### Abstract

We consider the semilinear parabolic equation ut = u_{xx} + f(u) on the real line, where f is a locally Lipschitz function on R. We prove that if a solution u of this equation is bounded and its initial value u(x, 0) has distinct limits at x = ±∞ , then the solution is quasiconvergent, that is, all its limit profles as t → ∞ are steady states.

Original language | English (US) |
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Pages (from-to) | 4423-4441 |

Number of pages | 19 |

Journal | Nonlinearity |

Volume | 31 |

Issue number | 9 |

DOIs | |

State | Published - Aug 7 2018 |

### Keywords

- Parabolic equations on the real line
- convergence
- convergent initial data
- quasiconvergence

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## Cite this

Pauthier, A., & Poláčik, P. (2018). Large-time behavior of solutions of parabolic equations on the real line with convergent initial data.

*Nonlinearity*,*31*(9), 4423-4441. https://doi.org/10.1088/1361-6544/aaced3