Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models

Abstract

Let J : R → R J: \mathbb {R} \to \mathbb {R} be a nonnegative, smooth function with ∫ R J ( r ) d r = 1 \int _{\mathbb {R}} J(r)dr =1 , supported in [ − 1 , 1 ] [-1,1] , symmetric, J ( r ) = J ( − r ) J(r)=J(-r) , and strictly increasing in [ − 1 , 0 ] [-1,0] . We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation \[ u t ( x , t ) = ∫ − L L ( J ( x − y u ( y , t ) ) − J ( x − y u ( x , t ) ) ) d y , x ∈ [ − L , L ] . \displaystyle u_t (x,t)= \int _{-L}^{L} \left (J\left (\dfrac {x-y}{u(y,t)}\right ) - J\left (\dfrac {x-y}{u(x,t)}\right ) \right ) \, dy, \quad x \in [-L,L]. \] We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as t → ∞ t\to \infty : they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments.