \(p(x, t) > \frac{2(N+1)}{N+2}\), \(f \in L^2 (Q_T)\), \(|\nabla u_0|^{p(x, 0)} \in L^1 (\Omega)\), then the problem has a solution \(u \in C^0 ([0,T]; \, L^2(\Omega))\) with \(|\nabla u|^{p(x,t)} \in L^{\infty} (0, T; \, L^1(\Omega))\), \(u_t \in L^2 (Q_T)\), obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties:
\[
|\nabla u|^{2(p(x,t)-1)+r} \in L^1 (Q_T), \quad \text{for any } 0 < r < \frac{4}{N+2}, \quad |\nabla u|^{p(x,t)-2} \nabla u \in L^2 (0, T; W^{1,2} (\Omega))^N.
\]On the regularity theory for mixed local and nonlocal quasilinear parabolic equationshttps://zbmath.org/1544.350682024-11-01T15:51:55.949586Z"Garain, Prashanta"https://zbmath.org/authors/?q=ai:garain.prashanta"Kinnunen, Juha"https://zbmath.org/authors/?q=ai:kinnunen.juhaThe weak subsolutions and supersolutions of very general parabolic nonlocal equations
\[
\frac{\partial u}{\partial t}+ \mathcal{L}_p u(x,t)- \operatorname{div}\mathcal{B}_p(x,t,u,\nabla u)= g(x,t,u)
\]
are studied in space \(\times\) time cylinders \(\Omega\times(0,T)\), where \(\Omega\subset\mathbb{R}^N\). A special case is the equation with the operators
\[
\mathcal{L}_p u(x,t)= \int_{\mathbb{R}^N} \frac{|u(x,t)- u(y,t)|^{p-2}(u(x,t)- u(y,t))}{|x-y|^{N+ps}}\,dy,
\]
where \(0<s<1\), \(1<p<\infty\) (the principal value of the integral is taken), and
\[
\mathcal{B}_p(x,t,u,\nabla u)= |\nabla u|^{p-2}\nabla u.
\]
(There is a misprint in formula (1.2), defining \(\mathcal{L}_p\).)
The weak subsolutions are proved to be locally bounded. The theorem comes with an estimate; the cases \(2N/(N+2)<p\) and \(1<p\le 2N/(N+2)\) have somewhat different bounds.
The semicontinuity and pointwise behaviour of the weak supersolutions is studied for the pointwise defined representation
\[
u(x,t)=\operatorname{ess}\liminf_{\substack{(y,\tau)\to(x,t)\\ \tau<t}} u(y,\tau).
\]
The assumption \(g=0\) is required for the deeper results. Energy estimates and a technically advanced variant of De Giorgi's method are used.
Reviewer: Peter Lindqvist (Trondheim)Self-similar solutions for the heat equation with a positive non-Lipschitz continuous, semilinear source termhttps://zbmath.org/1544.350732024-11-01T15:51:55.949586Z"Farina, A."https://zbmath.org/authors/?q=ai:farina.angiolo"Gianni, R."https://zbmath.org/authors/?q=ai:gianni.robertoSummary: We investigate the existence of self-similar solutions for the parabolic equation \(u_t = \varDelta u + u^m H(u)\), with \(0 \leq m < 1\) and \(H\) the Heaviside graph, coupled with the initial datum \(u(\boldsymbol{x}, 0) = -c(|\boldsymbol{x}|^2)^{\frac{1}{1 - m}}\), with \(c > 0\). We analyze two cases: the problem in \(\mathbb{R}^n\), \(n > 1\), with \(m = 0\) and the problem in \(\mathbb{R}\) when \(0 \leq m < 1\). In the first case we extend the result of
\textit{R. Gianni} and \textit{J. Hulshof} [Eur. J. Appl. Math. 3, No. 4, 367--379 (1992; Zbl 0789.35088)]
and show that there exist only two self-similar solutions changing sign, provided \(0 < c < c_{cr}\), with \(c_{cr}\) obtained solving a specific algebraic equation depending on \(n\). In the second case we prove that there exist at least two self-similar solutions of problem \(u_t = u_{xx} + u^m H(u)\), \(u(x, 0) = - c(x^2)^{\frac{1}{1 - m}}\), changing sign and evolving region where \(u > 0\). These solutions are of great interest. Indeed, on one hand they prove that the problem does not admit uniqueness and on the other they prove that a single point where \(u(x, 0) = 0\), for an initial datum which is otherwise negative, can generate a region where \(u(x, t)\) is positive.Multiple-peak traveling waves of the Gray-Scott modelhttps://zbmath.org/1544.350752024-11-01T15:51:55.949586Z"Chen, Xinfu"https://zbmath.org/authors/?q=ai:chen.xinfu"Lai, Xin"https://zbmath.org/authors/?q=ai:lai.xin"Qin, Cong"https://zbmath.org/authors/?q=ai:qin.cong"Qi, Yuanwei"https://zbmath.org/authors/?q=ai:qi.yuanwei"Zhang, Yajing"https://zbmath.org/authors/?q=ai:zhang.yajingSummary: We study a reaction-diffusion system which models the pre-mixed isothermal autocatalytic chemical reaction of order \(m\) \((m>1)\) between two chemical species, a reactant \(A\) and an auto-catalyst \(B\), \(A+mB\to(m+1)B\), and a linear decay \(B\to C\), where \(C\) is an inert product. The special case of \(m=2\) is the much studied Gray-Scott model, but without feeding. We prove existence of multiple traveling waves which have distinctive number of local maximaor peaks. It shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different traveling pulses than related models in literature such as isothermal autocatalysis without decay, or a bio-reactor model with isothermal autocatalysis of order \(m+1\) with \(m\)-th order of decay.Stability of a traveling wave on a saddle-node trajectoryhttps://zbmath.org/1544.350772024-11-01T15:51:55.949586Z"Kalyakin, L. A."https://zbmath.org/authors/?q=ai:kalyakin.leonid-anatolevichSummary: For semilinear partial differential equations, we consider the solution in the form of a plane wave traveling with a constant velocity. This solution is determined from an ordinary differential equation. A wave that stabilizes at infinity to equilibria corresponds to a phase trajectory connecting fixed points. The fundamental problem of the possibility of using such solutions in applications is their stability in the linear approximation. The stability problem is solved for a wave that corresponds to a trajectory from a saddle to a node. It is known that the velocity is determined ambiguously in this case. In this paper, a method is indicated for finding the limit of the velocity of stable waves for parabolic and hyperbolic equations, which can easily be implemented numerically.Nonlinear stability of shock-fronted travelling waves in reaction-nonlinear diffusion equationshttps://zbmath.org/1544.350782024-11-01T15:51:55.949586Z"Lizarraga, Ian"https://zbmath.org/authors/?q=ai:lizarraga.ian"Marangell, Robert"https://zbmath.org/authors/?q=ai:marangell.robertIn this paper, the stability of shock-fronted traveling waves is studied in the 4th-order PDE with nonlinear diffusion of the form
\[
\frac {\partial \bar U}{\partial t} = \frac {\partial }{\partial t} \left(D(\bar U)\frac {\partial \bar U}{\partial x}\right )+R(\bar U)-\epsilon^2 \frac {\partial^4 \bar U}{\partial t^4}
\]
with polynomial diffusion and reaction terms \(D\) and \(R\). When \(\epsilon \ll 1\) the system exhibits a multi-scale behavior. The shock-fronted traveling waves exist due to this multi-scale structure. For calculating the point spectrum of the operator of the linearization of the PDE about the wave, one typically performs direct computations of the eigenvalue problem along the wave or computations of the winding number of the Evans function. However, at the computational level there are complications and difficulties associated with both approaches, so, instead, the authors propose to use a recently developed technique based on the computation of the winding number of the Riccati-Evans function. Furthermore, the authors explain the output of the Riccati-Evans function calculation with the geometric-topologic framework developed by Alexander, Gardner, and Jones in 1990. In multi-scale problems the augmented unstable bundle which is in the center of this framework allows a splitting on the fast and slow subbundles. The authors use numerical computations to visualize the the fast and slow subbundles explicitly and calculate their Churn numbers. Lastly, the linearized operator is analyzed and shown to be sectorial. Thus the semigroup generated by this operator is analytic. The absence of the unstable spectrum then implies the nonlinear stability of the shock-fronted traveling waves.
Reviewer: Anna Ghazaryan (Oxford)Time-periodic traveling wave solutions of a reaction-diffusion Zika epidemic model with seasonalityhttps://zbmath.org/1544.350792024-11-01T15:51:55.949586Z"Zhao, Lin"https://zbmath.org/authors/?q=ai:zhao.linSummary: In this paper, the full information about the existence and nonexistence of a time-periodic traveling wave solution of a reaction-diffusion Zika epidemic model with seasonality, which is non-monotonic, is investigated. More precisely, if the basic reproduction number, denoted by \(R_0\), is larger than one, there exists a minimal wave speed \(c^* >0\) satisfying for each \(c>c^*\), the system admits a nontrivial time-periodic traveling wave solution with wave speed \(c\), and for \(c<c^*\), there exist no nontrivial time-periodic traveling waves such that if \(R_0 \leqslant 1\), the system admits no nontrivial time-periodic traveling waves.Global entropy solutions to a degenerate parabolic-parabolic chemotaxis system for flux-limited dispersalhttps://zbmath.org/1544.350822024-11-01T15:51:55.949586Z"Zhigun, Anna"https://zbmath.org/authors/?q=ai:zhigun.annaSummary: Existence of global finite-time bounded entropy solutions to a parabolic-parabolic system proposed in [\textit{N. Bellomo} et al., Math. Models Methods Appl. Sci. 20, No. 7, 1179--1207 (2010; Zbl 1402.92065)] is established in bounded domains under no-flux boundary conditions for nonnegative bounded initial data. This modification of the classical Keller-Segel model features degenerate diffusion and chemotaxis that are both subject to flux-saturation. The approach is based on Schauder's fixed point theorem and calculus of functions of bounded variation.On the boundary control problem associated with a fourth order parabolic equation in a two-dimensional domainhttps://zbmath.org/1544.350972024-11-01T15:51:55.949586Z"Dekhkonov, Farrukh"https://zbmath.org/authors/?q=ai:dekhkonov.farrukh-n"Li, Wenke"https://zbmath.org/authors/?q=ai:li.wenkeSummary: In this paper, we consider a boundary control problem associated with a fourth order parabolic type equation in a bounded two-dimensional domain. The solution with the control function on the border of the considered domain is given. The constraints on the control are determined to ensure that the average value of the solution within the considered domain attains a given value. The initial-boundary problem is solved by the Fourier method, and the control problem under consideration is analyzed with the Volterra integral equation. The existence of admissible control is proved by the Laplace transform method.Front selection in reaction-diffusion systems via diffusive normal formshttps://zbmath.org/1544.350982024-11-01T15:51:55.949586Z"Avery, Montie"https://zbmath.org/authors/?q=ai:avery.montieSummary: We show that propagation speeds in invasion processes modeled by reaction-diffusion systems are determined by marginal spectral stability conditions, as predicted by the \textit{marginal stability conjecture}. This conjecture was recently settled in scalar equations; here we give a full proof for the multi-component case. The main new difficulty lies in precisely characterizing diffusive dynamics in the leading edge of invasion fronts. To overcome this, we introduce coordinate transformations which allow us to recognize a leading order diffusive equation relying only on an assumption of generic marginal pointwise stability. We are then able to use self-similar variables to give a detailed description of diffusive dynamics in the leading edge, which we match with a traveling invasion front in the wake. We then establish front selection by controlling these matching errors in a nonlinear iteration scheme, relying on sharp estimates on the linearization about the invasion front. We briefly discuss applications to parametrically forced amplitude equations, competitive Lotka-Volterra systems, and a tumor growth model.A nonlocal reaction-diffusion-advection model with free boundarieshttps://zbmath.org/1544.350992024-11-01T15:51:55.949586Z"Tang, Yaobin"https://zbmath.org/authors/?q=ai:tang.yaobin"Dai, Binxiang"https://zbmath.org/authors/?q=ai:dai.binxiangSummary: A nonlocal diffusion single population model with advection and free boundaries is considered. Our aim is to discuss how the advection rate affects dynamic behaviors of species under the case of small advection. Firstly, the well-posed global solution is obtained. Secondly, we apply the eigenvalue problem of integro-differential operator to obtain the dichotomy and sharp criteria for spreading and vanishing, which is determined by initial habitat and initial density. Further, the asymptotic spreading speed of species is estimated when spreading happens. Namely, we get the exact asymptotic spreading speed and find that if kernel function satisfies the certain condition, then the leftward asymptotic spreading speed is less than the rightward one due to the impact of advection rate. Meanwhile, we also observe that accelerated spreading happens if the certain condition does not be satisfied.Amplitude equations for wave bifurcations in reaction-diffusion systemshttps://zbmath.org/1544.351002024-11-01T15:51:55.949586Z"Villar-Sepúlveda, Edgardo"https://zbmath.org/authors/?q=ai:villar-sepulveda.edgardo"Champneys, Alan"https://zbmath.org/authors/?q=ai:champneys.alan-rSummary: A wave bifurcation is the counterpart to a Turing instability in reaction-diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary \(n\)-component systems \((n\geqslant 3)\) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction-diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.
{{\copyright} 2024 The Author(s). Published by IOP Publishing Ltd and the London Mathematical Society}Bifurcations and exact bounded solutions of some traveling wave systems determined by integrable nonlinear oscillators with \(q\)-degree dampinghttps://zbmath.org/1544.351012024-11-01T15:51:55.949586Z"Zhang, Lijun"https://zbmath.org/authors/?q=ai:zhang.lijun.2"Chen, Guanrong"https://zbmath.org/authors/?q=ai:chen.guanrong"Li, Jibin"https://zbmath.org/authors/?q=ai:li.jibin(no abstract)Global population propagation dynamics of reaction-diffusion models with shifting environment for non-monotone kinetics and birth pulsehttps://zbmath.org/1544.351022024-11-01T15:51:55.949586Z"Zhang, Yurong"https://zbmath.org/authors/?q=ai:zhang.yurong"Yi, Taishan"https://zbmath.org/authors/?q=ai:yi.taishan"Wu, Jianhong"https://zbmath.org/authors/?q=ai:wu.jianhongSummary: We consider a general impulsive reaction-diffusion equation with shifting environment and birth pulse, both induced by climate changes, to capture essential features of the dynamics of species exhibiting distinct stages of reproduction and dispersal. We convert this model into a discrete-time semiflow, and hence to a discrete-time recursion system. We establish the existence of forced waves and asymptotic spreading properties of solutions, and obtain sufficient conditions for the global asymptotic stability of the forced waves.Turing instability and amplitude equation of reaction-diffusion system with multivariablehttps://zbmath.org/1544.351032024-11-01T15:51:55.949586Z"Zheng, Qianqian"https://zbmath.org/authors/?q=ai:zheng.qianqian"Shen, Jianwei"https://zbmath.org/authors/?q=ai:shen.jianweiSummary: In this paper, we investigate pattern dynamics with multivariable by using the method of matrix analysis and obtain a condition under which the system loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation with multivariable. This is an effective tool to investigate multivariate pattern dynamics. The example and simulation used in this paper validate our theoretical results. The method presented is a novel approach to the investigation of specific real systems based on the model developed in this paper.An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimensionhttps://zbmath.org/1544.351042024-11-01T15:51:55.949586Z"Lappicy, Phillipo"https://zbmath.org/authors/?q=ai:lappicy.phillipo"Beatriz, Ester"https://zbmath.org/authors/?q=ai:beatriz.esterSummary: Energy (or Lyapunov) functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano constructed a Lyapunov function for quasilinear non-degenerate parabolic equations. We modify Matano's method to construct an energy formula for fully nonlinear degenerate parabolic equations. We provide several examples of formulae, and in particular, a new energy candidate for the porous medium equation.The fourth-order total variation flow in \(\mathbb{R}^n\)https://zbmath.org/1544.351052024-11-01T15:51:55.949586Z"Giga, Yoshikazu"https://zbmath.org/authors/?q=ai:giga.yoshikazu"Kuroda, Hirotoshi"https://zbmath.org/authors/?q=ai:kuroda.hirotoshi"Łasica, Michał"https://zbmath.org/authors/?q=ai:lasica.michalSummary: We define rigorously a solution to the fourth-order total variation flow equation in \(\mathbb{R}^n\). If \(n\geq3\), it can be understood as a gradient flow of the total variation energy in \(D^{-1}\), the dual space of \(D^1_0\), which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case \(n\leq2\), the space \(D^{-1}\) does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if \(n\neq2\). If \(n\neq2\), all annuli are calibrable, while in the case \(n = 2\), if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.On self-similar patterns in coupled parabolic systems as non-equilibrium steady stateshttps://zbmath.org/1544.351672024-11-01T15:51:55.949586Z"Mielke, Alexander"https://zbmath.org/authors/?q=ai:mielke.alexander"Schindler, Stefanie"https://zbmath.org/authors/?q=ai:schindler.stefanieThe authors investigate reaction-diffusion systems and other related dissipative systems on unbounded domains. They discuss the self-similarity and asymptotical behavior. Self-similar behavior is a known phenomenon in extended systems. The solutions are usually considered with trivial behavior at infinity in the case of finite mass or energy of the considered physical system.
The authors of the present paper discuss the asymptotic self-similar behavior and show that it can occur in three different ways. Given three cases which can be distinguished after transforming into scaling variables: (A) The transformed system is autonomous and its steady state is a classical self-similar solution. (B) The transformed system converges to an autonomous system having suitable steady states. (C) An exponentially growing term generates a constraint that generates a constrained steady state. Constrained self-similar profiles occur mainly in systems of PDEs where diffusion and reaction terms scale differently.
It seems that the authors suppose that case (A) often occurs in scalar equations while cases (B) and (C) are more common in coupled systems of equations. Moreover, they consider the case of nonzero boundary conditions at infinity, which leads to systems with infinite mass displaying a richer structure than finite-mass systems.
For the evolution equation \((1)\) \(\tilde{u}_t=\tilde{f}(t,x, \tilde{u},\tilde{\nabla}\tilde{u}. \ldots ,\tilde{\nabla}^k\tilde{u})\) per definition, the solution \(\tilde{u}\) is called self-similar, in the sense of Barenblatt, if it can be written in the form \(\tilde{u}(t,x)= (1+t)^{-\alpha}U(x/(1+t)^{\beta})\) for a function \(U\) and scaling exponents \(\alpha \) and \(\beta \), which are suitably chosen, for instance, in order to guarantee mass conservation. The function \(U\) is called the profile of the self-similar solution. This concept is well-known, as it already finds an application for simple problems. To classify different types of self-similarity one has to transform the system into suitable scaling coordinates in which the self-similar profile appears as a steady pattern. The new coordinates \((\tau ,y)\) can be obtained by \(\tau = \log{(1+t)}\) and \(y=x/(1+t)^{\beta}\). After replacing \(u(\tau ,y)= (1+t)^{\alpha}\tilde{u}(t,x)\) in \((1)\) then obtain \((2)\) \(u_{\tau}=f(\tau ,y,u,\nabla u, \ldots ,\nabla^{k}u)\). Here \(\nabla\) concerns spatial derivatives w.r.t. \(y\). When the general structure of the transformed system has the form \(\boldsymbol{u}_{\tau}=\boldsymbol{f} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})+ e^{\gamma\tau} \boldsymbol{g} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})\), \(\gamma > 0\), and if \(\boldsymbol{u}(\tau ,y)\to \boldsymbol{u}(y)\) as \(\tau\to\infty\), then the constrained self-similar profile \(\boldsymbol{u}\) should satisfy \(\boldsymbol{g} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})=0\) and \(\boldsymbol{f} (y,\boldsymbol{u},\nabla\boldsymbol{u}, \ldots ,\nabla^k\boldsymbol{u})+ \boldsymbol{\lambda }(y)=0\), \(y\in\mathbb{R}^d\).
