### 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 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Fingerprint Dive into the research topics of 'Blow-up directions for quasilinear parabolic equations'. Together they form a unique fingerprint.

## Cite this

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

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