Global existence and optimal decay rates for the Timoshenko system: The case of equal wave speeds

We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L1(R)-L2(R) type is also shown.