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Global solutions of the wave-Schrödinger system with rough data
Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients
| 1. | Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730 |
| 2. | Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO-0316 Oslo, Norway |
$\partial_tu + \sum_{i=1}^d\partial_{x_i}f_i(u,t,x)=\sum_{i,j=1}^d\partial_{x_j}(a_{ij}(u,t,x)\partial_{x_i}u),$
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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