On a nonlocal diffusion model with Neumann boundary conditions

Abstract

We study a nonlocal diffusion model analogous to heat equation with Neumann boundary conditions. We prove the existence and uniqueness of solutions and a comparison principle. Furthermore, we analyze the asymptotic behavior of the solutions as the temporal variable goes to infinity and the boundary datum depends only on a spacial variable.

Introduction

The mathematical description of a great variety of phenomena that appear in many sciences such as Physics, Chemistry, Biology, can be done using linear and nonlinear partial differential equations. Some of the models that appear are the so called diffusion models. The most important of these models consists of the heat equation with Neumann or Dirichlet boundary conditions. In the case of Neumann boundary conditions, see [1], the simplest model is given by:

$$u_t(x,t)=\Delta u(x,t)\text{,}(x,t)\in \Omega \times (0,T)\text{,}$$$$\frac{\partial u}{\partial n}(x,t)=0\text{,}(x,t)\in \partial \Omega \times (0,T)\text{,}$$$$u(x,0)=u_0\left(x\right)\text{,}x\in \Omega \text{.}$$

In this kind of model, the diffusion of the density $u$ at a point $x$ and time $t$ depends locally on $u(x,t)$, reason for which this is known as the local model. However, in the past years some interest has been paid to models described by equations of the form

$$u_t(x,t)={\int }_{\Omega }J(x-y)[u(y,t)-u(x,t)]dy\text{,}(x,t)\in \Omega \times (0,T)\text{,}$$$$u(x,0)=u_0\left(x\right)\text{,}x\in \Omega \text{,}$$

and variations of it. Here, $J$ is a symmetric continuous nonnegative real function defined on ${\mathbb{R}}^N$, compactly supported on the unit ball, such that ${\int }_{{\mathbb{R}}^N}J\left(x\right)dx=1$. This equation is known as a nonlocal model since the diffusion of the density $u$ at a point $x$ and time $t$ does not only depend on $u(x,t)$, but also on all values of $u$ through the convolution term $J\ast u$ (see below).

Some nonlocal models have been studied recently. See for instance [2], [3], [4], [5], [6], [7], and the book [8]. Following [9], model (1.2) can be interpreted as follows. If $u(x,t)$ is the density of a population at point $x$ and time $t$, and $J(x-y)$ is thought as the probability distribution of jumping from location $y$ to location $x$, then the convolution

$$(J\ast u)(x,t)={\int }_{{\mathbb{R}}^N}J(y-x)u(y,t)dy\text{,}$$

is the rate at which the individuals are arriving to location $x$ from all other places. In the same way,

$$-{\int }_{{\mathbb{R}}^N}J(y-x)u(x,t)dy=-u(x,t)\text{,}$$

is the rate at which the points are leaving the location $x$ to travel to other sites. So, in the absence of external or internal sources, the density $u$ satisfies the nonlocal equation (1.2). These equations which include nonlocal terms have been used in several applications, for instance in Biology [10], image processing [11], systems of particles [12], optimal control theory [13] etc; see also [14], [15], [16]. For the mathematical analysis of these models the list of references is large and we refer to [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], and again to the book [8] and references therein.

The main goal of this paper is to introduce and analyze a new nonlocal diffusion model with Neumann boundary conditions. We deal with the following problem

$$u_t(x,t)={\int }_{\Omega }J(x-y)(u(y,t)-u(x,t))dy+{\int }_{\partial \Omega }G(x-y)g(y,t)d{S}_y\text{,}(x,t)\in \overline{\Omega }\times (0,T)\text{,}$$$$u(x,0)=u_0\left(x\right)\text{,}x\in \overline{\Omega }\text{,}$$

where, $\Omega$ is a smooth and bounded domain, $J:{\mathbb{R}}^N\mapsto \mathbb{R}$ and $G:{\mathbb{R}}^N\mapsto \mathbb{R}$ are continuous, nonnegative, radially symmetric functions compactly supported in the unit ball such that ${\int }_{{\mathbb{R}}^N}J\left(z\right)dz=1,{\int }_{{\mathbb{R}}^N}G\left(z\right)dz=1$, and $g\in L_{\mathrm{loc}}^{\infty }[(0,\infty );L^1\left(\partial \Omega \right)]$. The second integral in (1.3) takes into account the prescribed flux of individuals that enter or leave the domain according to the sign of $g$. This is what is called Neumann boundary conditions [7].

Our main results can be summarized as follows: we show the existence and uniqueness of solutions for $u_0\in L^1\left(\Omega \right)$, prove a comparison principle and study the asymptotic behavior of the solutions as $t\to \infty$.

Concerning Neumann boundary conditions for nonlocal diffusion, Cortazar et al. in [7], [6] (see also [8]) studied a closely related problem of the form

$$u_t(x,t)={\int }_{\Omega }J(x-y)(u(y,t)-u(x,t))dy+{\int }_{{\mathbb{R}}^N\setminus \Omega }G(x-y)g(y,t)dy\text{.}$$

If one deals with this type of problem, when the boundary data is given by a nonlinear function, $g(y,t)=f\left(u(y,t)\right)$, it is necessary to use an extension of the solution outside the domain, that is, from $\Omega$ to ${\mathbb{R}}^N\setminus \Omega$. In our case we can show that this extension is not necessary if we deal with solutions that are continuous in $\overline{\Omega }$. This is the main advantage of (1.3) compared with previous models.

The paper is organized as follows. In Section 2, we prove the existence and uniqueness of solutions. In Section 3, we give a comparison principle. In Section 4, we study the asymptotic behavior of solutions as $t\to \infty$. Finally, some conclusions are given.