# American Institute of Mathematical Sciences

November  2011, 10(6): 1663-1686. doi: 10.3934/cpaa.2011.10.1663

## Unbounded solutions of the nonlocal heat equation

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

Received  February 2010 Revised  January 2011 Published  May 2011

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.
Citation: C. Brändle, E. Chasseigne, Raúl Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1663-1686. doi: 10.3934/cpaa.2011.10.1663
##### References:
 [1] N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data,, Trans. Amer. Math. Soc., 361 (2009), 2527. doi: 10.1090/S0002-9947-08-04758-2. Google Scholar [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. doi: 10.1016/j.na.2009.04.059. Google Scholar [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, (). Google Scholar [4] P. Carr, H. Geman, D. B. Madan and M. Yor, Stochastic volatility for Lévy processes,, Math. Finance, 13 (2003), 345. doi: 10.1111/1467-9965.00020. Google Scholar [5] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. Google Scholar [6] E. Chasseigne and R. Ferreira, Isothermalization for a Non-local Heat Equation,, preprint, (). Google Scholar [7] F. John, "Partial Differential Equations,", 4$^{nd}$ edition, (1982). Google Scholar

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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. doi: 10.1090/S0002-9947-08-04758-2. Google Scholar [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. doi: 10.1016/j.na.2009.04.059. Google Scholar [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, (). Google Scholar [4] P. Carr, H. Geman, D. B. Madan and M. Yor, Stochastic volatility for Lévy processes,, Math. Finance, 13 (2003), 345. doi: 10.1111/1467-9965.00020. Google Scholar [5] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl., 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. Google Scholar [6] E. Chasseigne and R. Ferreira, Isothermalization for a Non-local Heat Equation,, preprint, (). Google Scholar [7] F. John, "Partial Differential Equations,", 4$^{nd}$ edition, (1982). Google Scholar
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