Well-posedness and energy decay of solutions for a quasilinear Petrovsky with a localized nonlinear dissipation involving the p- Laplacian

  • Ahmed Chahtou Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes 22000, Algeria; Mustapha Stambouli University, Mascara, (Algeria)
  • Mama Abdelli Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes 22000, Algeria.; Mustapha Stambouli University, Mascara, (Algeria)
  • Ali Hakem Djillali Liabes University, P. O. Box 89, Sidi Bel Abbes 22000, Algeria.

Abstract

We prove the existence and uniqueness of global solutions by Faedo-Galerkin method for the Cauchy problem concerning the evolution equation  $$ u_{tt} + \Delta^2 u - \Phi( \| \nabla u \|^2_2 ) \Delta u - \mathrm{div}( a(x) | \nabla u_t |^{p-2} \nabla u_t) =0,$$ suggested by the study of plates and beams, where $p \geq 2$ and $\Phi$ is a real function. We also investigate the asymptotic behavior of the solutions, under suitable growth assumptions. We will use for this task
an appropriate perturbed energy coupled with multiplier technique.

Published
2020-11-24