## Differential and Integral Equations

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

#### 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

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