Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data

Abstract

We study the large time behavior of nonnegative solutions of the Cauchy problem ut =R J(x,y)(u(y; t),u(x; t)) dy,up, u(x; 0) = u0(x) 2 L∞, where |x|αu0(x) → A < 0 as |x| → 1. One of our main goals is the study of the critical case p = 1+2=ff for 0 > ff > N, left open in previous articles, for which we prove that tff=2ju(x; t) , U(x; t)j → 0 where U is the solution of the heat equation with absorption with initial datum U(x; 0) = CA;N |x|,α. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data u0 in the supercritical case and also in the critical case (p = 1 + 2=N) for bounded and integrable u0.