## Abstract

We consider the Cauchy problem (Formula presented.) where f is a C^{1} function on (Formula presented.) with (Formula presented.) and u_{0} is a nonnegative continuous function on (Formula presented.) whose limits at (Formula presented.) are equal to 0. Assuming that the solution u is bounded, we study its asymptotic behavior as (Formula presented.) In the first part of this study, we proved a general quasiconvergence result: as (Formula presented.) the solution approaches a set of steady states in the topology of (Formula presented.) In this paper, we show that under certain generic, explicitly formulated conditions on the nonlinearity f, the solution necessarily converges to a single steady state (Formula presented.) in (Formula presented.) Then, under the same conditions, we describe the global asymptotic shape of the solution: the graph of (Formula presented.) has a top part close to the graph of (Formula presented.) and two sides taking shapes of “terraces” moving in the opposite directions with precisely determined speeds.

Original language | English (US) |
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Pages (from-to) | 483-524 |

Number of pages | 42 |

Journal | Communications in Partial Differential Equations |

Volume | 45 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2 2020 |

### Bibliographical note

Funding Information:This work was supported in part by NSF Grant DMS–1565388.

Publisher Copyright:

© 2019, © 2019 Taylor & Francis Group, LLC.

## Keywords

- 35B40
- 35K15
- 35K57
- Convergence
- localized initial data
- parabolic equations on
- propagating terraces
- quasiconvergence
- traveling fronts