Asymptotic stability and explosion of solutions for a class of hyperbolic problem with delay time.

Abstract

In this work, we investigated the stability of solutions using the energy method, and finite time blow-up of solutions through a direct approach based on Lyapunov functionals, for certain classes of nonlinear evolution equations. First, the theoretical concepts and methodological tools introduced in this study provide a foundational framework for the detailed results and analyses presented in the subsequent chapters. Next, we established the asymptotic stability of the solution to a Kirchhoff type viscoelastic wave equation that includes Balakrishnan Taylor damping and a time delay term. By constructing an appropriate Lyapunov functional, employing differential inequalities, we proved that the total energy of the system decays to zero over time, despite the additional complexity introduced by the memory and delay effects. Finally, we examined the finite time blow up of solutions for a problem involving a time delay term. Our objective was to determine sufficient conditions under which the solution either decays to zero as time approaches infinity or blows up in finite time, depending on the interplay between the source and damping terms in the evolution equation. From our literature review, we observed that classical models often incorporate nonlinear damping and first order perturbation terms, both of which have been extensively studied.