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Existence and stability results of a plate equation with nonlinear damping and source term


  • Received: 22 June 2022 Revised: 29 August 2022 Accepted: 31 August 2022 Published: 13 September 2022
  • The main goal of this work is to investigate the following nonlinear plate equation

    $ u_{tt}+\Delta ^2 u +\alpha(t) g(u_t) = u \vert u\vert ^{\beta}, $

    which models suspension bridges. Firstly, we prove the local existence using Faedo-Galerkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions $ \alpha $ and $ g $ and obtained without any restriction growth assumption on $ g $ at the origin. The multiplier method, properties of the convex functions, Jensen's inequality and the generalized Young inequality are used to establish the stability results.

    Citation: Mohammad M. Al-Gharabli, Adel M. Al-Mahdi. Existence and stability results of a plate equation with nonlinear damping and source term[J]. Electronic Research Archive, 2022, 30(11): 4038-4065. doi: 10.3934/era.2022205

    Related Papers:

  • The main goal of this work is to investigate the following nonlinear plate equation

    $ u_{tt}+\Delta ^2 u +\alpha(t) g(u_t) = u \vert u\vert ^{\beta}, $

    which models suspension bridges. Firstly, we prove the local existence using Faedo-Galerkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions $ \alpha $ and $ g $ and obtained without any restriction growth assumption on $ g $ at the origin. The multiplier method, properties of the convex functions, Jensen's inequality and the generalized Young inequality are used to establish the stability results.



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