抄録
We consider the Cauchy problem for quasilinear parabolic equations u • = Δ φ (u) + f(u), with the bounded non-negative initial data u0(x) (u0(x) ≢ 0), where f(ξ) is a positive function in ξ > 0 satisfying a blow-up condition ∫1∞ 1/f(ξ) dξ < ∞. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation dv/dt = f(v) with the initial data ||u0|| • ∞ (ℝN) > 0. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on u0 for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on u_0 for blow-up with the least blow-up time, provided that f(ξ) grows more rapidly than φ(ξ).
| 元の言語 | 英語 |
|---|---|
| ページ（範囲） | 379-405 |
| ページ数 | 27 |
| ジャーナル | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |
| 巻 | 138 |
| 発行部数 | 2 |
| DOI | |
| 出版物ステータス | 出版済み - 4 1 2008 |
| 外部発表 | Yes |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
これを引用
Blow-up directions for quasilinear parabolic equations. / Seki, Yukihiro; Umeda, Noriaki; Suzuki, Ryuichi.
：: Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 巻 138, 番号 2, 01.04.2008, p. 379-405.研究成果: ジャーナルへの寄稿 › 記事
}
TY - JOUR
T1 - Blow-up directions for quasilinear parabolic equations
AU - Seki, Yukihiro
AU - Umeda, Noriaki
AU - Suzuki, Ryuichi
PY - 2008/4/1
Y1 - 2008/4/1
N2 - We consider the Cauchy problem for quasilinear parabolic equations u • = Δ φ (u) + f(u), with the bounded non-negative initial data u0(x) (u0(x) ≢ 0), where f(ξ) is a positive function in ξ > 0 satisfying a blow-up condition ∫1∞ 1/f(ξ) dξ < ∞. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation dv/dt = f(v) with the initial data ||u0|| • ∞ (ℝN) > 0. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on u0 for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on u_0 for blow-up with the least blow-up time, provided that f(ξ) grows more rapidly than φ(ξ).
AB - We consider the Cauchy problem for quasilinear parabolic equations u • = Δ φ (u) + f(u), with the bounded non-negative initial data u0(x) (u0(x) ≢ 0), where f(ξ) is a positive function in ξ > 0 satisfying a blow-up condition ∫1∞ 1/f(ξ) dξ < ∞. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation dv/dt = f(v) with the initial data ||u0|| • ∞ (ℝN) > 0. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on u0 for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on u_0 for blow-up with the least blow-up time, provided that f(ξ) grows more rapidly than φ(ξ).
UR - http://www.scopus.com/inward/record.url?scp=48849110801&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=48849110801&partnerID=8YFLogxK
U2 - 10.1017/S0308210506000801
DO - 10.1017/S0308210506000801
M3 - Article
AN - SCOPUS:48849110801
VL - 138
SP - 379
EP - 405
JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
SN - 0308-2105
IS - 2
ER -