Large time asymptotic behavior for the dissipative Timoshenko system and its application

In this paper, we study large time behavior for the dissipative Timoshenko system in the whole space [Formula: see text], particularly, on the transversal displacement [Formula: see text] and the rotation angle [Formula: see text] of the filament for the beam. Different from decay properties of the energy term derived by Ide et al. [Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18(5) (2008) 647–667], we discover new optimal growth [Formula: see text] estimates for the solutions themselves. Under the non-trivial mean condition on the initial data [Formula: see text], the unknowns [Formula: see text] and [Formula: see text] grow polynomially with the optimal rates [Formula: see text] and [Formula: see text], respectively, as large time. Furthermore, asymptotic profiles of them are introduced by the diffusion plate function, which explains a hidden cancellation mechanism in the shear stress [Formula: see text]. As an application of our results, we study the semilinear dissipative Timoshenko system with a power nonlinearity. Precisely, if the power is greater than the Fujita exponent, then the global in time existence of Sobolev solution is proved for the case of equal wave speeds, which partly gives a positive answer to the open problem in Racke and Said–Houari [Decay rates and global existence for semilinear dissipative Timoshenko systems, Q. Appl. Math. 71(2) (2013) 229–266].