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

Descret Solution for a Nonlinear Parabolic Equations with Diffusion Terms in Museilak-Spaces