Nonlocal heat equations: decay estimates and Nash inequalities

Abstract

We obtain $L^q$--$L^p$ decay estimates, $1\le q<p<\infty$ for solutions of nonlocal heat equations of the form ${\partial }_{t}u+\mathcal{L}u=0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of Lévy type. We obtain these estimates in terms only of the behaviour of the kernel at infinity, without any information of its behaviour at the origin. This includes bounded and unbounded transition probability densities. An equivalence between the decay and a restricted Nash inequality is shown. We also prove that ${\lim }_{t\to \infty }\Vert u(t){\Vert }_{\infty }=0$. Finally we deal with nonlinear nonlocal equations of porous medium type ${\partial }_{t}u+\mathcal{L}\phi (u)=0$.

References in corpus (1)

• Intrinsic scaling properties for nonlocal operators II

Cited by in corpus (4)

• Nonlocal filtration equations with rough kernels
• Improved energy methods for nonlocal diffusion problems
• Decay estimates for time-fractional and other non-local in time subdiffusion equations in ${\mathbb{R}}^d$
• Mixed Lebesgue space norm Strichartz type estimation for solution of inhomogeneous parabolic equation, with constants evaluation