Exponentially stable positive solutions to a forced semilinear parabolic equation

Article type: Research Article

Authors: Haraux, A.

Affiliations: Analyse Numérique, T. 55–65, 5ème étage, Université P. et M. Curie, 4 place Jussieu, 75252 Paris Cedex 05, France

Abstract: Let Ω be a bounded connected open subset of RN 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 ut−Δu+f(u)=h(t,x) in R×Ω; u=0 on R×∂Ω has one and only one nonnegative global solution ω∈CB(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 ut−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.

DOI: 10.3233/ASY-1993-7102

Journal: Asymptotic Analysis, vol. 7, no. 1, pp. 3-13, 1993