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 a class of models of the form $\mathbf{u}_{tt} = \mathrm{div}(\mathbb{T}) + \mathbf{f}$ for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor $\boldsymbol{\epsilon}(\mathbf{u})$ to the Cauchy stress tensor $\mathbb{T}$, is assumed to be of the form $\boldsymbol{\epsilon}(\mathbf{u}_t) +\alpha \boldsymbol{\epsilon}(\mathbf{u})= F(\mathbb{T})$, where we define $F(\mathbb{T}) = (1 + |\mathbb{T}|^a)^{-\frac{1}{a}}\mathbb{T}$, for constant parameters $\alpha \in (0, \infty)$ and $a\in (0, \infty)$, in any number $d$ of space dimensions, with periodic boundary conditions. The Cauchy stress $\mathbb{T}$ is show to belong to $L^1(Q)^{d\times d}$ over the space-time domain $Q$. In particular, in three space dimensions, if $a\in (0, \frac{2}{7})$, then in fact $\mathbb{T}\in L^{1+\delta}(Q)^{d\times d}$ for a $\delta>0$, the value of which depends only on $a$.