Critical exponents for the fast diffusion equation with a nonlinear boundary condition

Ryuichi Sato; Jin Takahashi

doi:10.1016/j.jmaa.2019.123526

Critical exponents for the fast diffusion equation with a nonlinear boundary condition

Ryuichi Sato, Jin Takahashi

SCI - Mathematics

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

Abstract

In this paper we consider the fast diffusion equation ∂_tu=Δ(u^m) (x∈Ω, t>0) with a nonlinear boundary condition ∂_νu^m=u^p (x∈∂Ω, t>0), where 0<m<1, p>0, Ω⊂R^N is a smooth domain and N≥1. We prove that p₀=(m+1)/2 is the critical global existence exponent for the cases Ω=R^N∖B₁‾ (N≥2) and Ω=B₁:={x∈R^N:|x|<1} (N≥1).

Original language	English
Article number	123526
Journal	Journal of Mathematical Analysis and Applications
Volume	482
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2019.123526
Publication status	Published - 2020 Feb 1

Keywords

Blow-up
Critical exponents
Fast diffusion equation
Nonlinear boundary condition

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2019.123526

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title = "Critical exponents for the fast diffusion equation with a nonlinear boundary condition",

abstract = "In this paper we consider the fast diffusion equation ∂tu=Δ(um) (x∈Ω, t>0) with a nonlinear boundary condition ∂νum=up (x∈∂Ω, t>0), where 00, Ω⊂RN is a smooth domain and N≥1. We prove that p0=(m+1)/2 is the critical global existence exponent for the cases Ω=RN∖B1‾ (N≥2) and Ω=B1:={x∈RN:|x|<1} (N≥1).",

keywords = "Blow-up, Critical exponents, Fast diffusion equation, Nonlinear boundary condition",

author = "Ryuichi Sato and Jin Takahashi",

note = "Funding Information: The first author was supported by JSPS KAKENHI Grant Number 18K13435 . The second author was supported by JSPS KAKENHI Grant Number 19K14567 . Publisher Copyright: {\textcopyright} 2019 Elsevier Inc.",

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AU - Sato, Ryuichi

AU - Takahashi, Jin

N1 - Funding Information: The first author was supported by JSPS KAKENHI Grant Number 18K13435 . The second author was supported by JSPS KAKENHI Grant Number 19K14567 . Publisher Copyright: © 2019 Elsevier Inc.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - In this paper we consider the fast diffusion equation ∂tu=Δ(um) (x∈Ω, t>0) with a nonlinear boundary condition ∂νum=up (x∈∂Ω, t>0), where 00, Ω⊂RN is a smooth domain and N≥1. We prove that p0=(m+1)/2 is the critical global existence exponent for the cases Ω=RN∖B1‾ (N≥2) and Ω=B1:={x∈RN:|x|<1} (N≥1).

AB - In this paper we consider the fast diffusion equation ∂tu=Δ(um) (x∈Ω, t>0) with a nonlinear boundary condition ∂νum=up (x∈∂Ω, t>0), where 00, Ω⊂RN is a smooth domain and N≥1. We prove that p0=(m+1)/2 is the critical global existence exponent for the cases Ω=RN∖B1‾ (N≥2) and Ω=B1:={x∈RN:|x|<1} (N≥1).

KW - Blow-up

KW - Critical exponents

KW - Fast diffusion equation

KW - Nonlinear boundary condition

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JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

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Critical exponents for the fast diffusion equation with a nonlinear boundary condition

Abstract

Keywords

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