Publication: Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities
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2018-05
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American Institute of Mathematical Sciences (AIMS)
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Abstract
We study the short and large time behaviour of solutions of nonlocal heat equations of the form ∂tu+Lu=0. Here 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 Lq-Lp 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 ∂tu+LΦ(u)=0.
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Nonlocal diffusion equations, Integral operators, Asymptotic behaviour, Nash inequalities
Bibliographic citation
Brändle, C. & Pablo, A. de (2018). Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities. Communications on Pure & Applied Analysis, 17(3), 1161–1178.