Communications on Pure and Applied Analysis (CPAA)

Unbounded solutions of the nonlocal heat equation

Pages: 1663 - 1686, Volume 10, Issue 6, November 2011      doi:10.3934/cpaa.2011.10.1663

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C. Brändle - Departamento de Matemáticas, U. Carlos III de Madrid, 28911 Leganés, Spain (email)
E. Chasseigne - Laboratoire de Mathématiques et Physique Théorique, U. F. Rabelais, Parc de Grandmont, 37200 Tours, France (email)
Raúl Ferreira - Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain (email)

Abstract: We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: $ u_t = J\ast u -u, $ where $J$ is a symmetric continuous probability density. Depending on the tail of $J$, we give a rather complete picture of the problem in optimal classes of data by: $(i)$ estimating the initial trace of (possibly unbounded) solutions; $(ii)$ showing existence and uniqueness results in a suitable class; $(iii)$ proving blow-up in finite time in the case of some critical growths; $(iv)$ giving explicit unbounded polynomial solutions.

Keywords:  Non-local diffusion, initial trace, optimal classes of data.
Mathematics Subject Classification:  35A01, 35A02, 45A05.

Received: February 2010;      Revised: January 2011;      Available Online: May 2011.