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.

27 pages, 5 figures