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Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations

Authors
  • Ghisi, Marina
  • Gobbino, Massimo
  • Haraux, Alain
Type
Preprint
Publication Date
Jun 16, 2013
Submission Date
Jun 16, 2013
Identifiers
arXiv ID: 1306.3644
Source
arXiv
License
Yellow
External links

Abstract

We consider a class of semi-linear dissipative hyperbolic equations in which the operator associated to the linear part has a nontrivial kernel. Under appropriate assumptions on the nonlinear term, we prove that all solutions decay to 0, as t -> +infinity, at least as fast as a suitable negative power of t. Moreover, we prove that this decay rate is optimal in the sense that there exists a nonempty open set of initial data for which the corresponding solutions decay exactly as that negative power of t. Our results are stated and proved in an abstract Hilbert space setting, and then applied to partial differential equations.

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