Decay estimates for nonlocal problems via energy methods

Liviu I Ignat,Julio D Rossi

doi:10.1016/j.matpur.2009.04.009

Liviu I Ignat, Julio D Rossi

Open Access

https://doi.org/10.1016/j.matpur.2009.04.009

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Abstract

In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form u t ( x , t ) = ∫ R d G ( x − y ) ( u ( y , t ) − u ( x , t ) ) d y . For example, we will consider equations like, u t ( x , t ) = ∫ R d J ( x , y ) ( u ( y , t ) − u ( x , t ) ) d y + f ( u ) ( x , t ) , and a nonlocal analogous to the p-Laplacian, u t ( x , t ) = ∫ R d J ( x , y ) | u ( y , t ) − u ( x , t ) | p − 2 ( u ( y , t ) − u ( x , t ) ) d y . The energy method developed here allows us to obtain decay rates of the form, ‖ u ( ⋅ , t ) ‖ L q ( R d ) ⩽ C t − α , for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration.

Highlights

In this paper our main aim is to apply energy methods to obtain decay estimates for solutions to nonlocal evolution equations
Let us introduce the prototype of nonlocal equation that we have in mind
As stated in [20], if u(x, t) is thought of as a density at the point x at time t and G(x − y) is thought of as the probability distribution of jumping from location y to location x, Rd G(y − x)u(y, t) dy = (G ∗ u)(x, t) is the rate at which individuals are arriving at position x from all other places and −u(x, t) = − Rd G(y − x)u(x, t) dy is the rate at which they are leaving location x to travel to all other sites

Summary

Introduction

In this paper our main aim is to apply energy methods to obtain decay estimates for solutions to nonlocal evolution equations. This problem, with a convolution kernel, J (x, y) = G(x − y) was considered in [3] and [2] where the authors found existence, uniqueness and the convergence of the solutions to solutions of the local p-Laplacian evolution problem, vt = div(|∇v|p−2∇v) when a rescaling parameter (that measures the size of the support of the convolution kernel G) goes to zero In this case the asymptotic decay is described as follows: given u0 ∈ L1(Rd ) ∩ L∞(Rd ) there exists a unique solution to (1.3). We have to mention that we are assuming the following hypothesis on the kernel J (x; ·) ∈ L1(Rd ) This excludes the analysis of the possibility of a faster decay for u if for example J has fat tails, as happens for equations involving generators of Levy processes.

Preliminaries

Decay estimates for the linear diffusion problem with a nonlinear source

Decay estimates for the nonlocal p-Laplacian

Examples of exponential decay