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

We study the well-posedness ofdiscontinuous entropy solutions to quasilinearanisotropic degenerate parabolic equations withexplicit $(t,x)$--dependence:∂ₜu + ∑ᵢ₌₁ᵈ∂xᵢfᵢ(u,t,x)=∑i,j=1ᵈ∂xⱼ(aᵢⱼ(u,t,x)∂xᵢu),where a(u,t,x)=(aᵢⱼ(u,t,x))=σᵃ(u,t,x)σᵃ(u,t,x)^ isnonnegative definite and each x↦ fᵢ(u,t,x) is Lipschitz continuous.We establish a well-posedness theoryfor the Cauchy problem for such degenerate parabolic equationsvia Kruzkov's device of doubling variables,provided σᵃ(u,t,·)∈ W2,∞ for the general caseand the weaker condition σᵃ(u,t,·)∈ W1,∞for the case that a is a diagonal matrix.We also establish a continuous dependence estimate forperturbations of the diffusionand convection functions.