QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS WITH TIME-SPACE DEPENDENT DIFFUSION COEFFICIENTS

Abstract

We study the well-posedness of discontinuous entropy solutions to quasilinear aniso-tropic degenerate parabolic equations with explicit $(t,x)$--dependence: $$ \pt u + \si \pxi f_i(u,t,x)=\sij \pxj\left(\aij(u,t,x)\pxi u\right), $$ where $a(u,t,x)=(\aij(u,t,x))=\sa(u,t,x)\sa(u,t,x)^\top$ 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 $\sa(u,t,\cdot)\in W^{2,\infty}$ for the general case and the weaker condition $\sa(u,t,\cdot)\in W^{1,\infty}$ for the case that $a$ is a diagonal matrix. We also establish a continuous dependence estimate for perturbations of the diffusion and convection functions.