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 ${\partial }_{t}u-\Delta u+u^p=0$ in exterior domains $\Omega$ in ${\mathbb{R}}^N$ ($N\ge 2$). 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 $\varphi (x)dx$, where $\varphi$ is a positive harmonic function satisfying the Dirichlet boundary condition. This paper clarifies that the mass of solutions with respect to the measure $\varphi (x)dx$ vanishes as $t\to \infty$ if and only if $1<p\le \min \{2,1+\frac{2}{N}\}$. 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 $N\ge 3$.

19 pages