## Abstract

We continue our study of bounded solutions of the semilinear parabolic equation u_{t}=u_{xx}+f(u) on the real line, where f is a locally Lipschitz function on R. Assuming that the initial value u_{0}=u(⋅,0) of the solution has finite limits θ^{±} as x→±∞, our goal is to describe the asymptotic behavior of u(x,t) as t→∞. In a prior work, we showed that if the two limits are distinct, then the solution is quasiconvergent, that is, all its locally uniform limit profiles as t→∞ are steady states. It is known that this result is not valid in general if the limits are equal: θ^{±}=θ_{0}. In the present paper, we have a closer look at the equal-limits case. Under minor non-degeneracy assumptions on the nonlinearity, we show that the solution is quasiconvergent if either f(θ_{0})≠0, or f(θ_{0})=0 and θ_{0} is a stable equilibrium of the equation ξ˙=f(ξ). If f(θ_{0})=0 and θ_{0} is an unstable equilibrium of the equation ξ˙=f(ξ), we also prove some quasiconvergence theorem making (necessarily) additional assumptions on u_{0}. A major ingredient of our proofs of the quasiconvergence theorems—and a result of independent interest—is the classification of entire solutions of a certain type as steady states and heteroclinic connections between two disjoint sets of steady states.

Original language | English (US) |
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Pages (from-to) | 137-186 |

Number of pages | 50 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 153 |

DOIs | |

State | Published - Sep 2021 |

### Bibliographical note

Funding Information:Supported in part by the NSF Grant DMS-1856491.

Publisher Copyright:

© 2021 Elsevier Masson SAS

## Keywords

- Convergent initial data
- Entire solutions
- Parabolic equations on the real line
- Quasiconvergence