A nonlocal p-Laplacian evolution equation with Neumann boundary conditions

F Andreu,J.M Mazón,J.D Rossi,J Toledo

doi:10.1016/j.matpur.2008.04.003

Abstract

In this paper we study the nonlocal p-Laplacian type diffusion equation, u t ( t , x ) = ∫ Ω J ( x − y ) | u ( t , y ) − u ( t , x ) | p − 2 ( u ( t , y ) − u ( t , x ) ) d y . If p > 1 , this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation u t = div ( | ∇ u | p − 2 ∇ u ) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L ∞ ( 0 , T ; L p ( Ω ) ) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1 , that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition.

Highlights

Our main goal in this paper is to study the following nonlocal nonlinear diffusion problem, which we call the nonlocal p-Laplacian problem
As stated in [22], if u(t, x) is thought of as the density of a single population at the point x at time t, and J (x − y) is thought of as the probability distribution of jumping from location y to location x, the convolution (J ∗ u)(t, x) = RN J (y − x)u(t, y) dy is the rate at which individuals are arriving to position x from all other places and −u(t, x) = − RN J (y − x)u(t, x) dy is the rate at which they are leaving location x to travel to all other sites
Our main objective in this paper is to study the nonlocal equation PpJ, that is, the nonlocal analogous to the p-Laplacian evolution

Summary

Introduction and presentation of results

Our main goal in this paper is to study the following nonlocal nonlinear diffusion problem, which we call the nonlocal p-Laplacian problem (with homogeneous Neumann boundary conditions),. J (z)|zN |p dz is a normalizing constant in order to obtain the p-Laplacian in the limit instead a multiple of it Associated with these rescaled kernels we have solutions uε to the equation in PpJ with J replaced by Jp,ε and the same initial condition u0 (we shall call this problem PpJp,ε ). The rest of the paper is organized as follows: In Section 2 we prove the existence and uniqueness of strong solutions for the nonlocal problems for p > 1 and p = 1.

Existence of solutions for the nonlocal problems

Convergence to the p-Laplacian

Asymptotic behavior