Asymptotic behaviour for a semilinear nonlocal equation

Article type: Research Article

Authors: Pazoto, Ademir F. | Rossi, Julio D.

Affiliations: Instituto de Matemática, Universidade Federal do Rio de Janeiro (UFRJ), Cidade Universitária, P.O. Box 68530, CEP 21945-970, Rio de Janeiro, Brasil E-mail: ademir@acd.ufrj.br | Depto. Matemática, FCEyN UBA, Ciudad Universitária, Pab I, 1428 Buenos Aires, Argentina E-mail: jrossi@dm.uba.ar

Abstract: We study the semilinear nonlocal equation ut=J*u−u−up in the whole ${\mathbb{R}}^{N}$. First, we prove the global well-posedness for initial conditions $u(x,0)=u_{0}(x)\in L^{1}({\mathbb{R}}^{N})\cap L^{\infty}({\mathbb{R}}^{N})$. Next, we obtain the long time behaviour of the solutions. We show that different behaviours are possible depending on the exponent p and the kernel J: finite time extinction for p<1, faster than exponential decay for the linear case p=1, a weakly nonlinear behaviour for p large enough and a decay governed by the nonlinear term when p is greater than one but not so large.

Keywords: nonlocal diffusion, semilinear problems, asymptotic behaviour

Journal: Asymptotic Analysis, vol. 52, no. 1-2, pp. 143-155, 2007