Asymptotic behavior for a nonlocal diffusion equation with absorption and nonintegrable initial data. The supercritical case

Abstract:

In this paper we study the asymptotic behavior as time goes to infinity of the solution to a nonlocal diffusion equation with absorption modeled by a powerlike reaction $-u^p$, $p>1$ and set in $\mathbb {R}^N$. We consider a bounded, nonnegative initial datum $u_0$ that behaves like a negative power at infinity. That is, $|x|^\alpha u_0(x)\to A>0$ as $|x|\to \infty$ with $0<\alpha \le N$. We prove that, in the supercritical case $p>1+2/\alpha$, the solution behaves asymptotically as that of the heat equation (with diffusivity $\mathfrak {a}$ related to the nonlocal operator) with the same initial datum.