## Abstract

We consider the Cauchy problem (Formula presented.) where (Formula presented.), f is a (Formula presented.) function satisfying minor nondegeneracy conditions, and (Formula presented.) is a radially symmetric function having a finite limit (Formula presented.) as (Formula presented.). We have previously proved that if (Formula presented.) is a stable equilibrium of the equation (Formula presented.) and the solution u is bounded, then u is quasiconvergent: its (Formula presented.) -limit set with respect to the topology of (Formula presented.) consists of steady states. In the present paper, we consider the case when (Formula presented.) is linearly stable: (Formula presented.) and (Formula presented.). Under this condition, we show that if the solution of the above Cauchy problem is bounded, then it converges, locally uniformly with respect to (Formula presented.), to a single steady state.

Original language | English (US) |
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Journal | Sao Paulo Journal of Mathematical Sciences |

DOIs | |

State | Accepted/In press - 2024 |

### Bibliographical note

Publisher Copyright:© Instituto de Matemática e Estatística da Universidade de São Paulo 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

## Keywords

- 35B40
- 35K15
- 35K57
- Cauchy problem
- Convergence
- Normal hyperbolicity
- Radial solutions
- Semilinear parabolic equations

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