### 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 language | English |
---|---|

Pages (from-to) | 109-129 |

Number of pages | 21 |

Journal | Mathematische Nachrichten |

Volume | 198 |

Publication status | Published - 1999 |

### Keywords

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

### Cite this

*Mathematische Nachrichten*,

*198*, 109-129.

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*Mathematische Nachrichten*, vol. 198, pp. 109-129.

**Local solutions of weakly parabolic quasilinear differential equations.** / Dreher, M; Pluschke, V.

Research output: Contribution to journal › Article

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 -