Next the authors study systems on the unbounded real line that have the property that their restriction to a finite domain has a Lyapunov function and a gradient structure. Then the system reach local equilibrium on a rather fast time scale, but on unbounded domains with an infinite amount of mass or energy, it leads to a persistent mass or energy flow. It turns out that no true equilibrium is reached globally. In suitably rescaled variables, however, the solutions to the transformed system converge to non-equilibrium steady states that correspond to asymptotically self-similar behavior in the original system.
Reviewer: Dimitar A. Kolev (Sofia)Finite-strain poro-visco-elasticity with degenerate mobilityhttps://zbmath.org/1544.351712024-11-01T15:51:55.949586Z"van Oosterhout, Willem J. M."https://zbmath.org/authors/?q=ai:van-oosterhout.willem-j-m"Liero, Matthias"https://zbmath.org/authors/?q=ai:liero.matthiasThe authors consider a time horizon \(T>0\), a bounded and open domain \(\Omega \subseteq \mathbb{R}^{d}\), and the quasi-static system: \(-\operatorname{div}(\sigma _{el}(\nabla \chi ,c)+\sigma _{vi}(\nabla \chi ,\nabla \overset{.}{\chi } ,c)-\operatorname{div}\mathfrak{h}(D^{2}\chi ))=f(t)\), \(\overset{.}{c}-\operatorname{div}(\mathcal{M} (\nabla \chi ,c)\nabla \mu )=0\), in \([0,T]\times \Omega \), the total stress \( \Sigma _{tot}=\sigma _{el}+\sigma _{vi}-\operatorname{div}\mathfrak{h}\) consisting of the elastic stress \(\sigma _{el}(F,c)=\partial _{F}\Phi (F,c)\), the viscous stress \(\sigma _{vi}(F,\overset{.}{F},c)=\partial _{\overset{.}{F}}\zeta (F, \overset{.}{F},c)\), and the hyperstress \(\mathfrak{h}(G)=\partial _{G} \mathcal{H}(G)\), \(f\) is a body force, \(\mathcal{M}\) is the mobility tensor, and \(\mu (F,c)=\partial _{c}\Phi (F,c)\) is the chemical potential. The authors give two examples of such poro-visco-elastic materials. The boundary conditions \(\chi =Id\), on \(\Gamma _{D}\), \((\sigma _{el}(\nabla \chi ,c)+\sigma _{vi}(\nabla \chi ,\nabla \overset{.}{\chi },c)-\operatorname{div}_{x}(\mathfrak{ h}(D^{2}\chi ))\cdot \overrightarrow{n}=g(t)\), on \(\Gamma _{N}\), \(\mathfrak{h }(D^{2}\chi ):(\overrightarrow{n}\otimes \overrightarrow{n})=0\), on \( \partial \Omega \), and \(\mathcal{M}(\nabla \chi ,c)\nabla \mu \cdot \overrightarrow{n}=\kappa (\mu _{ext}(t)-\mu )\), on \(\partial \Omega \), are added, where \(\overrightarrow{n}\) denotes the unit normal vector on \( \partial \Omega \), \(\kappa \geq 0\) is a given permeability and \(\mu _{ext}\) is an external potential. The authors define a weak solution to this problem as a pair \((\chi ,c)\) with \(\chi \in L^{\infty }(0,T;W_{id}^{2,p}(\Omega ; \mathbb{R}^{d}))\), \(\overset{.}{\chi }\in L^{2}(0,T;H^{1}(\Omega ;\mathbb{R} ^{d}))\) and \(c\in L^{\infty }(0,T;L\log L(\Omega )))\), \(\overset{.}{c}\in L^{s}(0,T;W^{1,ss}(\Omega )^{\ast })\), and \(\nabla c^{m/2}\in L^{2}(0,T;L^{2}(\Omega ))\), and depending on the hypotheses additionally \( c\in L^{\infty }(0,T;L^{2+r}(\Omega ))\) and \(\nabla c^{m/2+1+r}\in L^{2}(0,T;L^{2}(\Omega ))\), the pair \((\chi ,c)\) satisfying variational formulations associated with the previous equations. The main result proves under hypotheses on the data the existence of a weak solution to this problem. For the proof, the authors introduce a regularized problem and a time discretization through a modified variational formulation. They prove the existence of a solution to this time-discretized and regularized problem, applying Schauder's fixed-point theorem. They then prove an energy-dissipation inequality and uniform estimates with respect to the regularizing and time discretization parameters, which allow passing to the limit.
Reviewer: Alain Brillard (Riedisheim)On non-autonomous fractional evolution equations and applicationshttps://zbmath.org/1544.351832024-11-01T15:51:55.949586Z"Achache, Mahdi"https://zbmath.org/authors/?q=ai:achache.mahdiSummary: We consider the problem of maximal regularity for semilinear non-autonomous fractional equations
\[
\sum_{i=1}^n \lambda_i \partial^{\alpha_i} (u-u_0)(t)+{\mathscr{A}}(t)u(t)=F(t,u(t))\quad t {\text{-a.e.}}, \,\lambda_i\in{\mathbb{C}}, \,n\in{\mathbb{N}}.
\]
Here, \( \partial^{\alpha_i}\) denotes the Riemann-Liouville fractional derivative of order \(\alpha_i \in (0,1)\) w.r.t. time and each operator \({\mathscr{A}}(t)\) arises from a time depending sesquilinear form \(\mathfrak{a}(t)\) on a Hilbert space \({\mathscr{H}}\) with constant domain \({\mathscr{V}}\), such that \({\mathscr{V}}\) is continuously and densely embedded into \({\mathscr{H}}\). We prove non-autonomous maximal \(L^p\)-regularity results on \({\mathscr{V}}^\prime\) and other regularity properties for the solutions of the above equation under minimal regularity assumptions on the forms, the initial data \(u_0\) and the inhomogeneous term \(F\).The fractional porous medium equation on noncompact Riemannian manifoldshttps://zbmath.org/1544.351862024-11-01T15:51:55.949586Z"Berchio, Elvise"https://zbmath.org/authors/?q=ai:berchio.elvise"Bonforte, Matteo"https://zbmath.org/authors/?q=ai:bonforte.matteo"Grillo, Gabriele"https://zbmath.org/authors/?q=ai:grillo.gabriele"Muratori, Matteo"https://zbmath.org/authors/?q=ai:muratori.matteoSummary: We study nonnegative solutions to the fractional porous medium equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for data belonging either to the usual \(L^{1}\) space or to a considerably larger weighted space determined in terms of the fractional Green function. The class of manifolds for which the results hold includes both the Euclidean and the hyperbolic spaces and even in the Euclidean situation involves a class of data which is larger than the previously known one.On a non-local Kirchhoff type equation with random terminal observationhttps://zbmath.org/1544.351892024-11-01T15:51:55.949586Z"Duc, Phuong Nguyen"https://zbmath.org/authors/?q=ai:duc.phuong-nguyen"Van, Tien Nguyen"https://zbmath.org/authors/?q=ai:van.tien-nguyen"Anh, Tuan Nguyen"https://zbmath.org/authors/?q=ai:anh.tuan-nguyenSummary: In this work, we are concerned with the terminal value problem for the time fractional equation (in the sense of Conformable fractional derivative) with a nonlocal term of the Kirchhoff type
\[
\partial_t^\alpha u = K\Big(\|\nabla u\|_{L^2(\mathcal{D})}\Big)\Delta u + f(x,t), \quad (x,t) \in (0,T)\times \mathcal{D}
\]
subject to the final data which is blurred by random Gaussian white noise. The principal goal of this article is to recover the solution \(u \). This problem is severely ill-posed in the sense of Hadamard, because of the violation of the continuous dependence of the solution on the data (the solution's behavior does not change continuously with the final condition). By applying non-parametric estimates of the value data from observation data and the truncation method for the Fourier series, we obtain a regularized solution. Under some priori assumptions, we derive an error estimate between a mild solution and its regularized solution.A volume constraint problem for the nonlocal doubly nonlinear parabolic equationhttps://zbmath.org/1544.351942024-11-01T15:51:55.949586Z"Misawa, Masashi"https://zbmath.org/authors/?q=ai:misawa.masashi"Nakamura, Kenta"https://zbmath.org/authors/?q=ai:nakamura.kenta"Yamaura, Yoshihiko"https://zbmath.org/authors/?q=ai:yamaura.yoshihikoSummary: We consider a volume constraint problem for the nonlocal doubly nonlinear parabolic equation, called the nonlocal \(p\)-Sobolev flow, and introduce a nonlinear intrinsic scaling, converting a prototype nonlocal doubly nonlinear parabolic equation into the nonlocal \(p\)-Sobolev flow. This paper is dedicated to Giuseppe Mingione on the occasion of his 50th birthday, who is a maestro in the regularity theory of PDEs.Initial value and terminal value problems for distributed order fractional diffusion equationshttps://zbmath.org/1544.351952024-11-01T15:51:55.949586Z"Peng, Li"https://zbmath.org/authors/?q=ai:peng.li"Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yongSummary: In this work, we introduce and study two problems for diffusion equations with the distributed order fractional derivatives including the initial value problem and the terminal value problem. For the initial value problem, we establish some existence results and Hölder regularity for the mild solution. On the other hand, we also show the existence results and a decay estimate of the mild solution for the terminal value problems. Especially, the polynomial decay of the solutions to the terminal value problems is firstly included when the source function is equal to zero.Shape and location recovery of laser excitation sources in photoacoustic imaging using topological gradient optimizationhttps://zbmath.org/1544.352032024-11-01T15:51:55.949586Z"BenSalah, Mohamed"https://zbmath.org/authors/?q=ai:salah.mohamed-ben|bensalah.mohamed-oudiSummary: This study focuses on source term inversion in fractional partial differential equations, specifically applied to photoacoustic imaging. This work contributes to advancing imaging techniques and provides practical insights for medical diagnostics and materials characterization. Our aim in this paper is to develop an accurate method for recovering the location and the shape of a laser excitation source from partial boundary data. To achieve this, we reformulate our inverse problem as an optimization challenge. We utilize topological sensitivity analysis to establish an asymptotic expansion of a relevant shape function. These theoretical findings form the basis for a