Local Solutions of Weakly Parabolic Semilinear Differential Equations

Abstract

We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to nd a{priori{estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time{discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coecien ts and right{hand side, since we are able to prove L1{estimates.