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Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations

  • Fatiha Alabau-BoussouiraEmail author
Article

Abstract

The purpose of this paper is to establish strong lower energy estimates for strong solutions of nonlinearly damped Timoshenko beams, Petrowsky equations in two and three dimensions and wave-like equations for bounded one-dimensional domains or annulus domains in two or three dimensions. We also establish weak lower velocity estimates for strong solutions of the nonlinearly damped Petrowsky equation in two and three dimensions. The feedbacks in consideration have arbitrary growth close to the origin. These results improve the strong lower energy decay rates obtained in our previous papers (Alabau-Boussouira in J Differ Equ 249:1145–1178, 2010; J Differ Equ 248:1473–1517, 2010) for strong solutions of the nonlinearly locally damped wave equation and extend to systems and to Petrowsky equation the method of Alabau-Boussouira (J Differ Equ 249:1145–1178, 2010; J Differ Equ 248:1473–1517, 2010). These results are the first ones for Timoshenko beams and Petrowsky equations.

Mathematics Subject Classification (2000)

34G10 35B35 35B37 35L90 93D15 93D20 

Keywords

Timoshenko beams Wave equation Petrowsky equation Nonlinear dissipation Boundary damping Locally distributed feedback Hyperbolic equations Optimality Asymptotic behavior Lower energy estimates Regularity Energy comparison principles 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Present position Délégation, CNRS at MAPMO, UMR 6628Orléans Cedex 2France
  2. 2.Université Paul Verlaine-Metz, LMAM UMR 7122Metz Cedex 1France

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