Museilak-spaces

In this paper, a  class of nonlinear evolution equations with damping arising in fluid dynamics and rheology is studied.  The nonlinear term is monotone and possesses a convex potential but exhibits non-standard growth.  The appropriate functional framework for such equations is the modularly Museilak–spaces.  The existence and uniqueness of a weak solution are proved 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.