Stability analysis of positive solutions to classes of reaction-diffusion systems. (English) Zbl 1174.35320

Summary: We analyze the stability of positive solutions to systems of the form \[ \begin{cases} -\Delta u_i=f_i(u_1,u_2,\dots ,u_m) & \text{ in } \Omega \\ u_i=0 & \text{ on } \partial \Omega , \end{cases} \] where \(\Omega \) is a bounded region in \(\mathbb R^n\) \((n\geq 1)\) with smooth boundary \(\partial \Omega \), and \(f_i\:[0,\infty )^m \to \mathbb R\) 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 (\(\frac {\partial f_i}{\partial u_j}\geq 0\), \(i\neq j\), \(\sum _{j=i,j\neq i}^m(\frac {\partial f_i}{\partial u_j})^2>0\)). 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.


35B35 Stability in context of PDEs