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

Mitsuhiro Nakao

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Abstract

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.

Original languageEnglish
Pages (from-to)1585-1604
Number of pages20
JournalJournal of Mathematical Analysis and Applications
Volume462
Issue number2
DOIs
Publication statusPublished - 2018 Jun 15

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Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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