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Approximate Inertial Manifold for a Class of the Kirchhoff Wave Equations with Nonlinear Strongly Damped Terms
Department of Mathematics, Yunnan University, Kunming, China
Department of Mathematics, Yunnan University, Kunming, China
Department of Mathematics, Yunnan University, Kunming, China
- 1 Department of Mathematics, Yunnan University, Kunming, China
- 2 Department of Mathematics, Yunnan University, Kunming, China
- 3 Department of Mathematics, Yunnan University, Kunming, China
International Journal of Modern Nonlinear Theory and Application·Volume 05 (2016)·Pages 218–234·Published 10 November 2016·DOI10.4236/ijmnta.2016.54020
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Abstract
This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.
KeywordsKirchhoff Wave EquationGlobal AttractorThe Smoothing EffectThe RegularityApproximate Inertial Manifold
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