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Unbounded solutions of the nonlocal heat equation

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  • 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.


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    F. John, "Partial Differential Equations," 4nd edition, Applied Mathematical Sciences, 1, Springer-Verlag, New York, 1982.

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