Global existence and decay property of the Timoshenko system in thermoelasticity with second sound

Abstract

Our main focus in the present paper is to study the asymptotic behavior of a nonlinear version of the Timoshenko system in thermoelasticity with second sound. As it has been already proved in \cite{SaidKasi_2011}, the linear version of this system is of regularity-loss type. It is well known (\cite{HKa06}, \cite%b{IK08}, \cite{KK09}) that the regularity-loss property of the linear problem creates difficulties when dealing with the nonlinear problem. In fact, the dissipative property of the problem becomes very weak in the high frequency region and as a result the classical energy method fails. To overcome this difficulty and following \cite{IK08} and \cite{Ikehata_2002}, we use an energy method with negative weights to create an artificial damping which allows us to control the nonlinearity. We prove that for $0\leq k\leq [s/2]-2 $ with $s\geq 8$, the solution of our problem is global in time and decays as $\left\Vert \partial _{x}^{k}U\left( t\right) \right\Vert _{2}\leq C\left( 1+t\right) ^{-1/4-k/2}, $ provided that the initial datum $U_0\in H^s(\mathbb{R})\cap L^1(\mathbb{R}) $.