Journal of the Mathematical Society of Japan

Blow-up profile of a solution for a nonlinear heat equation with small diffusion

Hiroki YAGISITA

Full-text: Open access

Abstract

This paper is concerned with positive solutions of semilinear diffusion equations ut=ε2u+up in Ω with small diffusion under the Neumann boundary condition, where p>1 is a constant and Ω is a bounded domain in RN with C2 boundary. For the ordinary differential equation ut=up, the solution u0 with positive initial data u0C(Ω¯) has a blow-up set S0={xΩ¯|u0(x)=maxyΩ¯u0(y)} and a blowup profile u*0(x)=(u0(x)-(p-1)-(maxyΩ¯u0(y))-(p-1))-1/(p-1) outside the blow-up set S0. For the diffusion equation ut=ε2u+up in Ω under the boundary condition u/v=0 on Ω, it is shown that if a positive function u0C2(Ω¯) satisfies u0/v=0 on Ω, then the blow-up profile u*ε(x) of the solution uε with initial data u0 approaches u*0(x) uniformly on compact sets of Ω¯S0 as ε+0.

Article information

Source
J. Math. Soc. Japan, Volume 56, Number 4 (2004), 993-1005.

Dates
First available in Project Euclid: 27 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1190905445

Digital Object Identifier
doi:10.2969/jmsj/1190905445

Mathematical Reviews number (MathSciNet)
MR2091413

Zentralblatt MATH identifier
1065.35154

Subjects
Primary: 35B25: Singular perturbations 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B40: Asymptotic behavior of solutions 35B50: Maximum principles

Keywords
nonlinear diffusion equation blow-up profile

Citation

YAGISITA, Hiroki. Blow-up profile of a solution for a nonlinear heat equation with small diffusion. J. Math. Soc. Japan 56 (2004), no. 4, 993--1005. doi:10.2969/jmsj/1190905445. https://projecteuclid.org/euclid.jmsj/1190905445


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