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: February 2010

Revised: January 2011

Published: November 2011

Abstract / Introduction Related Papers Cited by

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

[1]	N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data, Trans. Amer. Math. Soc., 361 (2009), 2527-2566.doi: 10.1090/S0002-9947-08-04758-2.
[2]	C. Brändle and E. Chasseigne, Large deviations estimates for some non-local equations. Fast decaying kernels and explicit bounds, Nonlinear Analysis, 71 (2009), 5572-5586.doi: 10.1016/j.na.2009.04.059.
[3]	C. Brändle and E. Chasseigne, Large Deviations estimates for some non-local equations. General bounds and applications, to appear in Trans. Amer. Math. Soc, arXiv:0909.1467
[4]	P. Carr, H. Geman, D. B. Madan and M. Yor, Stochastic volatility for Lévy processes, Math. Finance, 13 (2003), 345-382.doi: 10.1111/1467-9965.00020.
[5]	E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.doi: 10.1016/j.matpur.2006.04.005.
[6]	E. Chasseigne and R. Ferreira, Isothermalization for a Non-local Heat Equation, preprint, arXiv:0912.3332
[7]	F. John, "Partial Differential Equations," 4nd edition, Applied Mathematical Sciences, 1, Springer-Verlag, New York, 1982.