rapid and precise detection algorithm. Additionally, we present several numerical experiments that demonstrate the effectiveness and accuracy of our proposed approach.Determination of the time-dependent effective ion collision frequency from an integral observationhttps://zbmath.org/1544.352042024-11-01T15:51:55.949586Z"Cao, Kai"https://zbmath.org/authors/?q=ai:cao.kai"Lesnic, Daniel"https://zbmath.org/authors/?q=ai:lesnic.danielSummary: Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn-Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition.Determination of unknown time-dependent heat source in inverse problems under nonlocal boundary conditions by finite integration methodhttps://zbmath.org/1544.352082024-11-01T15:51:55.949586Z"Hazanee, Areena"https://zbmath.org/authors/?q=ai:hazanee.areena"Makaje, Nifatamah"https://zbmath.org/authors/?q=ai:makaje.nifatamahSummary: In this study, we investigate the unknown time-dependent heat source function in inverse problems. We consider three general nonlocal conditions; two classical boundary conditions and one nonlocal over-determination, condition, these genereate six different cases. The finite integration method (FIM), based on numerical integration, has been adapted to solve PDEs, and we use it to discretize the spatial domain; we use backward differences for the time variable. Since the inverse problem is ill-posed with instability, we apply regularization to reduce the instability. We use the first-order Tikhonov's regularization together with the minimization process to solve the inverse source problem. Test examples in all six cases are presented in order to illustrate the accuracy and stability of the numerical solutions.Simultaneous uniqueness for the diffusion coefficient and initial value identification in a time-fractional diffusion equationhttps://zbmath.org/1544.352092024-11-01T15:51:55.949586Z"Jing, Xiaohua"https://zbmath.org/authors/?q=ai:jing.xiaohua"Jia, Junxiong"https://zbmath.org/authors/?q=ai:jia.junxiong"Song, Xueli"https://zbmath.org/authors/?q=ai:song.xueliSummary: This article investigates the uniqueness of simultaneously determining the diffusion coefficient and initial value in a time-fractional diffusion equation with derivative order \(\alpha\in(0, 1)\). By additional boundary measurements and a priori assumption on the diffusion coefficient, the uniqueness of the eigenvalues and an associated integral equation for the diffusion coefficient are firstly established. The proof is based on the Laplace transform and the expansion of eigenfunctions for the solution to the initial value/boundary value problem. Furthermore, by using these two results, the simultaneous uniqueness in determining the diffusion coefficient and initial value is demonstrated from the Liouville transform and Gelfand-Levitan theory. The result shows that the uniqueness in simultaneous identification can be achieved, provided the initial values non-orthogonality to the eigenfunction of differential operators, which incorporates only one diffusion coefficient rather than scenarios involving two diffusion coefficients.Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equationhttps://zbmath.org/1544.352112024-11-01T15:51:55.949586Z"Qiao, Yu"https://zbmath.org/authors/?q=ai:qiao.yu.2|qiao.yu|qiao.yu.1"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuanSummary: In this paper, an inverse problem of a two-dimensional diffusion equation is considered. The purpose here includes recovering not only the source term but also the initial value from given observations at two fixed times \(t = T_1\) and \(t = T_2\). The uniqueness of the solutions for the inverse problem is given. Instead of the classical Tikhonov regularization strategy, we propose an iterated fractional Tikhonov regularization method to solve this problem, an \textit{a priori} and an \textit{a posterior} parameter selection rules and corresponding convergence rates are derived. For verification of the theoretical estimates, several numerical examples are constructed and compared with the standard iterated Tikhonov regularization method.Variational-hemivariational system for contaminant convection-reaction-diffusion model of recovered fracturing fluidhttps://zbmath.org/1544.490042024-11-01T15:51:55.949586Z"Cen, Jinxia"https://zbmath.org/authors/?q=ai:cen.jinxia"Migórski, Stanisław"https://zbmath.org/authors/?q=ai:migorski.stanislaw"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chih"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: This work is devoted to study the convection-reaction-diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection-reaction-diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.Analysis on ultra-metric spaces via heat kernelshttps://zbmath.org/1544.580092024-11-01T15:51:55.949586Z"Grigor'yan, Alexander"https://zbmath.org/authors/?q=ai:grigoryan.alexanderThe classical Laplace operator \(\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}\) on \(\mathbb{R}^n\) is associated with the Dirichlet integral by the Green formula. It is known that \(-\Delta\) is a non-negative definite self-adjoint operator on \(L^2(\mathbb{R}^n)\).
The associated heat equation \(\partial_tu-\Delta u=0\) has a fundamental solution \(p_t(x,y)\).
For any \(\beta\in (0,2)\), the operator \((-\Delta)^{\beta/2}\) determines a non-local Dirichlet form and its associated heat equation has a non-negative fundamental solution. It is known that for \(\beta=1\) the fundamental solution \(p_t^{(1)}(x,y)\) is the Cauchy distribution.
Definition and main properties of ultra-metric spaces are recalled.
In the paper under review, the author presents a construction of a natural class of random walks on any ultra-metric space \((X, d)\) that satisfies in addition the following conditions: it is separable, proper (that is, all balls are compact), and non-compact.
This construction is very natural, takes full advantage of the ultra-metric property and does not use Fourier Analysis.
In the case of \(\mathbb{Q}_p\) this class of processes coincides with the one constructed by Albeverio and Karwowski, and their generators coincide with the operators of Taibleson and Vladimirov.
Interesting examples are given. \par In Section 3, results in the analysis on \(\mathbb{Q}^n_p\) are presented. In Section 4, examples of heat kernels are given. Heat kernel estimates on Riemannian manifolds and for diffusions on fractals, respectively, are established. The walk dimension is defined. \par The main results are obtained for heat kernels on \(\alpha\)-regular ultra-metric spaces. The example of jump measure on products is provided. \par The proofs are detailed in the last section.
Reviewer: Ion Mihai (Bucureşti)Infinite-time blowing-up solutions to small perturbations of the Yamabe flowhttps://zbmath.org/1544.580102024-11-01T15:51:55.949586Z"Kim, Seunghyeok"https://zbmath.org/authors/?q=ai:kim.seunghyeok"Musso, Monica"https://zbmath.org/authors/?q=ai:musso.monicaIt is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold \(M\) exists for all time \(t\) and uniformly converges to a solution to the Yamabe problem on \(M\) as \(t \to +\infty\). The paper shows that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it. To prove their results the authors apply a modulation argument combined with the inner-outer gluing procedure to construct solutions to the perturbed flow that blow up at multiple points.
Reviewer: Fabrice Baudoin (Aarhus)Reconstructions of the asymptotics of an integral determined by a hyperbolic unimodal singularityhttps://zbmath.org/1544.580192024-11-01T15:51:55.949586Z"Zakharov, S. V."https://zbmath.org/authors/?q=ai:zakharov.sergei-viktorovichAuthor's abstract: The asymptotic behavior of an exponential integral is studied in which the phase function has the form of a special deformation of the germ of a hyperbolic unimodal singularity of type \(T_{4,4,4}.\) The integral under examination satisfies the heat equation, its Cole-Hopf transformation gives a solution of the vector Burgers equation in four-dimensional space-time, and its principal asymptotic approximations are expressed in terms of real solutions of systems of third-degree algebraic equations. The obtained analytical results make it possible to trace the bifurcations of an asymptotic structure depending on the parameter of the modulus of the singularity.
Reviewer: Maria Aparecida Soares Ruas (São Carlos)Schauder estimates for nonlocal equations with singular Lévy measureshttps://zbmath.org/1544.600532024-11-01T15:51:55.949586Z"Hao, Zimo"https://zbmath.org/authors/?q=ai:hao.zimo"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen.7|wang.zhen.20|wang.zhen.5|wang.zhen.10|wang.zhen.2|wang.zhen.1|wang.zhen.12|wang.zhen.14|wang.zhen.3|wang.zhen|wang.zhen.17|wang.zhen.13|wang.zhen.8|wang.zhen.9|wang.zhen.23"Wu, Mingyan"https://zbmath.org/authors/?q=ai:wu.mingyanThe authors ``establish Schauder's estimates for the following non-local equations in \(\mathbb{R}^d\):
\[
\partial_t u=\mathcal{L}_{k,\sigma}^{(\alpha)}u +b \cdot \nabla u +f , u(0)=0
\]
where \(\alpha\in(1/2,2)\) and \(b:\mathbb{R}_+ \times \mathbb{R}^d\) is an unbounded local \(\beta\)-order Hölder function in \(x\) uniformly in \(t\)', and \(\mathcal{L}_{k,\sigma}^{(\alpha)}\) is a certain non-local \(\alpha\)-stable-like operator.'' ``It is well-known that Schauder's estimates play a basic role in constructing the classical solution for quasilinear partial differential equations (...), and also give an approach to show the well-posedness of stochastic differential equations (...).''
