## 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.

### MSC:

 35B35 Stability in context of PDEs