International Monthly Seminar on Time Scales Analysis #3

In this work, the uniform stabilization of certain hyperbolic systems with Wentzell boundary conditions is considered, and a uniform energy decay rate for the problem is established, taking into account both internal localized damping and boundary feedback. The exponential stabilization is attained by constructing a new multiplier and using multiplier methods.

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.