Note on a Hyperbolic Equation having p -Laplacian and Kirchhoff Damping

In this paper, we investigate a high-order quasilinear hyperbolic equation that involves Kirchhoff damping and logarithmic source,in (0, Tmax), subject to null Neumann boundary value conditions, where R n is an open bounded domain with smooth boundary and p, q > 2 and ( u 2 2 ) is the Kirchhofftype coefficient of the damping ut.Through the utilization of the Faedo-Galerkin approximation, we gain the well-posedness of local weak solutions.When q p, we construct algebraic and exponential decay estimates for the energy of global weak solutions.At the same time, relying on the contradiction argument and the auxiliary function method, we prove that the weak solutions blow up with negative initial energy for q > p.When q < p, we achieve that the weak solutions are globally bounded.