Local solutions of weakly parabolic semilinear differential equations

M Dreher, Volker Pluschke

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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 Lm-estimates for the solutions of the approximate problems.

Original languageEnglish
Pages (from-to)5-20
Number of pages16
JournalMathematische Nachrichten
Volume200
Publication statusPublished - 1999

Keywords

  • semidiscretization in time
  • semilinear degenerated parabolic equations
  • local solutions
  • ORLICZ-SOBOLEV SPACES
  • SYSTEMS

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