Differential and Integral Equations

Stability analysis of positive solutions to classes of reaction-diffusion systems

Alfonso Castro, Maya Chhetri, and R. Shivaji

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We analyze the stability of positive solutions to systems of the form \[ \begin{cases} - \Delta u_{i} = f_{i}(u_{1},u_{2},\dots,u_{m}) & \quad \mbox{ in }\ \Omega \\ u_{i} = 0 & \quad \mbox{ on }\ \partial \Omega \end{cases} \] where $ \Omega $ is a bounded region in $ {\mathbb R}^{n}\, (n \geq 1) $ with smooth boundary $ \partial \Omega $, and $ f_{i} : [0,\infty)^m \rightarrow {\mathbb R} $ are $ C^{1} $ functions for $ i=1,\dots, m $. In particular, we establish conditions for stability/instability when the system is cooperative and strictly coupled ($ \frac{\partial f_{i}}{\partial u_{j}} \geq 0, \ i \neq j,\ \sum_{j=1,j \neq i}^m(\frac{\partial f_i}{\partial u_j})^2 > 0$). When $ m=2, $ we extend this analysis for strictly coupled competitive systems ($ \frac{\partial f_{i}}{\partial u_{j}} < 0, \ i \neq j $). We apply our results to various examples, each one of different characteristics, and further analyze systems with unequal diffusion coefficients.

Article information

Differential Integral Equations Volume 17, Number 3-4 (2004), 391-406.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B35: Stability 35K50


Castro, Alfonso; Chhetri, Maya; Shivaji, R. Stability analysis of positive solutions to classes of reaction-diffusion systems. Differential Integral Equations 17 (2004), no. 3-4, 391--406. https://projecteuclid.org/euclid.die/1356060438.

Export citation