Large-time behavior in a nonlocal heat equation with absorption. The absorption dominated case with fast decaying initial data

TL;DR

The paper analyzes the large-time behavior of nonnegative solutions to a nonlocal diffusion equation with absorption in $\mathbb{R}^N$, where the initial data decay as a power at infinity. By a careful scaling that preserves the absorption term, the authors prove that diffusion is captured by a local semilinear heat equation and that the large-time profile is determined by the asymptotic constant $A$ of $|x|^{\frac{2}{p-1}}u_0(x)$. When $A=0$, the limit is the very singular solution of the local problem; when $A>0$, the limit is the corresponding local solution with initial trace $A|x|^{-\frac{2}{p-1}}$. The results provide a sharp, diffusion-absorption interplay description in the nonlocal setting and yield precise large-time convergence on diffusive scales.

Abstract

We study the large-time behavior of nonnegative solutions to a nonlocal dispersal equation in $\mathbb R^N$ with an absorption term modeled by $-u^p$, with $1<p<1+\frac2N$. The initial datum $u_0$ is assumed to be bounded, and to satisfy $|x|^{\frac2{p-1}}u_0(x)\to A\ge0$ as $|x|\to\infty$. Under these assumptions, we prove that the decay rate is that of the purely absorbing problem, while the limit profile is a very singular solution to a local diffusion problem with absorption if $A=0$, and a solution to this same local problem with initial datum $A|x|^{-\frac2{p-1}}$ if $A>0$.

Theorems & Definitions (32)

• Theorem 2.1
• proof
• Proposition 2.1
• proof
• Corollary 2.1
• Corollary 2.2
• Remark 2.1
• Proposition 2.2
• Proposition 3.1
• proof