Reviewer: Alexander Schnurr (Siegen)Weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic SPDEs driven by additive noisehttps://zbmath.org/1544.650252024-11-01T15:51:55.949586Z"Mukam, Jean Daniel"https://zbmath.org/authors/?q=ai:mukam.jean-daniel"Tambue, Antoine"https://zbmath.org/authors/?q=ai:tambue.antoineSummary: This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.On median filters for motion by mean curvaturehttps://zbmath.org/1544.651312024-11-01T15:51:55.949586Z"Esedoḡlu, Selim"https://zbmath.org/authors/?q=ai:esedoglu.selim"Guo, Jiajia"https://zbmath.org/authors/?q=ai:guo.jiajia"Li, David"https://zbmath.org/authors/?q=ai:li.david-z|li.david-day-uei|li.david-s|li.david-xSummary: The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters in light of recent progress on the threshold dynamics method. In particular, we give a variational formulation of, and exhibit a Lyapunov function for, median filters, resulting in energy based unconditional stability properties. The connection also yields analogues of median filters in the multiphase setting of mean curvature flow of networks. These new multiphase level set methods do not require frequent redistancing, and can accommodate a wide range of surface tensions.Realistic pattern formations on surfaces by adding arbitrary roughnesshttps://zbmath.org/1544.651382024-11-01T15:51:55.949586Z"Li, Siqing"https://zbmath.org/authors/?q=ai:li.siqing"Ling, Leevan"https://zbmath.org/authors/?q=ai:ling.leevan"Ruuth, Steven J."https://zbmath.org/authors/?q=ai:ruuth.steven-j"Wang, Xuemeng"https://zbmath.org/authors/?q=ai:wang.xuemengSummary: We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface construction methods yield comparable patterns to those observed in real-life animals.Positivity preserving and mass conservative projection method for the Poisson-Nernst-Planck equationhttps://zbmath.org/1544.651432024-11-01T15:51:55.949586Z"Tong, Fenghua"https://zbmath.org/authors/?q=ai:tong.fenghua"Cai, Yongyong"https://zbmath.org/authors/?q=ai:cai.yongyongSummary: We propose and analyze a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the \(L^2\) projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient \(L^2\) projection part), positivity preserving, and mass conserving. Rigorous error estimates in the \(L^2\) norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., \(H^1\) projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.Unconditional superconvergence analysis of a structure-preserving finite element method for the Poisson-Nernst-Planck equationshttps://zbmath.org/1544.651782024-11-01T15:51:55.949586Z"Yang, Huaijun"https://zbmath.org/authors/?q=ai:yang.huaijun"Li, Meng"https://zbmath.org/authors/?q=ai:li.meng.1|li.meng.2|li.meng.3Summary: In this paper, a linearized structure-preserving Galerkin finite element method is investigated for Poisson-Nernst-Planck (PNP) equations. By making full use of the high accuracy estimation of the bilinear element, the mean value technique and rigorously dealing with the coupled nonlinear term, not only the unconditionally optimal error estimate in \(L^2\)-norm but also the unconditionally superclose error estimate in \(H^1\)-norm for the related variables are obtained. Then, the unconditionally global superconvergence error estimate in \(H^1\)-norm is derived by a simple and efficient interpolation post-processing approach, without any coupling restriction condition between the time step size and the space mesh width. Finally, numerical results are provided to confirm the theoretical findings. The numerical scheme preserves the global mass conservation and the electric energy decay, and this work has a great improvement of the error estimate results given in [\textit{A. Prohl} and \textit{M. Schmuck}, Numer. Math. 111, No. 4, 591--630 (2009; Zbl 1178.65106)] and [\textit{H. Gao} and \textit{D. He}, J. Sci. Comput. 72, No. 3, 1269--1289 (2017; Zbl 1378.65168)].Preconditioning techniques of all-at-once systems for multi-term time-fractional diffusion equationshttps://zbmath.org/1544.651832024-11-01T15:51:55.949586Z"Gan, Di"https://zbmath.org/authors/?q=ai:gan.di"Zhang, Guo-Feng"https://zbmath.org/authors/?q=ai:zhang.guofeng"Liang, Zhao-Zheng"https://zbmath.org/authors/?q=ai:liang.zhaozhengSummary: In this paper, we consider solutions for discrete systems arising from multi-term time-fractional diffusion equations. Using discrete sine transform techniques, we find that all-at-once systems of such equations have a structure similar to that of diagonal-plus-Toeplitz matrices. We establish a generalized circulant approximate inverse preconditioner for the all-at-once systems. Through a detailed analysis of the preconditioned matrices, we show that the spectrum of the obtained preconditioned matrices is clustered around one. We give some numerical examples to demonstrate the effectiveness of the proposed preconditioner.Swarm gradient dynamics for global optimization: the mean-field limit casehttps://zbmath.org/1544.901562024-11-01T15:51:55.949586Z"Bolte, Jérôme"https://zbmath.org/authors/?q=ai:bolte.jerome"Miclo, Laurent"https://zbmath.org/authors/?q=ai:miclo.laurent"Villeneuve, Stéphane"https://zbmath.org/authors/?q=ai:villeneuve.stephaneSummary: Using jointly geometric and stochastic reformulations of nonconvex problems and exploiting a Monge-Kantorovich (or Wasserstein) gradient system formulation with vanishing forces, we formally extend the simulated annealing method to a wide range of global optimization methods. Due to the built-in combination of a gradient-like strategy and particle interactions, we call them swarm gradient dynamics. As in the original paper by Holley-Kusuoka-Stroock, a functional inequality is the key to the existence of a schedule that ensures convergence to a global minimizer. One of our central theoretical contributions is proving such an inequality for one-dimensional compact manifolds. We conjecture that the inequality holds true in a much broader setting. Additionally, we describe a general method for global optimization that highlights the essential role of functional inequalities la Łojasiewicz.Hysteresis and bistability in synaptic transmission modeled as a chain of biochemical reactions with a positive feedbackhttps://zbmath.org/1544.920162024-11-01T15:51:55.949586Z"Katauskis, Pranas"https://zbmath.org/authors/?q=ai:katauskis.pranas"Ivanauskas, Feliksas"https://zbmath.org/authors/?q=ai:ivanauskas.feliksas-f"Alaburda, Aidas"https://zbmath.org/authors/?q=ai:alaburda.aidasSummary: In this paper, we employ computational analysis to investigate the long-term potentiation (LTP) and memory formation in synapses between neurons. We use a mathematical model describing the synaptic transmission as a signal transduction pathway with a positive feedback loop formed by diffusion of nitric oxide (NO) to the presynaptic site. We found that the model of synaptic transmission exhibits a hysteresis-like behavior, where the strength of synaptic transmission depends not just on instantaneous interstimulus intervals, but also on the history of activity. The switching between resting and memory states can be induced by physiologically relevant and moderate (less than 50\%) changes in the duration of interstimulus intervals.Oscillations in a system modelling somite formationhttps://zbmath.org/1544.920272024-11-01T15:51:55.949586Z"Kovács, Sándor"https://zbmath.org/authors/?q=ai:kovacs.sandor"György, Szilvia"https://zbmath.org/authors/?q=ai:gyorgy.szilvia"Gyúró, Noémi"https://zbmath.org/authors/?q=ai:gyuro.noemiSummary: A minimal models of vertebrae formation were studied in concerning periodic structures formation. The authors proposed two kinds of reaction-diffusion models, from which one is of clock-and-wavefront type and the other one is of Turing type. Our goal is to show that in case of the Turing type model the kinetic system as well the reaction-diffusion system exhibit oscillating solutions. The chapter is organised as follows. In the next section we introduce the model. In the section that follows we examine the existence and stability of some equilibria. In the third section we show the occurrence of Hopf bifurcation in the kinetic system as well as in the parabolic system.
For the entire collection see [Zbl 1515.92004].New results on traveling waves for the Keller-Segel model with logistic sourcehttps://zbmath.org/1544.920322024-11-01T15:51:55.949586Z"Wang, Yahui"https://zbmath.org/authors/?q=ai:wang.yahui"Ou, Chunhua"https://zbmath.org/authors/?q=ai:ou.chunhuaSummary: We study the existence of traveling waves for the Keller-Segel chemotaxis system with logistic source in parabolic-parabolic and parabolic-elliptic cases. By a unified approach, we obtain new results on the traveling waves of these systems, via a novel constructive method and applying Schauder's fixed point theorem. These findings can be further applied to study the spreading behaviors and estimate the spreading speed of the systems.Critical patch size of a two-population reaction diffusion model describing brain tumor growthhttps://zbmath.org/1544.920432024-11-01T15:51:55.949586Z"Harris, Duane C."https://zbmath.org/authors/?q=ai:harris.duane-c"He, Changhan"https://zbmath.org/authors/?q=ai:he.changhan"Preul, Mark C."https://zbmath.org/authors/?q=ai:preul.mark-c"Kostelich, Eric J."https://zbmath.org/authors/?q=ai:kostelich.eric-j"Kuang, Yang"https://zbmath.org/authors/?q=ai:kuang.yangSummary: The critical patch (KISS) size is the minimum habitat size needed for a population to survive in a region. Habitats larger than the critical patch size allow a population to persist, while smaller habitats lead to extinction. We perform a rigorous derivation of the critical patch size associated with a 2-population glioblastoma multiforme (GBM) model that divides the tumor cells into proliferating and quiescent/necrotic populations. We determine that the critical patch size of our model is consistent with that of the Fisher-Kolmogorov-Petrovsky-Piskunov equation, one of the first reaction-diffusion models proposed for GBM, and does not depend on parameters pertaining to the quiescent/necrotic population. The critical patch size may indicate that GBM tumors have a minimum size depending on the location in the brain. We also derive a theoretical relationship between the size of a GBM tumor at steady-state and its maximum cell density, which has potential applications for patient-specific parameter estimation based on magnetic resonance imaging data. Finally, we identify a positively invariant region for our model, which guarantees that solutions remain positive and bounded from above for all time.Oscillations in neuronal activity: a neuron-centered spatiotemporal model of the unfolded protein response in prion diseaseshttps://zbmath.org/1544.920452024-11-01T15:51:55.949586Z"Miller, Elliot M."https://zbmath.org/authors/?q=ai:miller.elliot-m"Chan, Tat Chung D."https://zbmath.org/authors/?q=ai:chan.tat-chung-d"Montes-Matamoros, Carlos"https://zbmath.org/authors/?q=ai:montes-matamoros.carlos"Sharif, Omar"https://zbmath.org/authors/?q=ai:sharif.omar"Pujo-Menjouet, Laurent"https://zbmath.org/authors/?q=ai:pujo-menjouet.laurent"Lindstrom, Michael R."https://zbmath.org/authors/?q=ai:lindstrom.michael-rSummary: Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the unfolded protein response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P (healthy) and S (toxic) interacting with heterodimer dynamics (S interacts with P to form two S's). The model takes the form of a coupled system of nonlinear reaction-diffusion equations with a delayed, nonlinear flux for P (delay from the UPR). Through the delay, we find parameter regimes that exhibit oscillations in the P- and S-protein levels. We find that oscillations are more pronounced when the S-clearance rate and S-diffusivity are small in comparison to the P-clearance rate and P-diffusivity, respectively. The oscillations become more pronounced as delays in initiating the UPR increase. We also consider quasi-realistic clinical parameters to understand how possible drug therapies can alter the course of a prion disease. We find that decreasing the production of P, decreasing the recruitment rate, increasing the diffusivity of S, increasing the UPR S-threshold, and increasing the S clearance rate appear to be the most powerful modifications to reduce the mean UPR intensity and potentially moderate the disease progression.Computational modeling of membrane blockage via precipitation: a 2D extended Poisson-Nernst-Planck modelhttps://zbmath.org/1544.920562024-11-01T15:51:55.949586Z"Lefraich, H."https://zbmath.org/authors/?q=ai:lefraich.hamidSummary: The mathematical modeling and simulation of ion transport through biological membranes are a challenging problem with direct applications in biology and biophysics. The most important aim of this research is to develop a generalized mathematical model for a diffusion-migration-reaction system, which incorporates a pore blockage effect caused by the generation of insoluble precipitates in a porous membrane. While creating such model, both the electrostatic interactions of ions and the effects due to the buildup of solid reaction products in the membrane have to be taken into account. Consequently, the system behavior is investigated via the numerical simulation of an extended, highly nonlinear equation set based on the classical Poisson-Nernst-Planck equations for ion transport. This formulation incorporates both a reaction term and a space-and time-dependent diffusivity which often seem to be ignored and are not computed in currently used models. The need of incorporating nonlinear mobilities is demonstrated in an investigation of the time-dependent concentration profiles for all ions in the membrane and also the effects of precipitate buildup in the pore space.
