Open Access
Translator Disclaimer
2004 Stability analysis of positive solutions to classes of reaction-diffusion systems
Alfonso Castro, Maya Chhetri, R. Shivaji
Differential Integral Equations 17(3-4): 391-406 (2004).

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.

Citation

Download Citation

Alfonso Castro. Maya Chhetri. R. Shivaji. "Stability analysis of positive solutions to classes of reaction-diffusion systems." Differential Integral Equations 17 (3-4) 391 - 406, 2004.

Information

Published: 2004
First available in Project Euclid: 21 December 2012

zbMATH: 1174.35320
MathSciNet: MR2037983

Subjects:
Primary: 35K57
Secondary: 35B35 , 35K50

Rights: Copyright © 2004 Khayyam Publishing, Inc.

JOURNAL ARTICLE
16 PAGES


SHARE
Vol.17 • No. 3-4 • 2004
Back to Top