  In this paper, we study a class of nonlinear evolution equations with damping
arising in fluid dynamics and rheology. The nonlinear term is monotone and
possesses a convex potential but exhibits non-standard growth. The appropriate
functional framework for such equations is the modular Museilak-Spaces. We
prove the existence and uniqueness of a weak solution using an approximation
approach by combining an internal approximation with the backward Euler scheme,
also a priori error estimate for the temporal semi-discretization is given.
