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Fractional decay bounds for nonlocal zero order heat equations. (English) Zbl 1302.35052

Summary: In this paper, we obtain bounds for the decay rate for solutions to the nonlocal problem \({\partial}_t u(t,x) = \int _{\mathbb {R} ^n} J(x,y)[u(t,y) - u(t,x)]\,dy\). Here we deal with bounded kernels \(J\) but with polynomial tails, that is, we assume a lower bound of the form \(J(x,y) \geq c_1|x-y|^{-(n +2\sigma )}\), for \(|x - y| >c_2\). Our estimates takes the form \(\|u(t)\|_{L^q(\mathbb {R} ^n)} \leq C t^{-({n}/{2\sigma }) (1 - {1}/{q})}\) for \(t\) large.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
45P05 Integral operators
35R09 Integro-partial differential equations
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References:

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