Existence of positive stationary solutions and threshold results for a reaction-diffusion system. (English) Zbl 0858.35059

Gu, Yonggeng; Wang, Mingxin

J. Differ. Equations 130, No. 2, 277-291 (1996).

Summary: The system \[ u_{1t}-\Delta u_1=u_1u_2-bu_1, \quad u_{2t}-\Delta u_2=au_1\quad \text{in }\Omega\times(0,T), \] where \(\Omega\subset \mathbb{R}^n\) is a smooth bounded domain, with homogeneous Dirichlet boundary conditions \(u_1=u_2=0\) on \(\partial\Omega\times (0,T)\) and initial conditions \(u_1(x,0)\), \(u_2(x,0)\), is studied. First, it is proved that there is at least one positive stationary solution if \(2\leq n<6\). Second, it is proved that every positive stationary solution is a threshold when \(\Omega\) is a ball.

MSC:

35K57	Reaction-diffusion equations
35J65	Nonlinear boundary value problems for linear elliptic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

initial-boundary-value problem