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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 L∞‐estimates for the solutions of the approximate problems.