Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body

We prove the existence of a unique large-data global-in-time weak solution to class models form \begin{document}$ \boldsymbol{u}_{tt} = \mbox{div }\mathbb{T} + \boldsymbol{f} $\end{document} for viscoelastic bodies exhibiting strain-limiting behaviour, where constitutive equation, relating linearised strain tensor id="M2">\begin{document}$ \boldsymbol{\varepsilon}( \boldsymbol{u}) Cauchy stress id="M3">\begin{document}$ \mathbb{T} $\end{document}, is assumed be id="M4">\begin{document}$ \boldsymbol{u}_t) \alpha F( \mathbb{T}) we define \(F(\mathbb{T}) (1 | \mathbb{T}|^a)^{-\frac{1}{a}} \mathbb{T}\), constant parameters id="M5">\begin{document}$ \in (0,\infty) and id="M6">\begin{document}$ in any number id="M7">\begin{document}$ d space dimensions, with periodic boundary conditions. The id="M8">\begin{document}$ shown belong id="M9">\begin{document}$ L^{1}(Q)^{d \times d} over space-time domain id="M10">\begin{document}$ Q $\end{document}. In particular, three if id="M11">\begin{document}$ (0,\frac{2}{7}) then fact id="M12">\begin{document}$ L^{1+\delta}(Q)^{d id="M13">\begin{document}$ \delta > 0 value which depends only on id="M14">\begin{document}$