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 u t = R J ( x − y )( u ( y,t ) − u ( x,t )) dy − u p , u ( x, 0) = u 0 ( x ) ∈ L ∞ , where | x | α u 0 ( x ) → A > 0 as | x |→∞ . One of our main goals is the study of the critical case p = 1 + 2 /α for 0 < α < N , left open in previous articles, for which we prove that t α/ 2 | u ( x,t ) − U ( x,t ) | → 0 where U is the solution of the heat equation with absorption with initial datum U ( x, 0) = C A,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 u 0 in the supercritical case and also in the critical case ( p = 1 + 2 /N ) for bounded and integrable u 0 .