Unbounded solutions of the nonlocal heat equation

C. Brändle; E. Chasseigne; Raúl Ferreira

doi:10.3934/cpaa.2011.10.1663

Article Contents

2011, Volume 10, Issue 6: 1663-1686. Doi: 10.3934/cpaa.2011.10.1663

This issue Previous Article Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains Next Article Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

Unbounded solutions of the nonlocal heat equation

1.
Departamento de Matemáticas, U. Carlos III de Madrid, 28911 Leganés
2.
Laboratoire de Mathématiques et Physique Théorique, U. F. Rabelais, Parc de Grandmont, 37200 Tours
3.
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid

Received: January 31, 2010

Revised: December 31, 2010

Published: October 2011

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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.

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Mathematics Subject Classification: 35A01, 35A02, 45A05.

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References

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