### Abstract

We consider the Cauchy problem for quasilinear parabolic equations u _{•} = Δ φ (u) + f(u), with the bounded non-negative initial data u_{0}(x) (u_{0}(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 ||u_{0}|| _{• ∞ (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 u_{0} 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).

Original language | English |
---|---|

Pages (from-to) | 379-405 |

Number of pages | 27 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 138 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 1 2008 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Proceedings of the Royal Society of Edinburgh Section A: Mathematics*,

*138*(2), 379-405. https://doi.org/10.1017/S0308210506000801

**Blow-up directions for quasilinear parabolic equations.** / Seki, Yukihiro; Umeda, Noriaki; Suzuki, Ryuichi.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society of Edinburgh Section A: Mathematics*, vol. 138, no. 2, pp. 379-405. https://doi.org/10.1017/S0308210506000801

}

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

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 -