For the entire collection see [Zbl 1531.92006].Oscillations in biological systemshttps://zbmath.org/1544.920682024-11-01T15:51:55.949586Z"Kovács, Sándor"https://zbmath.org/authors/?q=ai:kovacs.sandor-j.1|kovacs.sandor|kovacs.sandor-j.2Summary: As it is well known, many physical, chemical and biological phenomena are modelled by parabolic equations, among these one of the most frequently examined type is the reaction-diffusion equation. One of the fascinating features of these equations is the variety of special types of solutions they exhibit.
For the entire collection see [Zbl 1515.92004].A reaction-diffusion fractional model for cancer virotherapy with immune response and Hattaf time-fractional derivativehttps://zbmath.org/1544.920772024-11-01T15:51:55.949586Z"El Younoussi, Majda"https://zbmath.org/authors/?q=ai:el-younoussi.majda"Hajhouji, Zakaria"https://zbmath.org/authors/?q=ai:hajhouji.zakaria"Hattaf, Khalid"https://zbmath.org/authors/?q=ai:hattaf.khalid"Yousfi, Noura"https://zbmath.org/authors/?q=ai:yousfi.nouraSummary: Recently, fractional partial differential equations (FPDEs) play a crucial role in the modeling of the dynamics of many systems arising from biology and other fields of science and engineering. The aim of this work is to model the interaction between nutrient, normal cells, tumor cells, M1 virus, and cytotoxic T lymphocyte (CTL) cells by using the new generalized Hattaf fractional (GHF) derivative. The mathematical model describing such type of interaction is rigorously formulated. Furthermore, the dynamical behaviors of the model are determined by three threshold parameters, namely, the ability of absorbing nutrient by normal cells, the ability of absorbing nutrient by tumor cells, and the reproduction number for cellular immunity.
For the entire collection see [Zbl 1531.92006].Is maximum tolerated dose (MTD) chemotherapy scheduling optimal for glioblastoma multiforme?https://zbmath.org/1544.920802024-11-01T15:51:55.949586Z"Kao, Chiu-Yen"https://zbmath.org/authors/?q=ai:kao.chiu-yen"Mohammadi, Seyyed Abbas"https://zbmath.org/authors/?q=ai:mohammadi.seyyed-abbas"Yousefnezhad, Mohsen"https://zbmath.org/authors/?q=ai:yousefnezhad.mohsenSummary: In this study, we investigate a control problem involving a reaction-diffusion partial differential equation (PDE). Specifically, the focus is on optimizing the chemotherapy scheduling for brain tumor treatment to minimize the remaining tumor cells post-chemotherapy. Our findings establish that a bang-bang increasing function is the unique solution, affirming the MTD scheduling as the optimal chemotherapy profile. Several numerical experiments on a real brain image with parameters from clinics are conducted for tumors located in the frontal lobe, temporal lobe, or occipital lobe. They confirm our theoretical results and suggest a correlation between the proliferation rate of the tumor and the effectiveness of the optimal treatment.Global bounded classical solutions for a gradient-driven mathematical model of antiangiogenesis in tumor growthhttps://zbmath.org/1544.920882024-11-01T15:51:55.949586Z"Yang, Xiaofei"https://zbmath.org/authors/?q=ai:yang.xiaofei"Lu, Bo"https://zbmath.org/authors/?q=ai:lu.boSummary: In this paper, we consider a gradient-driven mathematical model of antiangiogenesis in tumor growth. In the model, the movement of endothelial cells is governed by diffusion of themselves and chemotaxis in response to gradients of tumor angiogenic factors and angiostatin. The concentration of tumor angiogenic factors and angiostatin is assumed to diffuse and decay. The resulting system consists of three parabolic partial differential equations. In the present paper, we study the global existence and boundedness of classical solutions of the system under homogeneous Neumann boundary conditions.Exploring aeration strategies for enhanced simultaneous nitrification and denitrification in membrane aerated bioreactors: a computational approachhttps://zbmath.org/1544.921032024-11-01T15:51:55.949586Z"Ghasemi, Maryam"https://zbmath.org/authors/?q=ai:ghasemi.maryam"Chang, Sheng"https://zbmath.org/authors/?q=ai:chang.sheng"Sivaloganathan, Sivabal"https://zbmath.org/authors/?q=ai:sivaloganathan.sivabalSummary: In this study we employ computational methods to investigate the influence of aeration strategies on simultaneous nitrification-denitrification processes. Specifically, we explore the impact of periodic and intermittent aeration on denitrification rates, which typically lag behind nitrification rates under identical environmental conditions. A two-dimensional deterministic multi-scale model is employed to elucidate the fundamental processes governing the behavior of membrane aerated biofilm reactors (MABRs). We aim to identify key factors that promote denitrification under varying aeration strategies. Our findings indicate that the concentration of oxygen during the off phase and the duration of the off interval play crucial roles in controlling denitrification. Complete discontinuation of oxygen is not advisable, as it inhibits the formation of anaerobic heterotrophic bacteria, thereby impeding denitrification. Extending the length of the off interval, however, enhances denitrification. Furthermore, we demonstrate that the initial inoculation of the substratum (membrane in this study) influences substrate degradation under periodic aeration, with implications for both nitrification and denitrification. Comparison between continuous and periodic/intermittent aeration scenarios reveals that the latter can extend the operational cycle of MABRs. This extension is attributed to relatively low biofilm growth rates associated with non-continuous aeration strategies. Consequently, our study provides a comprehensive understanding of the intricate interplay between aeration strategies and simultaneous nitrification-denitrification in MABRs. The insights presented herein can contribute significantly to the optimization of MABR performance in wastewater treatment applications.Asymptotic regimes of an integro-difference equation with discontinuous kernelhttps://zbmath.org/1544.921332024-11-01T15:51:55.949586Z"Halim, Omar Abdul"https://zbmath.org/authors/?q=ai:halim.omar-abdul"El Smaily, Mohammad"https://zbmath.org/authors/?q=ai:el-smaily.mohammad-ibrahimThe authors study the integro-difference equation
\[
u_{n+1}(x)=\int_\Omega k(x,y)F(u_n(y)) \ dy,
\]
where \(\delta<k=k(x,y)\leq \Lambda\), with \(\delta>0\) for \(x,y\in \Omega=(-a,a)\), is continuous in \(\bigcup_{i,j}\Omega_i\times \Omega_j\) for
\[
\Omega_i=(a_i, a_{i+1}), \ 0\leq i\leq m, \quad a_0=-a, \quad a_{m+1}=a
\]
The nonlinearity \(0\leq F=F(u)\leq M\), \(u\in {\mathbf R}\) is continuous, strictly increasing on \([0,\infty)\) and vanishes elsewhere. It is assumed that \(F\) is differentiable at \(0\), \(r_0=F'(0)>1\), and
\[
u\in (0,\infty) \ \mapsto \ F(u)/u
\]
is strictly decreasing. The linearized operator
\[
(T_0)u(x)=r_0\int_\Omega k(x,y)u(y) \ dy
\]
is positive in \(L^2(\Omega)\), and its principal eigenvalue is denoted by \(\lambda_0\). Then the sequence \(u_n\) is constructed in \(X\), which denotes the set of \(L^2(\Omega\) functions continuous except finite many points. First, if \(F(0)=0\) and \(\lambda_0\leq 1\), then \(0\) is the only stationary solution and \((u_n)\) converges \(0\) in \(L^2(\Omega)\). Second, if either \(F(0)=0\) and \(\lambda_0>1\) or \(F(0)>0\), there is a unique positive stationary solution \(w\), and if \(u_0\geq 0\) is not identical to \(0\), \(u_n\) converges to this \(w\) in \(L^2(\Omega)\).
