## Abstract

We consider parabolic equations of the form. u_{t}=δu+f(u)+h(x,t),(x,t)∈R{double-struck}^{N}×(0,∞), where f is a C1 function with f(0)=0, f<(0)<0, and h is a suitable function on RN×[0,∈) which decays to zero as t→∈ (hence the equation is asymptotically autonomous). We show that, as t→∈, each bounded localized solution u≥0 approaches a set of steady states of the limit autonomous equation u_{t}=δu+f(u). Moreover, if the decay of h is exponential, then u converges to a single steady state. We also prove a convergence result for abstract asymptotically autonomous parabolic equations.

Original language | English (US) |
---|---|

Pages (from-to) | 1903-1922 |

Number of pages | 20 |

Journal | Journal of Differential Equations |

Volume | 251 |

Issue number | 7 |

DOIs | |

State | Published - Oct 1 2011 |

### Bibliographical note

Funding Information:E-mail address: polacik@math.umn.edu (P. Polácˇik). 1 Supported in part by NSF Grant DMS-0900947.

## Keywords

- Asymptotically autonomous
- Convergence
- Parabolic equation
- Quasiconvergence

## Fingerprint

Dive into the research topics of 'Convergence to a steady state for asymptotically autonomous semilinear heat equations on R{double-struck}^{N}'. Together they form a unique fingerprint.