Local solutions of weakly parabolic semilinear differential equations
M. Dreher and V. Pluschke
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^\infty-estimates for the solutions of the approximate problems.
Math. Nachr.
200
(1999),
5-20
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