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 $\partial_t u = \text{div}(k(x)\nabla G(u))$, $u|_{t=0}=u_0$ with Neumann boundary conditions $k(x)\nabla G(u)\cdot \nu = 0$. Here $x\in B\subset \mathbb{R}^d$, a bounded open set with locally Lipchitz boundary, and with $\nu$ as the unit outer normal. The function $G$ is Lipschitz continuous and nondecreasing, while $k(x)$ is diagonal matrix. We show that any two weak entropy solutions $u$ and $v$ satisfy $\Vert{u(t)-v(t)}\Vert_{L^1(B)}\le \Vert{u|_{t=0}-v|_{t=0}}\Vert_{L^1(B)}e^{Ct}$, for almost every $t\ge 0$, 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, $k_i=k_i(x_i)$, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.