## Abstract

We give a necessary and sufficient condition for the existence of L ^{1}-connections between equilibria of a semilinear parabolic equation. By an L^{1}-connection from an equilibrium φ^{-} to an equilibrium φ^{+} we mean a function u(·, t) which is a classical solution on the interval (-∞, T) for some T ∈ ℝ and blows up at t = T but continues to exist in the space L^{1} for t ∈ [T, ∞) and satisfies u(·, t) → φ^{±} (in a suitable sense) as t → ± ∞. The main tool in our analysis is the zero number.

Original language | English (US) |
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Pages (from-to) | 463-491 |

Number of pages | 29 |

Journal | Journal of Dynamics and Differential Equations |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - 2002 |

### Bibliographical note

Funding Information:All three authors were partially supported by JSPS Grant 4023. M.F. and P.P. were partially supported by VEGA Grant 1/7677/20.

## Keywords

- Blow-up
- Connecting orbits
- L-connections
- Nonlinear heat equation
- Semilinear parabolic equation
- Zero number

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