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

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.