Decay rates and global existence for semilinear dissipative Timoshenko systems

Abstract

We prove new decay estimates for the dissipative Timoshenko system in the one-dimensional whole space, and a global existence theorem for semilinear systems. More precisely, if we restrict the initial data ( ( φ 0 , ψ 0 ) , ( φ 1 , ψ 1 ) ) ( (\varphi _{0},\psi _0),(\varphi _{1},\psi _1)) ∈ ( H s + 1 ( R N ) ∩ L 1 , γ ( R N ) ) × ( H s ( R N ) ) ∩ L 1 , γ ( R N ) ) \in \Big ( H^{s+1}( \mathbb {R}^{N}) \cap L^{1,\gamma }( \mathbb {R}^{N}) \Big ) \times \Big ( H^{s}( \mathbb {R}^{N})\Big ) \cap L^{1,\gamma }( \mathbb {R}^{N}) \Big ) with γ ∈ [ 0 , 1 ] \gamma \in \left [ 0,1\right ] , then we can derive faster decay estimates than those given by Ide, Haramoto and Kawashima. In addition, we use these decay estimates of the linear problem combined with the weighted energy method introduced by Todorova and Yordanov with the special weight given by Ikehata and Inoue to solve a semilinear problem with low regularity.