Reviewer: Takashi Suzuki (Ōsaka)Nonlinear dynamics and pattern formation in a space-time discrete diffusive intraguild predation modelhttps://zbmath.org/1544.921342024-11-01T15:51:55.949586Z"Han, Renji"https://zbmath.org/authors/?q=ai:han.renji"Salman, Sanaa Moussa"https://zbmath.org/authors/?q=ai:salman.sanaa-moussaSummary: In this paper, the spatiotemporal dynamics and pattern formation of a space-time discrete intraguild predation model with self-diffusion are investigated. The model is obtained by applying a coupled map lattice (CML) method. First, using linear stability analysis, the existence and stability conditions for fixed points are determined. Second, using the center manifold theorem and the bifurcation theory, the occurrence of flip, Neimark-Sacker, and Turing bifurcations are discussed. It is shown that the patterns obtained are results of Turing, flip, and Neimark-Sacker instabilities. Numerical simulations are performed to verify the theoretical analysis and to reveal complex and rich dynamics of the model, such as times series, maximal Lyapunov exponent, bifurcation diagrams, and phase portraits. Interesting patterns like spiral pattern, polygonal pattern, and the combinations of patterns of spiral waves and stripes are formed. The CML model's results help to understand how a spatially extended, discrete intraguild predation model forms complex patterns. Notably, the continuous reaction-diffusion counterpart of the model under study is incapable of experiencing Turing instability.Multistability for a mathematical model of the dynamics of predators and prey in a heterogeneous areahttps://zbmath.org/1544.921352024-11-01T15:51:55.949586Z"Ha, T. D."https://zbmath.org/authors/?q=ai:ha.t-d"Tsybulin, V. G."https://zbmath.org/authors/?q=ai:tsybulin.vyacheslav-gSummary: We consider the system of reaction-diffusion-advection equations describing the evolution of the spatial distributions of two populations of predators and two prey populations. This model allows us to consider directed migration, the Holling functional response of the second kind, and the hyperbolic prey growth function. We obtain conditions on the parameters under which cosymmetries exist. As a result, multistability is realized, i.e., the one- and two-parameter families of stationary solutions appear. For a homogeneous environment, we analytically derive explicit formulas for equilibria. With a heterogeneous habitat, we computed distributions of species using the method of lines and the scheme of staggered grids. We present the results of violation of cosymmetry and transformation of the family in the case of invasion of a predator.Hopf bifurcation analysis in a diffusive predator-prey model with repulsive predator-taxis and digestion delayhttps://zbmath.org/1544.921442024-11-01T15:51:55.949586Z"Liu, Moqing"https://zbmath.org/authors/?q=ai:liu.moqing"Jiang, Jiao"https://zbmath.org/authors/?q=ai:jiang.jiaoSummary: In this paper, we investigate the joint effect of repulsive predator-taxis and digestion delay in a diffusive predator-prey model. Our theoretical findings reveal that, when both species exhibit logistic growth, the predator-taxis does not lead to Turing bifurcation. However, the introduction of digestion delay gives rise to the occurrence of spatially homogeneous and inhomogeneous Hopf bifurcations which result in spatially homogeneous and inhomogeneous periodic solutions. In particular, the inhomogeneous Hopf bifurcations are often accompanied by rich and diverse pattern modes. Under certain conditions, the spatially homogeneous Hopf bifurcation disappears, while the spatially inhomogeneous Hopf bifurcations persist. Furthermore, our analysis reveals the existence of double Hopf bifurcation between homogeneous and inhomogeneous states in the bifurcation diagram. Finally, the numerical simulations confirm the theoretical findings and demonstrate the emergence of various spatial patterns with different values of predator-taxis coefficients and delays.On the structural sensitivity of some diffusion-reaction models of population dynamicshttps://zbmath.org/1544.921472024-11-01T15:51:55.949586Z"Manna, Kalyan"https://zbmath.org/authors/?q=ai:manna.kalyan"Banerjee, Malay"https://zbmath.org/authors/?q=ai:banerjee.malay"Petrovskii, Sergei"https://zbmath.org/authors/?q=ai:petrovskii.sergei-vSummary: In mathematical ecology, it is often assumed that properties of a mathematical model are robust to specific parameterization of functional responses, in particular preserving the bifurcation structure of the system, as long as different functions are qualitatively similar. This intuitive assumption has been challenged recently
[\textit{G. F. Fussmann} and \textit{B. Blasius}, Biol. Lett. 1, No. 1, 9--12 (2005; \url{doi:10.1098/rsbl.2004.0246})]. Having considered the prey-predator system as a paradigm of nonlinear population dynamics, it has been shown that in fact both the bifurcation structure and the structure of the phase space can be rather different even when the component functions are apparently close to each other. However, these observations have so far been largely limited to nonspatial systems described by ODEs. In this paper, our main interest is to investigate whether such structural sensitivity occurs in spatially explicit models of population dynamics, in particular those that are described by PDEs. We consider a prey-predator model described by a system of two nonlinear reaction-diffusion-advection equations where the predation term is parameterized by three different yet numerically close functions. Using some analytical tools along with numerical simulations, we show that the properties of spatiotemporal dynamics are rather different between the three cases, so that patterns observed for one parameterization may not occur for the other two ones.Spreading dynamics for an epidemic model of West-Nile virus with shifting environmenthttps://zbmath.org/1544.921632024-11-01T15:51:55.949586Z"Ahn, Inkyung"https://zbmath.org/authors/?q=ai:ahn.inkyung"Choi, Wonhyung"https://zbmath.org/authors/?q=ai:choi.wonhyung"Guo, Jong-Shenq"https://zbmath.org/authors/?q=ai:guo.jong-shenqSummary: We study the disease-spreading dynamics of the West Nile virus (WNv) epidemic model under shifting climatic conditions. A WNv epidemic model is developed incorporating a shifting net growth term to depict the evolving mosquito habitat. First, we comprehensively characterize the spreading dynamics of mosquitoes for any given climate change speed compared with the intrinsic spreading speed of mosquitoes. Utilizing the results from mosquito dynamics, we determine the spreading dynamics of infected birds and mosquitoes, taking into account relationships among the shifting speed and the spreading speeds of mosquito and WNv. Ultimately, we find that infected mosquitoes and birds propagate, and their population densities converge to a stable positive endemic state. This paper provides crucial insights into the impact of climate change on the spread of vector-borne diseases such as WNv.A risk-induced dispersal strategy of the infected population for a disease-free state in the SIS epidemic modelhttps://zbmath.org/1544.921762024-11-01T15:51:55.949586Z"Choi, Wonhyung"https://zbmath.org/authors/?q=ai:choi.wonhyung"Ahn, Inkyung"https://zbmath.org/authors/?q=ai:ahn.inkyungSummary: This article proposes a dispersal strategy for infected individuals in a spatial susceptible-infected-susceptible (SIS) epidemic model. The presence of spatial heterogeneity and the movement of individuals play crucial roles in determining the persistence and eradication of infectious diseases. To capture these dynamics, we introduce a moving strategy called risk-induced dispersal (RID) for infected individuals in a continuous-time patch model of the SIS epidemic. First, we establish a continuous-time \(n\)-patch model and verify that the RID strategy is an effective approach for attaining a disease-free state. This is substantiated through simulations conducted on 7-patch models and analytical results derived from 2-patch models. Second, we extend our analysis by adapting the patch model into a diffusive epidemic model. This extension allows us to explore further the impact of the RID movement strategy on disease transmission and control. We validate our results through simulations, which provide the effects of the RID dispersal strategy.Global threshold analysis of an age-space structured disease model with relapsehttps://zbmath.org/1544.921942024-11-01T15:51:55.949586Z"Lyu, Guoyang"https://zbmath.org/authors/?q=ai:lyu.guoyang"Guo, Yutong"https://zbmath.org/authors/?q=ai:guo.yutong"Wang, Jinliang"https://zbmath.org/authors/?q=ai:wang.jinliangSummary: In this paper, an age-space structured disease model with age-dependent relapse rate is investigated. We first prove the well-posedness of the model including the existence and uniqueness of the solution, positivity, and boundedness. By performing the Laplace transformation to renewal equation, we derive the next generation operator, whose spectral radius is defined as the basic reproduction number. By checking the distribution of the roots of the characteristic equation, exploring the strong persistence property of the solution and designing the Lyapunov functionals, we establish the local and global dynamics of the model.\(H_\infty\) control design for non-linear distributed parameter systems with mobile actuators and sensorshttps://zbmath.org/1544.931552024-11-01T15:51:55.949586Z"Zhang, Xiao-Wei"https://zbmath.org/authors/?q=ai:zhang.xiaowei.2"Wu, Huai-Ning"https://zbmath.org/authors/?q=ai:wu.huainingSummary: This study deals with the \(H_\infty\) control design problem for a class of non-linear distributed parameter systems described by parabolic partial differential equations (PDEs) via mobile collocated actuators and sensors. Initially, the spatial domain is decomposed into multiple subdomains according to the number of actuator/sensor pairs and the projection modification algorithm is employed to guarantee each actuator/sensor pair is only capable of moving within the respective subdomain. Subsequently, the well-posedness of the closed-loop PDE system is analysed by means of the operator semigroup theory. Then, a control-plus-guidance design method for the non-linear PDE system is developed in the form of bilinear matrix inequalities, such that the resulting closed-loop system is exponentially stable while satisfying a prescribed \(H_\infty\) performance of disturbance attenuation, and the mobile actuator/sensor guidance can enhance the transient performance of the closed-loop system. Finally, a numerical example and a practical application example are respectively given to show the effectiveness of the proposed design method.
