Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Decay rates and global existence for semilinear dissipative Timoshenko systems

Authors: Reinhard Racke and Belkacem Said-Houari
Journal: Quart. Appl. Math. 71 (2013), 229-266
MSC (2010): Primary 35L71, 35B40
DOI: https://doi.org/10.1090/S0033-569X-2012-01280-8
Published electronically: October 3, 2012
MathSciNet review: 3087421
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Abstract: We prove new decay estimates for the dissipative Timoshenko system in the one-dimensional whole space, and a global existence theorem for semilinear systems. More precisely, if we restrict the initial data $ ( (\varphi _{0},\psi _0),(\varphi _{1},\psi _1)) $ $ \in \Big ( H^{s+1}( \mathbb{R}^{N}) \cap L^{1,\gamma }( \mathbb{R}^{N}) \Big )... ... \Big ( H^{s}( \mathbb{R}^{N})\Big ) \cap L^{1,\gamma }( \mathbb{R}^{N}) \Big )$ with $ \gamma \in \left [ 0,1\right ]$, then we can derive faster decay estimates than those given by Ide, Haramoto and Kawashima. In addition, we use these decay estimates of the linear problem combined with the weighted energy method introduced by Todorova and Yordanov with the special weight given by Ikehata and Inoue to solve a semilinear problem with low regularity.

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Additional Information

Reinhard Racke
Affiliation: Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany
Email: reinhard.racke@uni-konstanz.de

Belkacem Said-Houari
Affiliation: Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany
Email: belkacem.said-houari@kaust.edu.sa

DOI: https://doi.org/10.1090/S0033-569X-2012-01280-8
Keywords: Global existence, decay rate, Timoshenko system, semilinear wave equation
Received by editor(s): May 30, 2011
Published electronically: October 3, 2012
Article copyright: © Copyright 2012 Brown University

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