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

Abstract

The paper concerns with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation $\partial_tu-\Delta u+u^p=0$ in exterior domains $\Omega$ in $\mathbb{R}^N$ ($N\geq 2$). The problem for the one-dimensional case is formulated with $\Omega=(0,\infty)$ which is one of the representative of the connected components in $\mathbb{R}$. One can see that the $C_0$-semigroup for the corresponding linear problem possesses an invariant measure $\phi(x)\,dx$, where $\phi$ is a positive harmonic function satisfying the Dirichlet boundary condition. This paper clarifies that the mass of solutions with respect to the measure $\phi(x)\,dx$ vanishes as $t\to \infty$ if and only if $1<p\leq \min\{2,1+\frac{2}{N}\}$. In the other case $p>\min\{2,1+\frac{2}{N}\}$, we prove that all solutions are asymptotically free. The asymptotic profile is actually given by a modification with Gaussian when $N\geq 3$.