## Abstract

We examine the behavior of positive bounded, localized solutions of semilinear parabolic equations u_{t} = Δu + f(u) on ℝ^{N}. Here f ∈ C^{1}, f(0) = 0, and a localized solution refers to a solution u(x, t) which decays to 0 as x→∞ uniformly with respect to t > 0. In all previously known examples, bounded, localized solutions are convergent or at least quasi-convergent in the sense that all their limit profiles as t→∞are steady states. If N = 1, then all positive bounded, localized solutions are quasi-convergent. We show that such a general conclusion is not valid if N ≥ 3, even if the solutions in question are radially symmetric. Specifically, we give examples of positive bounded, localized solutions whose ω-limit set is infinite and contains only one equilibrium.

Original language | English (US) |
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Pages (from-to) | 3481-3496 |

Number of pages | 16 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 46 |

Issue number | 5 |

DOIs | |

State | Published - 2014 |

### Bibliographical note

Publisher Copyright:© 2014 Society for Industrial and Applied Mathematics.

## Keywords

- Asymptotic behavior
- Localized solutions
- Nonconvergent solutions
- Semilinear parabolic equation