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

Mitsuhiro Nakao

doi:10.1016/j.jmaa.2018.02.061

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

Mitsuhiro Nakao

Waseda Research Institute for Science and Engineering

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

We give existence theorems of global solutions in L_loc ^∞((0,∞);W₀ ^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(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 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	https://doi.org/10.1016/j.jmaa.2018.02.061
Publication status	Published - 2018 Jun 15

Keywords

Moser's method
Quasilinear parabolic equation
Smoothing effects

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2018.02.061

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title = "Smoothing effects of the initial-boundary value problem for logarithmic type quasilinear parabolic equations",

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.",

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author = "Mitsuhiro Nakao",

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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

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JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

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ER -

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

Abstract

Keywords

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