Singular diffusion with Neumann boundary conditions

In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion αtu = div (k(x)ΛG(u)), u =0 = u0 with Neumann boundary conditions k(x)ΛG(u) · τ = 0. Here x ∈ B Rd, a bounded open set with C3 boundary, and with τ 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 ≈u(t) - v(t) ≈ L1(B) ≪ ≈u =0 - v =0≈ L1(B)eCt, for almost every t ≫ 0, and a constant C = C(k,G,B). If we restrict to the case when the entries ki of k depend only on the corresponding component, ki = ki(xi), and αB is C2, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.