### Abstract

The paper is devoted to the following problem: w″ (x) + cw′(x) +F (w(x), x) = 0, x ∈ ℝ^{1}, w(±∞) = w±, where the non-linear term F depends on the space variable x. A classification of non-linearities is given according to the behaviour of the function F (w, x) in a neighbourhood of the points w_{+} and w_{-}. The classical approach used in the Kolmogorov-Petrovsky-Piskunov paper [10] for an autonomous equation (where F = F (u) does not explicitly depend on x), which is based on the geometric analysis on the (w, w′)-plane, is extended and new methods are developed to analyse the existence and uniqueness of solutions in the non-autonomous case. In particular, we study the case where the function F (w, x) does not have limits as x → plusmn;∞.

Original language | English (US) |
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Pages (from-to) | 809-835 |

Number of pages | 27 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 19 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 1999 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics