International Monthly Seminar on Time Scales Analysis #3

In this work, we investigate the existence and decay properties of global solutions for a class of second-order evolution equations incorporating memory effects, a nonlinear delay term, and a time-varying weight function. The model reflects realistic dynamics observed in viscoelastic and thermoelastic systems with hereditary characteristics and delayed feedback. Using appropriate energy methods and the construction of a Lyapunov functional, we establish the global existence of solutions under suitable assumptions on the kernel, delay, and nonlinearity. Furthermore, we derive general decay estimates for the energy, which unify and extend various known exponential and polynomial decay results. These findings contribute to the understanding of long-term behavior in complex dynamical systems with combined memory and delay effects.