Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:436

Article

Cortazar, C.¹  Elgueta, M.¹  Quiros, F.²  Wolanski, N.^3,4 

[1] Pontificia Univ Catolica Chile, Dept Matemat, Alameda 340, Santiago, Chile

[2] Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain

[3] UBA, FCEyN, Dept Matemat, Pab 1, RA-1428 Buenos Aires, DF, Argentina

[4] Consejo Nacl Invest Cient & Tecn, IMAS, Ciudad Univ,Pab 1, RA-1428 Buenos Aires, DF, Argentina

关键词: Nonlocal diffusion;    Exterior domain;    Asymptotic behavior;    Matched asymptotics;   

DOI  :  10.1016/j.jmaa.2015.12.021

来源: Elsevier

【 摘 要 】

We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, partial derivative(t)u = J * u - u, where J is a smooth, radially symmetric kernel with support B-d(0) subset of R-2. The problem is set in an exterior two-dimensional domain which excludes a hole H, and with zero Dirichlet data on H. In the far field scale, xi(1) <= vertical bar x vertical bar t(-1/2) <= xi(2) with xi(1), xi(2) > 0, the scaled function log t u(x,t) behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by J. The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic 'logarithmic momentum' of the solution, lim(t ->infinity) integral(R2) u(x, t) log vertical bar x vertical bar dx. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, vertical bar x vertical bar <= t(1/2)h(t) with lim(t ->infinity)h(t) = 0, the scaled function t(logt)(2)u(x,t)/ log vertical bar x vertical bar converges to a multiple of phi(x)/ log vertical bar x vertical bar, where phi is the unique stationary solution of the problem that behaves as log vertical bar x vertical bar when vertical bar x vertical bar -> infinity. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, vertical bar x vertical bar >= t(1/2)g(t) with g(t) -> infinity, the solution is proved to be of order o((t logt)(-1)). (C) 2015 Elsevier Inc. All rights reserved.

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