Abstract
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 $\partial_tu+\mathcal{L}Φ(u) = 0$.
Highlights
The purpose of this work is to study the short time and long time behaviour of solutions to nonlocal evolution equations of the form∂tu + Lu = 0, x ∈ RN, t > 0. (1)Here the diffusion is driven by an integral operator L in the spatial variable defined byLf (x) = P.V. (f (x) − f (y))J(x − y) dy, (2)RN with J a Lévy kernel, i.e., satisfying RN J(z)(|z|2 ∧1) dz < ∞
When studying the long-time behaviour of solutions for problems like (1)–(2), it is common in the literature to impose a global behaviour on J, typically integrability on the whole space, or boundedness, or a power-type behaviour, which may be different at the origin and at infinity, see for instance [3, 7, 9, 13]
In Subsection 4.2 we prove that this borderline is precisely the function J(z) = |z|−N for |z| small, and it lies into the no regularizing effect side, though it is not integrable, see Theorem 4.4
Summary
When studying the long-time behaviour of solutions for problems like (1)–(2), it is common in the literature to impose a global behaviour on J, typically integrability on the whole space, or boundedness, or a power-type behaviour, which may be different at the origin and at infinity, see for instance [3, 7, 9, 13]. As we will see this is it not always necessary: in this paper we obtain estimates on the decay of solutions for large times in terms only of the behaviour of the kernel at infinity, without imposing any condition at the other end. To this purpose we establish some functional inequalities that might have independent interest. Work partially supported by Spanish project MTM2011-25287. ∗ Corresponding author
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