## Abstract

We consider the semilinear parabolic equation u_{t} = Δu +f(u) on ℝ^{N}, assuming that f is an arbitrary C^{1} function satisfying f(0) = 0 and f′(0) < 0. We prove that any bounded positive solution that decays to zero at spatial infinity, uniformly with respect to t, converges to a (single) stationary solution as t → ∞. Our proof combines energy and comparison techniques with dynamical system arguments. We first establish an asymptotic symmetrization result: as t → ∞, u(x, t) approaches a set of steady states that are radially symmetric about a common origin in ℝ^{N}. To this aim we introduce a new toot that we call first moments of energy. Having established the symmetrization, we apply a general convergence result for gradient-like dynamical systems. This amounts to showing that the dimension of the kernel of the linearized operator around an equilibrium w matches the dimension of a manifold of equilibria passing through w.

Original language | English (US) |
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Pages (from-to) | 1793-1814 |

Number of pages | 22 |

Journal | Communications in Partial Differential Equations |

Volume | 27 |

Issue number | 9-10 |

DOIs | |

State | Published - 2002 |

### Bibliographical note

Funding Information:*Supported in part by VEGA Grant 1/7677/20.

Copyright:

Copyright 2018 Elsevier B.V., All rights reserved.

## Keywords

- Asymptotic behavior
- Asymptotic symmetry
- Cauchy problem
- Convergence
- Parabolic equations