Résumé:
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.
