# American Institute of Mathematical Sciences

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Sign-changing solutions for non-local elliptic equations with asymptotically linear term
May  2018, 17(3): 1161-1178. doi: 10.3934/cpaa.2018056

## Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities

* Corresponding author

Received  January 2017 Revised  February 2017 Published  January 2018

Fund Project: Work partially supported by Spanish project MTM2011-25287.

We study the short and large time behaviour of solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u = 0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of Lévy type, that includes bounded and unbounded transition probability densities. We characterize when a regularizing effect occurs for small times and obtain $L^q$-$L^p$ decay estimates, $1≤ q < p < ∞$ when the time is large. These properties turn out to depend only on the behaviour of the kernel at the origin or at infinity, respectively, without need of any information at the other end. An equivalence between the decay and a restricted Nash inequality is shown. Finally we deal with the decay of nonlinear nonlocal equations of porous medium type $\partial_tu+\mathcal{L}Φ(u) = 0$.

Citation: Cristina Brändle, Arturo De Pablo. Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1161-1178. doi: 10.3934/cpaa.2018056
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