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

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  • 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.

    Mathematics Subject Classification: 35K65,35B35,35G25,35D99.

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