Finite-Time Blow-Up of Solutions to a Stochastic Quasilinear Viscoelastic Wave Equation with
Nonlinear Damping and Logarithmic Source

This paper investigates the occurence of a finite-time blow-up for a solutions of a problem of type stochastic quasilinear
viscoelastic wave equation posed in a given bounded domain. The studied model incorporates a memory
term that is governed by a decreasing relaxation kernel, a nonlinear damping mechanism, and a
logarithmic source nonlinearity, and it is also driven by an additive perturbation in form of a stochastic noise. In this problem we face the combinition of the influence of a viscoelastic memory and a nonlinear damping taht introduces the dissipative effects. Moreover, the combinition of
the logarithmic source term and the stochastic perturbation contributes to potential instability, which results in a delicate analytical balance.
Under appropriately choosen assumptions for each of our problem's component— the relaxation function, the damping exponent, and the
noise intensity—we construct a suitable energy functional and then derive refined energy estimates.
By introducing an appropriate Lyapunov functional and exploiting the non-convex structure
of the logarithmic potential, we establish sufficient conditions for finite-time blow-up with
positive probability. In particular, we prove that if the initial energy is below a critical
negative threshold depending explicitly on the noise intensity, then the corresponding solution
cannot exist globally in time. Moreover, an explicit upper bound for the blow-up time is
obtained in terms of the initial data and the system parameters.