Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients
Gui‐Qiang Chen, Kenneth Hvistendahl Karlsen · Communications on Pure & Applied Analysis · 2005
We study the well-posedness ofdiscontinuous entropy solutions to quasilinearanisotropic degenerate parabolic equations withexplicit $(t,x)$--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 $a(u,t,x)=(a_{ij}(u,t,x))=\sigma^a(u,t,x)\sigma^a(u,t,x)^\top$ isnonnegative definite and each $x\mapsto f_i(u,t,x)$ is Lipschitz continuous.We establish a well-posedness theoryfor the Cauchy problem for such degenerate parabolic equationsvia Kruzkov's device of doubling variables,provided $\sigma^a(u,t,\cdot)\in W^{2,\infty}$ for the general caseand 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 forperturbations of the diffusionand convection functions.