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 -