This talk investigates a system of nonlinear wave equations exhibiting time-varying delay and logarithmic nonlinearity. We first prove the
local existence and uniqueness of solutions using semigroup theory. Our main result establishes that solutions with negative initial energy undergo
finite-time blow-up, generalizing the scalar case. Our objective throughout this paper is to provide a comprehensive analysis
of the following coupled nonlinear wave system. This system models
the interaction of two nonlinear wave fields subject to internal damping and
logarithmic-type source terms. The nonlinear damping functions g(ut) and g(vt)
contribute to energy dissipation, whereas the coupled source terms may act as
energy generators, depending on the size and structure of the solution.
The strong coupling between u and v, together with the presence of logarithmic
nonlinearities, leads to a delicate competition between damping and source
effects. As a result, solutions may exhibit different qualitative behaviors, including
global existence with energy decay or finite-time blow-up for certain classes
of initial data. We introduce the functional setting, state the main assumptions, and establish the
well-posedness of the problem by proving the existence and uniqueness of weak solutions using semigroup theory and monotone operator techniques. In Section 3, we study the finite-time blow-up of solutions with negative initial energy by constructing an appropriate Lyapunov functional and employing nonlinear differential inequalities together with the concavity method.