Singular Diffusion with Neumann boundary conditions

04/26/2020

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by   Giuseppe Maria Coclite, et al.

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In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion ∂_t u = div(k(x)∇ G(u)), u|_t=0=u_0 with Neumann boundary conditions k(x)∇ G(u)·ν = 0. Here x∈ B⊂R^d, a bounded open set with locally Lipchitz boundary, and with ν 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 ‖u(t)-v(t)‖_L^1(B)<‖u|_t=0-v|_t=0‖_L^1(B)e^Ct, 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), we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.