On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains

2025/03/26 by Fino, Ahmad, Sobajima, Motohiro
#35B40 #35K20 #35K58 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.20480

Abstract

The paper concerns with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation ∂_tu-Δu+u^p=0 in exterior domains Ω in ℝ^N (N≥ 2). The problem for the one-dimensional case is formulated with Ω=(0,∞) which is one of the representative of the connected components in ℝ. One can see that the C₀-semigroup for the corresponding linear problem possesses an invariant measure ϕ(x) dx, where ϕ is a positive harmonic function satisfying the Dirichlet boundary condition. This paper clarifies that the mass of solutions with respect to the measure ϕ(x) dx vanishes as t→ ∞ if and only if 1min\2,1+(2)/(N)\, we prove that all solutions are asymptotically free. The asymptotic profile is actually given by a modification with Gaussian when N≥ 3.

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