{\copyright} 2021 The Authors. IET Control Theory \& Applications published by John Wiley \& Sons, Ltd. on behalf of The Institution of Engineering and TechnologyOutput-feedback adaptive boundary control of coupled equidiffusive parabolic PDE systemshttps://zbmath.org/1544.933052024-11-01T15:51:55.949586Z"Guo, Runsheng"https://zbmath.org/authors/?q=ai:guo.runsheng"Qiu, Jianbin"https://zbmath.org/authors/?q=ai:qiu.jianbin"Wang, Tong"https://zbmath.org/authors/?q=ai:wang.tong.2"Sun, Kangkang"https://zbmath.org/authors/?q=ai:sun.kangkangSummary: This paper investigates the output-feedback adaptive boundary control issue of the \(n\)-dimensional coupled parabolic PDE system (PPDES) with the same diffusivity parameters and the unknown spatially varying parameters. To further simplify the target system, the coupled PPDES is converted into its observer canonical form via a backstepping transformation. To estimate system states, an observer is constructed, which is expressed as a linear combination of three filters. To deal with unknown parameters, the swapping identifiers are given. Based on the observer and the swapping identifiers, a backstepping controller is designed to stabilize the target system. It is proved that all the system states will converge to zero with the proposed controller. Finally, a numerical example is given to illustrate the effectiveness of the proposed controller.Energy-shaping and entropy-assignment boundary control of the heat equationhttps://zbmath.org/1544.933082024-11-01T15:51:55.949586Z"Mora, Luis A."https://zbmath.org/authors/?q=ai:mora.luis-a"Le Gorrec, Yann"https://zbmath.org/authors/?q=ai:le-gorrec.yann"Ramirez, Hector"https://zbmath.org/authors/?q=ai:ramirez.hectorSummary: This paper shows a finite-dimensional controller design for the boundary control of the heat equation on a 1D spatial domain. The controller exponentially stabilizes the plant at the desired equilibrium profile. The controller is defined using irreversible port-Hamiltonian systems formulation, and it is motivated by passivity-based control techniques developed for port-Hamiltonian systems defined on 1D spatial domains. The boundary controller is designed to have an exponentially stabilizing energy-shaping and entropy-assignment effect. It works with an actuation at one boundary and a reflective boundary condition at the other. The controller can handle situations where measurements are available at only one or both boundaries. The paper characterizes the existence of structural invariant functions to shape the closed-loop energy and assign the required closed-loop entropy. The design approach is illustrated through numerical simulations.A solution of the complex fuzzy heat equation in terms of complex Dirichlet conditions using a modified Crank-Nicolson methodhttps://zbmath.org/1544.934452024-11-01T15:51:55.949586Z"Zureigat, Hamzeh"https://zbmath.org/authors/?q=ai:zureigat.hamzeh"Tashtoush, Mohammad A."https://zbmath.org/authors/?q=ai:tashtoush.mohammad-a"Jassar, Ali F. Al"https://zbmath.org/authors/?q=ai:jassar.ali-f-al"Az-Zo'bi, Emad A."https://zbmath.org/authors/?q=ai:az-zobi.emad-a"Alomari, Mohammad W."https://zbmath.org/authors/?q=ai:alomari.mohammad-wajeeh(no abstract)Event-triggered boundary feedback synchronisation control of nonlinear coupling reaction-diffusion neural networkshttps://zbmath.org/1544.934792024-11-01T15:51:55.949586Z"Fan, Xueru"https://zbmath.org/authors/?q=ai:fan.xueru"Kou, Chunhai"https://zbmath.org/authors/?q=ai:kou.chunhaiSummary: This paper aims to deal with Robin boundary synchronisation control for a kind of nonlinear coupling reaction-diffusion neural networks (RDNNs) with non-identical nodes. Some novel sufficient conditions involving the parameters of event-triggered mechanism are proposed to achieve synchronisation of this RDNNs. By utilising Lyapunov's method and Wiringer inequality, the error closed-loop systems with the designed event-triggered controller are proved to be well-posed and the RDNNs to be exponentially synchronised with target system. Moreover, Zeno behaviour of event-triggered control is avoided. Finally, the effectiveness of our theoretical results is verified by one numerical example.A note on boundary feedback stabilization for degenerate parabolic equations in multi-dimensional domainshttps://zbmath.org/1544.936212024-11-01T15:51:55.949586Z"Munteanu, Ionuţ"https://zbmath.org/authors/?q=ai:munteanu.ionutSummary: In this paper, we are concerned with the problem of stabilization of a degenerate parabolic equation with a Dirichlet control, evolving in bounded domain \(\mathcal{O}\subset\mathbb{R}^d\), \(d\geq 2\). We apply the proportional control design technique based on the spectrum of the linear operator which governs the evolution equation. The stabilizing feedback control, we design here, is linear, of finite-dimensional structure, easily manageable from the computational point of view.Observer-based stabilization of an unstable ODE-heat cascade system with multiple-point connectionhttps://zbmath.org/1544.936232024-11-01T15:51:55.949586Z"Pan, Lina"https://zbmath.org/authors/?q=ai:pan.lina"Wang, Jun-Min"https://zbmath.org/authors/?q=ai:wang.junminSummary: In this paper, we consider the output feedback stabilization for a cascade of ODE-heat system through multiple-point connection, where the point outputs of the heat are flowing into the ODE, and the control is proposed on the ODE and the boundary of the heat simultaneously. An observer system is presented to estimate the ODE and the heat variables, and the error system is showed to be exponentially convergent. Based on the estimated variables, an output feedback control is designed using two invertible transformations to exponentially stabilize the entire system. Some numerical simulation results are presented to validate the theoretical conclusions.Adaptive synchronization of quaternion-valued neural networks with reaction-diffusion and fractional orderhttps://zbmath.org/1544.936602024-11-01T15:51:55.949586Z"Zhang, Weiwei"https://zbmath.org/authors/?q=ai:zhang.weiwei.2"Zhao, Hongyong"https://zbmath.org/authors/?q=ai:zhao.hongyong"Sha, Chunlin"https://zbmath.org/authors/?q=ai:sha.chunlinSummary: This paper is dedicated to the study of adaptive finite-time synchronization (FTS) for generalized delayed fractional-order reaction-diffusion quaternion-valued neural networks (GDFORDQVNN). Utilizing the suitable Lyapunov functional, Green's formula, and inequalities skills, testable algebraic criteria for ensuring the FTS of GDFORDQVNN are established on the basis of two adaptive controllers. Moreover, the numerical examples validate that the obtained results are feasible. Furthermore, they are also verified in image encryption as the application.Global exponential synchronization of BAM memristive neural networks with mixed delays and reaction-diffusion termshttps://zbmath.org/1544.936612024-11-01T15:51:55.949586Z"Chen, Huihui"https://zbmath.org/authors/?q=ai:chen.huihui"Jiang, Minghui"https://zbmath.org/authors/?q=ai:jiang.minghui.1"Hu, Junhao"https://zbmath.org/authors/?q=ai:hu.junhaoSummary: Based on \(m\)-norm, this paper investigates global exponential synchronization (GES) for BAM memristive neural networks (BAMMNNs) with mixed delays and reaction-diffusion (RD) terms. Different from the existing literatures, this paper discusses the GES of the NNs based on a new integral inequality with infinite distributed delay. This method is based on inequality technique and comparison principle, which makes the form of Lyapunov function and controller more simple. Next, by introducing two different control strategies and the concept of driven response, two sufficient conditions are got to ensure GES of the proposed system. It is noteworthy that the results obtained by algebraic inequality are extension of the previous conclusions. Finally, two instances verify the correctness of the conclusions.The necessary and sufficient conditions of exponential stability for heat and wave equations with memoryhttps://zbmath.org/1544.936662024-11-01T15:51:55.949586Z"Li, Lingfei"https://zbmath.org/authors/?q=ai:li.lingfei"Zhang, Xiaoyi"https://zbmath.org/authors/?q=ai:zhang.xiaoyi"Zhou, Xiuxiang"https://zbmath.org/authors/?q=ai:zhou.xiuxiangSummary: This paper is addressed to a study of the stability of heat and wave equations with memory The necessary and sufficient conditions of the exponential stability are investigated by the theory of Laplace transform. The results show that the stability depends on the decay rate and the coefficient of the kernel functions of the memory. Besides, the feedback stabilization of the heat equation is obtained by constructing finite dimensional controller according to unstable eigenvalues. This stabilizing procedure is easy to operate and can be applicable for other parabolic equations with memory.Spatial domain decomposition approach to dynamic compensator design for linear space-varying parabolic MIMO PDEshttps://zbmath.org/1544.936712024-11-01T15:51:55.949586Z"Wang, Jun-Wei"https://zbmath.org/authors/?q=ai:wang.junweiSummary: This study addresses the problem of dynamic compensator design for exponential stabilisation of linear space-varying parabolic multiple-input-multiple-output (MIMO) partial differential equations (PDEs) subject to periodic boundary conditions. With the aid of the observer-based feedback control technique, an observer-based dynamic feedback compensator, whose implementation requires only a few actuators and sensors active over partial areas of the spatial domain, is constructed such that the resulting closed-loop coupled PDEs is exponentially stable. The spatial domain is divided into multiple subdomains according to the minimum of the actuators' number and the sensors' one. By Lyapunov direct method and two general variants of Poincaré-Wirtinger inequality at each subdomain, sufficient conditions for the existence of such feedback compensator are developed and presented in terms of algebraic linear matrix inequalities (LMIs) in space. Based on the extreme value theorem, LMI-based sufficient and necessary conditions are presented for the feasibility of algebraic LMIs in space. Finally, numerical simulation results are presented to support the proposed design method.
{\copyright} 2021 The Authors. IET Control Theory \& Applications published by John Wiley \& Sons, Ltd. on behalf of The Institution of Engineering and TechnologyExponential stability of large-scale stochastic reaction-diffusion equationshttps://zbmath.org/1544.936722024-11-01T15:51:55.949586Z"Wang, Yuan"https://zbmath.org/authors/?q=ai:wang.yuan.4|wang.yuan|wang.yuan.2|wang.yuan.3|wang.yuan.1"Ren, Yong"https://zbmath.org/authors/?q=ai:ren.yong.2|ren.yong.4|ren.yong|ren.yong.1Summary: In this paper, we consider a class of large-scale stochastic reaction-diffusion systems. To prove the exponential stability of the system, we introduce the corresponding isolated subsystems. We show that the exponential stability of the isolated systems implies the exponential stability of the large-scale stochastic reaction-diffusion system under some conditions. Furthermore, we discuss a special case where the large-scale stochastic reaction-diffusion system is described in a hierarchical form. In this case, we prove that the original system is exponentially stable if and only if the corresponding subsystems are exponentially stable.Finite-time local piecewise control for parabolic PDEs with ODE output feedbackhttps://zbmath.org/1544.937002024-11-01T15:51:55.949586Z"Li, Manna"https://zbmath.org/authors/?q=ai:li.manna"Mao, Weijie"https://zbmath.org/authors/?q=ai:mao.weijieSummary: This paper is devoted to the finite-time local piecewise control for parabolic partial differential equations (PDEs) by using dynamic output feedback control strategy, where the controller is designed as an ordinary differential equation (ODE). This makes the closed-loop system PDE-ODE coupled, which is employed to accurately describe the dynamics of the PDE system. According to the constructed PDE-ODE coupled model, a local piecewise dynamic feedback control law is first proposed. Sufficient conditions on finite-time stabilisation of the parabolic PDE-ODE coupled system by the suggested feedback controller are then developed in the sense of both complete spatial measurement and incomplete spatial measurement of the observed output of the PDE system, respectively. Finally, the issues regarding the finite-time stabilisation of the closed-loop system is converted into the feasibility of matrix inequalities, and some simulation studies are provided to verify the effectiveness of the proposed results.
{\copyright} 2021 The Authors. \textit{IET Control Theory \& Applications} published by John Wiley \& Sons Ltd on behalf of The Institution of Engineering and Technology