Article Contents
Article Contents

# Unbounded solutions of the nonlocal heat equation

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

 Citation:

•  [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.