## Differential and Integral Equations

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

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

Alfonso Castro, Maya Chhetri, and R. Shivaji

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060438

**Mathematical Reviews number (MathSciNet)**

MR2037983

**Zentralblatt MATH identifier**

1174.35320

**Subjects**

Primary: 35K57: Reaction-diffusion equations

Secondary: 35B35: Stability 35K50

#### Citation

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