A nonlocal nonlinear diffusion equation in higher space dimensions

Abstract

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole R N , N ⩾ 1 , or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In R N we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t → ∞ , they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t → ∞ , they converge to zero.