Local Existence and Blow-Up Analysis for a Damped Viscoelastic Kirchhoff-Type Equation with Logarithmic Power Source

We investigate the initial-boundary value problem for the nonlinear viscoelastic Kirchhoff-type wave equation\begin{equation*}u_{tt} - \left(1 + \|\nabla u(t)\|^2 \right) \Delta u + \int_0^t g(t - s) \Delta u(s) \, ds + \alpha u_t = |u|^{p-1} \ln |u|, \quad x \in \Omega, t > 0,\end{equation*}with homogeneous Dirichlet boundary conditions. Local existence and uniqueness of weak solutions are established via a Banach fixed-point argument. Moreover, we prove a finite-time blow-up result: if the initial energy is positive and the data satisfy a compatibility condition, the solution’s $H_0^1$-norm becomes unbounded in finite time. This work extends existing results by capturing the combined effects of Kirchhoff-type nonlinearity, viscoelastic damping, and a logarithmic source.