weak solution

The question of interest for the presented study is the mathematical modeling of wave propagation in dissipative media.  The generalized fractional Zener model in the case of dimension d (d=1,2,3) is considered.  This work is devoted to the mathematical analysis of such model: existence and uniqueness of the strong and weak solution and energy decay result which guarantees the wave dissipation.  The existence of the weak solution is shown using a priori estimates for solutions which are also presented.

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.