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.
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