Singular diffusion with Neumann boundary conditions

Abstract

In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion , u| t=0 = u 0 with Neumann boundary conditions k(x)∇G(u) ⋅ ν = 0. Here , a bounded open set with C 3 boundary, and with ν as the unit outer normal. The function G is Lipschitz continuous and nondecreasing, while k(x) is a diagonal matrix. We show that any two weak entropy solutions u and v satisfy , for almost every t ⩾ 0, and a constant C = C(k, G, B). If we restrict to the case when the entries k i of k depend only on the corresponding component, k i = k i (x i ), and ∂B is C 2, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.