Toka Diagana
The primary goal of this paper is to characterize solutions to coupled reaction-diffusion systems. Indeed, we use operators theory to show that under suitable assumptions, then the solutions to the reaction-diffusion equations exist. As applications, we consider a mathematical model arising in Biology and in Chemistry.
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We prove the existence of a compact L^2-H^1 attractor for a reaction-diffusion equation in R^n. This improves a previous result of B. Wang concerning the existence of a compact L^2-L^2 attractor for the same equation.
We study in this paper the periodic homogenization problem related to a strongly nonlinear reaction-diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that, the homogenized equation is exactly of a convection-diffusion type. The study relies on a suitable version of the well-known two-scale convergence method.
In this work we use functional methods to prove the boundedness and global existence of solutions for a class of strongly coupled parabolic systems. We apply the results to deduce the global existence of solutions for a classic Shigesada-Kawasaki-Teramoto (SKT) type model, for an extended range of the self-diffusion and cross-diffusion coefficients than those available in the current literature. We provide numerical simulations in 2D, via a spectral Galerkin method to verify our global existence results, as well as to visualize the dynamics of the system.
Some remarks on spatial uniformity of solutions of reaction-diffusion PDE's and a related synchronization problem for ODE's
本文研究了反应扩散PDE解的空间均匀性条件，基于雅可比矩阵和拉普拉斯算子的第一狄利克雷特征值，同时推导了ODE网络同步问题的类比结果。
Remarks on contractions of reaction-diffusion PDE's on weighted L^2 norms
本文扩展了先前关于反应扩散PDE在加权L^2范数下收缩性的结果，允许Q为正对称矩阵且满足Q^2D+DQ^2>0。
Spatial segregation occurs in population dynamics when $k$ species interact in a highly competitive way. As a model for the study of this phenomenon, we consider the competition-diffusion system of $k$ differential equations \[ -Δu_i(x)=-μu_i (x)\displaystyle{\sum_{j\neq i}} u_j (x) \quad i=1,...,k \] in a domain $D$ with appropriate boundary conditions. Any $u_i$ represents a population density and the parameter $μ$ determines the interaction strength between the populations. The purpose of this paper is to study the geometry of the limiting configuration as $μ\longrightarrow+\infty$ on a pla…
The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, growth of bacterial colonies. Since a scalar equation generates usually stable waves, the simplest mathematical description relies on two by two reaction-diffusion systems. Our interest is the extension of the Fisher/KPP equation to a two species reaction which represents reactant concentration and temperature when used for flame propagation, bacterial population and nutrient concentration when used in biology. We study in which …
We consider linear age-structured population equations with diffusion. Supposing maximal regularity of the diffusion operator, we characterize the generator and its spectral properties of the associated strongly continuous semigroup. In particular, we provide conditions for stability of the zero solution and for asynchronous exponential growth.
Some remarks about the Morse index for convex Hamiltonians systems
该研究聚焦凸哈密顿系统的变号径向解，针对强耦合系统刻画径向解形态并估计其线性化Morse指标，为相关哈密顿系统解的稳定性分析提供参考。
This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in \cite{KZI} and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed by periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or non-existence of inertial manifolds for…
The existence of positive solutions to the system of ordinary differential equations related to the Belousov-Zhabotinsky reaction is established. The key idea is to use successive approximation of solutions, ensuring its positivity. To obtain the positivity and invariant region for numerical solutions, the system is discretized as difference equations of explicit form, employing operator splitting methods with linear stability conditions.
In this paper we present a derivation and multiscale analysis of a mathematical model for plant cell wall biomechanics that takes into account both the microscopic structure of a cell wall coming from the cellulose microfibrils and the chemical reactions between the cell wall's constituents. Particular attention is paid to the role of pectin and the impact of calcium-pectin cross-linking chemistry on the mechanical properties of the cell wall. We prove the existence and uniqueness of the strongly coupled microscopic problem consisting of the equations of linear elasticity and a system of react…
The paper is concerned with a singular limit for the bistable traveling wave problem in a very large class of two-species fully nonlinear parabolic systems with competitive reaction terms. Assuming existence of traveling waves and enough compactness, we derive and characterize the limiting problem. The assumptions and results are discussed in detail. The free boundary problem obtained at the limit is specified for important applications.
A condition is identified that implies that solutions to the stochastic reaction-diffusion equation $\frac{\partial u}{\partial t} = \mathcal{A} u + f(u) + σ(u) \dot{W}$ on a bounded spatial domain never explode. We consider the case where $σ$ grows polynomially and $f$ is polynomially dissipative, meaning that $f$ strongly forces solutions toward finite values. This result demonstrates the role that the deterministic forcing term $f$ plays in preventing explosion.
A new variational formulation based on discontinuous Galerkin technique for a reaction-diffusion problem
提出一种基于间断伽辽金技术的反应扩散问题新变分公式，证明其适定性，并指出该公式有助于设计在空间变量上强稳定、时间变量上绝对稳定且最优收敛的混合数值方法。
Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains or in $\mathbb{R}^2$. We present a purely Lagrangian proof that the solution map is strongly continuous in $L^p$ for all $p\in [1, \infty)$ and is weakly-$*$ continuous in $L^\infty$.
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