Local Solutions of Weakly Parabolic Semilinear Differential Equations

✍ Scribed by Michael Dreher; Volker Pluschke

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 520 KB
• Volume: 200
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19992000102

✦ Synopsis

Semilinear parabolic boundary value problems with degenerated elliptic pert where the right-hand side depends on the solution an studied. We approximate the parabolic d l l n e a r problem by a system of linear degenerate elliptic problama by the aid of wmidiraatizstion in time.

Uilng weighted Sobolev spaces one derives aprioriatimatas for the approximate rdutions. These hpproximste solutions converge to a uniquely determined weak solution, if the time intend is sufti-Clsntly small. We point out that the nonlinear right-hand side is defined only in a neighbowhood of the initial data, therefore one has to p m LOD -estimates for the solutionr of the approximate problems.

I . Introduction

In this paper we will prove, by means of the Rothe method, the local existence of a fl&ltiOn of the weakly parabolic semilinear initial boundary value problem

Wo denote by R c IRN a bounded domain with boundary BR E C', T > 0, I = [0, TI, 0 t R x I , = BR x I and g(z) > o a. e. in R , g E L=(O), g-" E ~' ( 0 )

for some N' > N .