Existence and Asymptotic Behavior of Solutions for Degenerate Nonlinear Kirchhoff Strings with Variable-Exponent Nonlinearities

In this paper, we investigate the existence of a local solution in time and discuss the exponential asymptotic behavior to a weakly damped wave equation involving the variable-exponents u t t − M ∇ u t 2 Δ u + ∫ 0 t g t − s Δ u s d s + γ 1 u t + u t k x − 1 u t = u p x − 1 u in Ω × ℝ + with simply supported boundary condition, where Ω is a bounded domain of ℝ n , g > 0 is a memory kernel that decays exponentially, and M ( s ) is a locally Lipschitz function. This kind of problem without the memory term when k (.) and p (.) are constants models viscoelastic Kirchhoff equation.