Asymptotic behaviour for an equation of superslow diffusion in a bounded domain

Abstract

We study the large-time behaviour of the solution to the mixed problem: u_t=Δ(e^−1/u) in Q=Ω×(0,∞), with u(x, t) = 0 for x∈∂Ω, t≥0 and u(x,0)∈L^∞(Ω), u(x,0)≥0. Ω is a bounded domain with smooth boundary ∂Ω. We show that there exists a function F(x) > 0 in Ω such that as t→∞

t(log t)²e^−1/u→F(x)

uniformly in x∈Ω. The function F is uniquely determined as the solution of the problem: ΔF=−1 in Ω, F=0 on ∂Ω.