### Abstract

This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations u_{t} = div(u^{l}∇u) and the evolution equation governed by p-Laplacian u_{t} = div(|∇u|^{p-2}∇u). Because of the degeneracy or singularity caused by the terms u^{l} and |∇u|^{p-2}, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W^{1,∞}(Ω) is assured under the assumption that initial data are non-negative functions in W_{0}
^{1,∞}(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L^{∞}-energy method", which provides a main tool for our arguments and would be useful for other situations.

Original language | English |
---|---|

Pages (from-to) | 647-670 |

Number of pages | 24 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 8 |

Issue number | 3 |

Publication status | Published - 2002 |

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### Keywords

- Degenerate
- Doubly nonlinear
- Gradient estimate
- Parabolic

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete and Continuous Dynamical Systems*,

*8*(3), 647-670.

**Lipschitz continuous solutions of some doubly nonlinear parabolic equations.** / Otani, Mitsuharu; Sugiyama, Yoshie.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems*, vol. 8, no. 3, pp. 647-670.

}

TY - JOUR

T1 - Lipschitz continuous solutions of some doubly nonlinear parabolic equations

AU - Otani, Mitsuharu

AU - Sugiyama, Yoshie

PY - 2002

Y1 - 2002

N2 - This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W0 1,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L∞-energy method", which provides a main tool for our arguments and would be useful for other situations.

AB - This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W0 1,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L∞-energy method", which provides a main tool for our arguments and would be useful for other situations.

KW - Degenerate

KW - Doubly nonlinear

KW - Gradient estimate

KW - Parabolic

UR - http://www.scopus.com/inward/record.url?scp=0036014241&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036014241&partnerID=8YFLogxK

M3 - Article

VL - 8

SP - 647

EP - 670

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

SN - 1078-0947

IS - 3

ER -