TY - JOUR

T1 - Blow-up set for a semilinear heat equation and pointedness of the initial data

AU - Fujishima, Yohei

AU - Ishige, Kazuhiro

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - We consider the blow-up problem for a semilinear heat equation, (equation presented) where ε> 0, p> 1, N≥1, Ω is a domain in ℝN, and φ is a nonnegative smooth bounded function in Ω. It is known that, under suitable assumptions, if ε is sufficiently small, then the solution of (E) blows up only near the maximum points of the initial function φ (see, for example, [7]). In this paper, as a continuation of [7], we study the relationship between the location of the blow-up set and the level sets of the initial function φ. We also prove that the location of the blow-up set depends on the mean curvature of the graph of the initial function on its maximum points.

AB - We consider the blow-up problem for a semilinear heat equation, (equation presented) where ε> 0, p> 1, N≥1, Ω is a domain in ℝN, and φ is a nonnegative smooth bounded function in Ω. It is known that, under suitable assumptions, if ε is sufficiently small, then the solution of (E) blows up only near the maximum points of the initial function φ (see, for example, [7]). In this paper, as a continuation of [7], we study the relationship between the location of the blow-up set and the level sets of the initial function φ. We also prove that the location of the blow-up set depends on the mean curvature of the graph of the initial function on its maximum points.

KW - Blow-up set

KW - Mean curvature

KW - Small diffusion

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U2 - 10.1512/iumj.2012.61.4596

DO - 10.1512/iumj.2012.61.4596

M3 - Article

AN - SCOPUS:84877700241

VL - 61

SP - 627

EP - 663

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 2

ER -