# American Institute of Mathematical Sciences

June  2005, 4(2): 241-266. doi: 10.3934/cpaa.2005.4.241

## Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients

 1 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730 2 Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO-0316 Oslo, Norway

Received  June 2004 Revised  November 2004 Published  March 2005

We study the well-posedness of discontinuous entropy solutions to quasilinear anisotropic degenerate parabolic equations with explicit $(t,x)$--dependence:

$\partial_tu + \sum_{i=1}^d\partial_{x_i}f_i(u,t,x)=\sum_{i,j=1}^d\partial_{x_j}(a_{ij}(u,t,x)\partial_{x_i}u),$

where $a(u,t,x)=(a_{ij}(u,t,x))=\sigma^a(u,t,x)\sigma^a(u,t,x)^\top$ is nonnegative definite and each $x\mapsto f_i(u,t,x)$ is Lipschitz continuous. We establish a well-posedness theory for the Cauchy problem for such degenerate parabolic equations via Kruzkov's device of doubling variables, provided $\sigma^a(u,t,\cdot)\in W^{2,\infty}$ for the general case and the weaker condition $\sigma^a(u,t,\cdot)\in W^{1,\infty}$ for the case that $a$ is a diagonal matrix. We also establish a continuous dependence estimate for perturbations of the diffusion and convection functions.

Citation: Gui-Qiang Chen, Kenneth Hvistendahl Karlsen. Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients. Communications on Pure & Applied Analysis, 2005, 4 (2) : 241-266. doi: 10.3934/cpaa.2005.4.241
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