Local Solutions of Weakly Parabolic Semilinear Differential Equations
Michael Dreher, Volker Pluschke · Mathematische Nachrichten · 1999
Abstract 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.