Positive stationary solutions and threshold results for the non-homogeneous semilinear parabolic equation with Robin boundary conditions

Abstract

Let Ω be a smooth bounded domain in Rn. Considering the following Robin problem for a semilinear parabolic equation (0.1){ut−Δu=up+f(x),(x,t)∈Ω×(0,T),∂u∂ν+βu=0,(x,t)∈∂Ω×[0,T),u(x,0)=u0(x),x∈Ω, we show that for any function f(x) satisfying (F) which will be given in the introduction, there exists a positive number βf⋆ such that problem (0.1) has no stationary solution if β∈(0,βf⋆), and has at least two stationary solutions when β>βf⋆. Moreover, among all stationary solutions of problem (0.1) there is a minimal one. We prove further that the minimal stationary solution of problem (0.1) is stable, whereas, any other stationary solutions of problem (0.1) are initial datum thresholds for the existence and non-existence of a global solution to problem (0.1).