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) = Diu(partial derivative Phi/partial derivative F (Du) + Du(t)) with nonconvex stored-energy function Phi. Unlike previous methods [P. Rybka, Proc. Roy. Sec. Edinburgh Sect. A, 121 (1992), pp. 101-138], our result does not require that partial derivative Phi/partial derivative 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.