This paper addresses the issue of existence, uniqueness and long time existence of the Cauchy problem for the following semilinear Timoschenko system \[ \varphi_{tt}(t,x)-(\varphi_x-\psi)_x(t,x)=0, \]
\[ \psi_{tt}(t,x)-a^2\psi_{xx}(t,x)-(\varphi_x-\psi)(t,x)+\mu\psi_t(t,x)=|\psi(t,x)|^p, \] for \((t,x)\in\mathbb{R}^+\times\mathbb{R}\). This system is dissipative but not strictly so. The authors first consider the linear semigroup from \(H^s(\mathbb{R})\cap L^{1,\gamma}(\mathbb{R})\) to \(H^s(\mathbb{R})\), where \[ L^{1,\gamma}(\mathbb{R})=\{w\;|\int_{\mathbb{R}}(1+|x|^{\gamma})|w(x)|dx<\infty\}. \] As in [K. Ide et al., Math. Models Methods Appl. Sci. 18, No. 5, 647–667 (2008; Zbl 1153.35013)], the \(H^s\) norm of the semigroup is shown to decrease like a power of time, involving the parameter \(\gamma\).
The local existence for the semilinear system is given in a space of kind \(e^{-|x|^2/(1+t)^{\rho}}L^2(\mathbb{R})\cap H^1(\mathbb{R})\) as in [R. Ikehata and Y. Inoue, Nonlinear Anal., Theory Methods Appl. 69, No. 4, A, 1396–1401 (2008; Zbl 1149.35010)] without using the \(\gamma\)-linear estimate. Finally, a global existence result is proved for initial data with small norm in \(e^{-|x|^2}L^2(\mathbb{R})\cap L^{1,\gamma}(\mathbb{R})\). Because of non strictly dissipativeness, one additionally requires that \(p>12\).