Global Existence and Uniform Decay of Solutions for a Kirchhoff Beam Equation with Nonlinear Damping and Source Term

Abstract

An extensible beam equation of Kirchhoff type with internal damping and source term is investigated. We apply the potential well and establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations, taking into account that the initial data is located in a suitable set of stability created from the Nehari manifold. Moreover, by using Nakao’s method, we prove the exponential stability of the solution for \(p=1\) and the polynomial stability for \(p>1\).

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Correspondence to Carlos A. Raposo.

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Pereira, D.C., Raposo, C.A., Maranhão, C.H.M. et al. Global Existence and Uniform Decay of Solutions for a Kirchhoff Beam Equation with Nonlinear Damping and Source Term. Differ Equ Dyn Syst (2021). https://doi.org/10.1007/s12591-021-00563-x

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Keywords

  • Extensible beam
  • Well-posedness
  • Faedo-Galerkin
  • Potential well

Mathematics Subject Classification

  • 35L15
  • 35L70
  • 35B40
  • 35A01