## Abstract

We investigate the behaviour of some critical solutions of a non-local initial-boundary value problem for the equation u _{t}=?u+?f(u)/(?Of(u)dx) ^{2},O?R^{N},N=1,2. Under specific conditions on f, there exists a ?^{*} such that for each 0<?<?^{*} there corresponds a unique steady-state solution and u=u(x,t;?) is a global in time-bounded solution, which tends to the unique steady-state solution as t?8 uniformly in x. Whereas for ?=?^{*} there is no steady state and if ?>?^{*} then u blows up globally. Here, we show that when (a) N=1,O=(-1,1) and f(s)>0,f'(s)<0,s=0, or (b) N=2,O=B(0,1) and f(s)=e ^{-s}, the solution u^{*}=u(x,t;? ^{*}) is global in time and diverges in the sense ||u ^{*}(·,t)||_{8}?8, as t?8. Moreover, it is proved that this divergence is global i.e. u^{*}(x,t)?8 as t?8 for all x?O. The asymptotic form of divergence is also discussed for some special cases. © 2004 Elsevier Ltd. All rights reserved.

Original language | English |
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Pages (from-to) | 787-812 |

Number of pages | 26 |

Journal | Nonlinear Analysis: Theory, Methods and Applications |

Volume | 58 |

Issue number | 7-8 |

DOIs | |

Publication status | Published - Sep 2004 |

## Keywords

- Asymptotic behaviour
- Comparison methods
- Global-unbounded solutions
- Non-local parabolic problems