Local solutions of weakly parabolic quasilinear differential equations

M Dreher, V Pluschke

Research output: Contribution to journalArticle

Abstract

We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we are able to prove L-infinity-estimates.

Original languageEnglish
Pages (from-to)109-129
Number of pages21
JournalMathematische Nachrichten
Volume198
Publication statusPublished - 1999

Keywords

  • semidiscretization in time
  • quasilinear degenerate parabolic equations
  • local solutions
  • ORLICZ-SOBOLEV SPACES
  • SYSTEMS

Cite this

Dreher, M ; Pluschke, V. / Local solutions of weakly parabolic quasilinear differential equations. In: Mathematische Nachrichten. 1999 ; Vol. 198. pp. 109-129.
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Local solutions of weakly parabolic quasilinear differential equations. / Dreher, M; Pluschke, V.

In: Mathematische Nachrichten, Vol. 198, 1999, p. 109-129.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Local solutions of weakly parabolic quasilinear differential equations

AU - Dreher, M

AU - Pluschke, V

PY - 1999

Y1 - 1999

N2 - We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we are able to prove L-infinity-estimates.

AB - We consider a quasilinear parabolic boundary value problem, the elliptic part of which degenerates near the boundary. In order to solve this problem, we approximate it by a system of linear degenerate elliptic boundary value problems by means of semidiscretization with respect to time. We use the theory of degenerate elliptic operators and weighted Sobolev spaces to find a priori estimates for the solutions of the approximating problems. These solutions converge to a local solution, if the step size of the time-discretization goes to zero. It is worth pointing out that we do not require any growth conditions on the nonlinear coefficients and right-hand side, since we are able to prove L-infinity-estimates.

KW - semidiscretization in time

KW - quasilinear degenerate parabolic equations

KW - local solutions

KW - ORLICZ-SOBOLEV SPACES

KW - SYSTEMS

M3 - Article

VL - 198

SP - 109

EP - 129

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

ER -