### Abstract

We consider semilinear parabolic equations u_{t}= u_{xx}+ f(u) on R. We give an overview of results on the large time behavior of bounded solutions, focusing in particular on their limit profiles as t→ ∞ with respect to the locally uniform convergence. The collection of such limit profiles, or, the ω -limit set of the solution, always contains a steady state. Questions of interest then are whether—or under what conditions—the ω -limit set consists of steady states, or even a single steady state. We give several theorems and examples pertinent to these questions.

Original language | English (US) |
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Title of host publication | Patterns of Dynamics - In Honour of Bernold Fiedler’s 60th Birthday |

Editors | Pavel Gurevich, Juliette Hell, Arnd Scheel, Bjorn Sandstede |

Publisher | Springer New York LLC |

Pages | 172-183 |

Number of pages | 12 |

ISBN (Print) | 9783319641720 |

DOIs | |

State | Published - Jan 1 2017 |

Event | Conference on Patterns of Dynamics held in honor of Bernold Fiedler’s 60th Birthday, 2016 - Berlin, Germany Duration: Jul 25 2016 → Jul 29 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 205 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | Conference on Patterns of Dynamics held in honor of Bernold Fiedler’s 60th Birthday, 2016 |
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Country | Germany |

City | Berlin |

Period | 7/25/16 → 7/29/16 |

### Keywords

- Asymptotic behavior
- Convergence
- Entire solutions
- Quasiconvergence
- Semilinear heat equation on the real line

## Fingerprint Dive into the research topics of 'Convergence and Quasiconvergence Properties of Solutions of Parabolic Equations on the Real Line: An Overview'. Together they form a unique fingerprint.

## Cite this

Poláčik, P. (2017). Convergence and Quasiconvergence Properties of Solutions of Parabolic Equations on the Real Line: An Overview. In P. Gurevich, J. Hell, A. Scheel, & B. Sandstede (Eds.),

*Patterns of Dynamics - In Honour of Bernold Fiedler’s 60th Birthday*(pp. 172-183). (Springer Proceedings in Mathematics and Statistics; Vol. 205). Springer New York LLC. https://doi.org/10.1007/978-3-319-64173-7_11