Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients - Chen, G; Karlsen, K (2005)
We study the well-posed ness of discontinuous entropy solutions to quasilinear anisotropic degenerate parabolic equations with explicit (t,x)-dependence: ∂tu + ∑i=1d ∂xifi(u,t,x) = ∑i,j=1d ∂xj (aij (u,t,x)∂xi u), where a(u,t,x) = (aij(u,t,x)) = σa(u,t,x) σa(u,t,x)T is nonnegative definite and each x → fi(u,t,x) is Lipschitz continuous. We establish a well-posedness theory for the Cauchy problem for such degenerate parabolic equations via Kružkov's device of doubling variables, provided σa(u,t, ·) ∈ W2,∞ for the general case and the weaker condition σa(u,t,·) ∈ W1,∞ for the case that a is a diagonal matrix. We also establish a continuous dependence estimate for perturbations of the diffusion and convection functions.