In this paper, we concentrate on the model of Maxwell’s system in metamaterials with a nonlinear boundary condition, which has considered two partial differential equations of the currents in the free electron related to the polarization and magnetization in metamaterials. Then we attempt to provide a further study for the double convolution of Maxwell’s equations on time domain by viscoelasticity theory. Such a new initial boundary value problem is under the assumption of electrical sources and resistance effects are absent. Next the equations are transformed to an abstract nonlinear evolution equation. We proved the existence and uniqueness of solution with respect to the evolution equation by using nonlinear semi-group theory, which mainly explains the existence theory of nonlinear initial boundary value problem through the description of the basic theory of maximal monotone operators in reflexive Banach space. This provides a theoretical foundation for further research. Finally we give the energy dissipation estimate of the system with general nonlinear feedbacks on the all domain.