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

Official URL: https://dx.doi.org/10.3934/cpaa.2021053

We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form utt = div T + f for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor ε(u) to the Cauchy stress tensor T, is assumed to be of the form ε(ut) + αε(u) = F(T), where we define F(T) = (1 + |T|a)- a T, 1 for constant parameters α ∈ (0, ∞) and a ∈ (0, ∞), in any number d of space dimensions, with periodic boundary conditions. The Cauchy stress T is shown to belong to L1(Q)d×d over the space-time domain Q. In particular, in three space dimensions, if a ∈ (0, 27 ), then in fact T ∈ L1+δ(Q)d×d for a δ > 0, the value of which depends only on a.