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 $$\mathbf{u}$_{tt} = $\mathrm{d}\mathrm{i}\mathrm{v}$($\mathbb{T}$) + $\mathbf{f}$$ for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor $\mathit{ϵ}(\mathbf{u})$ 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}$)$,wherewedefine$F($\mathbb{T}$) = (1 + |$\mathbb{T}$|^a)^{-\frac{1}{a}}$\mathbb{T}$$, for constant parameters $\alpha \in (0,\mathrm{\infty })$ and $a\in (0,\mathrm{\infty })$, in any number $d$ of space dimensions, with periodic boundary conditions. The Cauchy stress $\mathbb{T}$ is show to belong to $L^1(Q{)}^{d\times d}$ over the space-time domain $Q$. In particular, in three space dimensions, if $a\in (0, \frac{2}{7})$,theninfact$$\mathbb{T}$\in L^{1+\delta}(Q)^{d\times d}$fora$\delta>0$,thevalueofwhichdependsonlyon$a$.