Global well-posedness and polynomial decay for a nonlinear Timoshenko-Cattaneo system under minimal Sobolev regularity

We consider the nonlinear stability of the Timoshenko-Cattaneo system in the one-dimensional whole space. The Timoshenko system consists of two coupled wave equations with non-symmetric relaxation, and describes vibrations of the beam with shear deformation and rotational inertia effect. Generally, if the relaxation is not symmetric, the dissipation is produced through the complicated interaction of the components of the system, and their decay estimates and the energy estimates are of regularity-loss type. In this paper, we introduce the mathematical method to control such a weak dissipativity by investigating the Timoshenko system with Cattaneo's law, which is the first order approximation of Fourier's law with its time-delay effect. Racke & Said-Houari (2012), showed the global existence and the decay estimate of solutions by assuming high regularity H8∩L1 on the small initial data to control their weak dissipativity. In contrast, we prove the global existence in H2 by energy methods without any negative weights. Our regularity assumption is the same as that needed to show the local existence. That is, we do not need to assume the extra higher regularity on the initial data. Besides, the optimal decay estimate in H2∩L1 is shown by using the time decay inequality of Lp-Lq-Lr type.