Local solutions of weakly parabolic semilinear differential equations

Semilinear parabolic boundary value problems with degenerated elliptic part where the right-hand side depends on the solution are studied. We approximate the parabolic semilinear problem by a system of linear degenerate elliptic problems by the aid of semidiscretization in time. Using weighted Sobolev spaces one derives a-priori-estimates for the approximate solutions. These approximate solutions converge to a uniquely determined weak solution, if the time interval is sufficiently small. We point out that the nonlinear right-hand side is defined only in a neighbourhood of the initial data, therefore one has to prove Lm-estimates for the solutions of the approximate problems.