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 uₜₜ=Div(∂ Φ/∂ F(Du)+Duₜ) with nonconvex stored-energy function Φ. Unlike previous methods [P. Rybka, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), pp. 101--138], our result does not require that ∂Φ/∂ 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.