A nonlocal nonlinear diffusion equation with blowing up boundary conditions

M Bogoya,R Ferreira,J.D Rossi

doi:10.1016/j.jmaa.2007.04.049

Abstract

We deal with boundary value problems (prescribing Dirichlet or Neumann boundary conditions) for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation. First, we prove existence, uniqueness and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.

Highlights

The aim of this paper is to study the asymptotic behavior of solutions of a nonlocal nonlinear diffusion operator under blowing up boundary conditions of Dirichlet or Neumann type
As stated in [11] if u(x, t) is thought of as a density 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, (J ∗ u)(x, t) is the rate at which individuals are arriving to position x from all other places and −u(x, t) = − R J(y − x)u(x, t)dy is the rate at which they are leaving location x to travel to all other sites
Equation (1), so called nonlocal diffusion equation, shares many properties with the classical heat equation, ut = Δu, such as: bounded stationary solutions are constant, a maximum principle holds for both of them and perturbations propagate with infinite speed

Summary

Introduction

The aim of this paper is to study the asymptotic behavior of solutions of a nonlocal nonlinear diffusion operator under blowing up boundary conditions of Dirichlet or Neumann type. In [9] a simple nonlocal model for diffusion that is analogous to the porous medium equation is studied In this model the probability distribution of jumping from location y to location x is given by J x−y u(y,t). We will look at the peaking phenomena, that is we impose that the boundary data blow up in finite time and study the asymptotic behavior of solutions. These type of boundary conditions appear in combustion processes, [14]. Remark 3 in this case the blow-up phenomena for our model is different from the one for the porous medium equation, see [8]

Existence and uniqueness

Blow-up analysis