Global Existence and Blowup of Solutions to a Kirchhoff-Type Viscoelastic Wave Equation With Variable-Exponent Damping and Logarithmic Nonlinearity

Motivated by the mathematical modeling of heterogeneous viscoelastic media with nonstandard damping mechanisms, this paper investigates a Kirchhoff-type viscoelastic wave equation involving a variable-exponent nonlinear damping term, a logarithmic source, and a memory kernel. The main novelty of the work lies in the simultaneous treatment of three nonlinear effects: the nonlocal Kirchhoff coefficient, the viscoelastic memory term, and the spatially dependent damping exponent. This combination creates several analytical difficulties, mainly due to the loss of homogeneity in variable-exponent spaces, the singular nature of the logarithmic source near the origin, and the interaction between the memory term and the Kirchhoff nonlinearity. Under suitable assumptions on the variable exponents and the relaxation kernel, we first establish the associated energy identity and show that the energy is nonincreasing. Then, by using potential-well-type arguments, Sobolev embeddings, and suitable auxiliary functionals, we prove the global existence and boundedness of solutions corresponding to initial data in the stable set. On the other hand, for initial data with negative initial energy, we construct a perturbed functional of Levine type and derive a differential inequality which implies finite-time blowup. The results provide a clear dichotomy between global existence and finite-time blowup for a Kirchhoff-type viscoelastic equation with logarithmic nonlinearity and variable-exponent damping.