On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains
Ahmad Fino, Motohiro Sobajima
TL;DR
This work analyzes the long-time behavior of nonnegative solutions to a semilinear heat equation in exterior domains using the invariant measure $d\mu=\phi(x)\,dx$, where $\phi$ is a positive harmonic function with $\phi=0$ on the boundary. A sharp threshold $p_c=\min\{2,1+\tfrac{2}{N}\}$ separates vanishing and non-vanishing mass with respect to $d\mu$: for $1<p\le p_c$, the weighted mass $M_{\Omega}(u(t))$ vanishes, while for $p>p_c$ it remains positive and the solution approaches a linear-dominated asymptotic profile. In the non-vanishing regime, the asymptotics are governed by $u_\infty=u_0-\int_0^\infty u^p$, with $u(t)$ approximated by $e^{t\Delta_{\Omega}}u_\infty$, and, in dimensions $N\ge3$, by the Gaussian-type profile $M_{\Omega}(u_\infty)\phi G(t,\cdot)$. The results extend to $N=1,2$ with dimension-specific decay: polynomial in $N=1$ and logarithmic in $N=2$, reflecting the underlying heat-kernel behavior on exterior domains.
Abstract
The paper concerns with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation $\partial_tu-Δu+u^p=0$ in exterior domains $Ω$ in $\mathbb{R}^N$ ($N\geq 2$). The problem for the one-dimensional case is formulated with $Ω=(0,\infty)$ 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 $φ(x)\,dx$, 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 $t\to \infty$ 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 $N\geq 3$.