Article ID Journal Published Year Pages File Type
840176 Nonlinear Analysis: Theory, Methods & Applications 2013 15 Pages PDF
Abstract

Let ΩΩ be a smooth bounded domain in RnRn. Considering the following Robin problem for a semilinear parabolic equation 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)f(x) satisfying (F)(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).

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