Exponentially stable positive solutions to a forced semilinear parabolic equation

Abstract

Let Ω be a bounded connected open subset of R^N with a Lipschitz continuous boundary and f:R→R a locally Lipschitz continuous function such that f(0)=0, f is strictly convex on [0,+∞) and f′_d(0)<−λ₁(Ω) where λ₁(Ω) is the smallest eigenvalue of (−Δ) in H¹₀(Ω), and f(s)→+∞ as s→+∞. For any h∈L^∞(R×Ω) with h(t,x)≥0, a.e. on R×Ω, the semilinear parabolic problem u_t−Δu+f(u)=h(t,x) in R×Ω; u=0 on R×∂Ω has one and only one nonnegative global solution ω∈C_B(R;H¹₀(Ω)∩L^∞(Ω)) such that w(t,·) does not tend to 0 as t→−∞. In addition, if h is T-periodic, then ω is T-periodic. If h:R→L^∞(Ω) is continuous and almost-periodic, then ω:R→H₀¹(Ω)∩L^∞(Ω) is almost-periodic. Finally assuming either that u₀≥0 is not identically 0 or that h(t,x)>0 on a subset of positive measure of R⁺×Ω, the unique solution u of u_t−Δu+f(u)=h(t,x) in R⁺×Ω; u=0 on R⁺×∂Ω such that u(0)=u₀ satisfies ‖u(t,·)−ω(t,·)‖_∞≤C(u₀)exp (−γt), for all t≥0, with γ>0 independent of u₀ and h.