### Abstract

We give existence theorems of global solutions in L_{loc} ^{∞}((0,∞);W_{0} ^{1,∞}) to the initial boundary value problem for quasilinear degenerate parabolic equations of the form u_{t}−div{σ(|∇u|^{2})∇u}=0, where the class of σ(v^{2}) includes the logarithmic case σ(|∇u|^{2})= log (1+|∇u|^{2}) for a typical example. We assume that the initial data belong to W_{0} ^{1,p0 },p_{0}≥2, or L^{r},r≥1, and we derive precise estimates for ‖∇u(t)‖_{∞} near t=0.

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

Pages (from-to) | 1585-1604 |

Number of pages | 20 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 462 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 Jun 15 |

### Fingerprint

### Keywords

- Moser's method
- Quasilinear parabolic equation
- Smoothing effects

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Smoothing effects of the initial-boundary value problem for logarithmic type quasilinear parabolic equations.** / Nakao, Mitsuhiro.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 462, no. 2, pp. 1585-1604. https://doi.org/10.1016/j.jmaa.2018.02.061

}

TY - JOUR

T1 - Smoothing effects of the initial-boundary value problem for logarithmic type quasilinear parabolic equations

AU - Nakao, Mitsuhiro

PY - 2018/6/15

Y1 - 2018/6/15

N2 - We give existence theorems of global solutions in Lloc ∞((0,∞);W0 1,∞) to the initial boundary value problem for quasilinear degenerate parabolic equations of the form ut−div{σ(|∇u|2)∇u}=0, where the class of σ(v2) includes the logarithmic case σ(|∇u|2)= log (1+|∇u|2) for a typical example. We assume that the initial data belong to W0 1,p0 ,p0≥2, or Lr,r≥1, and we derive precise estimates for ‖∇u(t)‖∞ near t=0.

AB - We give existence theorems of global solutions in Lloc ∞((0,∞);W0 1,∞) to the initial boundary value problem for quasilinear degenerate parabolic equations of the form ut−div{σ(|∇u|2)∇u}=0, where the class of σ(v2) includes the logarithmic case σ(|∇u|2)= log (1+|∇u|2) for a typical example. We assume that the initial data belong to W0 1,p0 ,p0≥2, or Lr,r≥1, and we derive precise estimates for ‖∇u(t)‖∞ near t=0.

KW - Moser's method

KW - Quasilinear parabolic equation

KW - Smoothing effects

UR - http://www.scopus.com/inward/record.url?scp=85042867170&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042867170&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2018.02.061

DO - 10.1016/j.jmaa.2018.02.061

M3 - Article

VL - 462

SP - 1585

EP - 1604

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -