Implicit Time Discretization and Global Existence for a Quasi-Linear Evolution Equation with Nonconvex Energy

We establish global existence of weak solutions for the viscoelastic system $u_{tt}=Div(\frac{\partial \Phi}{\partial F}(Du)+Du_t)$ with nonconvex stored-energy function $\Phi$. Unlike previous methods [P. Rybka, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), pp. 101--138], our result does not require that $\frac{\partial\Phi}{\partial F}$ be globally Lipschitz continuous. Our approach is based on implicit time discretization and a compactness property of the discrete dynamical scheme not shared by energy-minimizing sequences and not known to be shared by approximation schemes of Galerkin type.