Existence and stability of bounded solutions for a system of parabolic equations
type: Item
instance of: scholarly article
title: Existence and stability of bounded solutions for a system of parabolic equations
zbMATH Open document ID: 1019.35037
DOI: 10.1016/S0022-247X(03)00025-8
author: Hugo Leiva
author: Ibrain Sequera
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: In this paper the authors discuss the existence and the stability of bounded solutions of the following nonlinear system of parabolic equations with homogeneous Dirichlet boundary conditions: \[ \begin{cases} u_t = D\Delta u + f(t,u), & \text{in }\Omega \quad t \geq 0, \;u \in \mathbb{R}^n, \\ u=0 & \text{on }\partial \Omega. \end{cases} \] where \(f \in C^1\), \(D=\text{diag}(d_1,d_2,\dots,d_n)\) is a diagonal matrix with \(d_j>0\), \(i=1,2,\ldots, n\), and \( \Omega\) is a sufficiently regular bounded domain in \(\mathbb{R}^N (N = 1, 2, 3)\). The authors prove the following results: If \(f\) is globally Lipschitz with constant \(L\), \[ {3 \over 4} < \alpha < 1 \quad \text{and} \quad {(\lambda_1d)^{1-\alpha} \over \Gamma(1-\alpha)} > 6ML, \] then the system has a bounded solution which is stable, where \(2d = \min \{d_i: i=1,2,\ldots, n\}\), \((\lambda_jd_it)^\alpha e^{-\lambda_j(d_i/2)t}<M\), \(\lambda_i\) are the eigenvalues of \(-\Delta\), and \(\Gamma(\cdot)\) is the well-known gamma function. Also, the authors prove that for a large class of functions \(f\) this bounded solution is almost periodic.
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B15
zbMATH DE Number: 1915765
zbMATH Keywords: homogeneous Dirichlet boundary conditions
reviewed by: Zhang Wanguo
MaRDI profile type: Publication
cites work: Q4213963
cites work: Q4512789
cites work: Q5580482
cites work: Geometric theory of semilinear parabolic equations
cites work: Stability of a periodic solution for a system of parabolic equations
cites work: Existence of bounded solutions to a second-order system with dissipation
cites work: Coexistence in a simple food chain with diffusion
full work available at URL: s0022-247x(03)00025-8
OpenAlex ID: W2013117525
description: scientific article; zbMATH DE number 1915765