Exponential Energy Decay, Determining Functionals, and Numerical Simulations for a Nonlinear Non-autonomous Binary Mixture Model

Abstract In this work, we investigate the asymptotic behavior of solutions to a nonlinear model describing binary mixtures of solids. We establish the exponential decay of the energy in both the autonomous case (in the absence of external forces) and the non-autonomous case (under the influence of time-dependent external forces). For the autonomous system, we further analyze the observability properties of the associated dynamical system by providing a sufficient upper bound on the completeness defect that ensures a given set of bounded linear functionals constitutes a determining set. Numerical experiments are performed using the finite element method and Newmark’s algorithm to illustrate the conservative and dissipative behaviors of the proposed nonlinear coupled model. The results confirm the accuracy of the scheme and the exponential decay of the energy in dissipative cases.