Global existence and stability of solution for a p-Kirchhoff type hyperbolic equation with damping and source terms

"In this paper, we consider a nonlinear $p-$Kirchhoff type hyperbolic equation with damping and source terms $$u_{tt}-M\left( \underset{\Omega }{\int }\left\vert \nabla u\right\vert ^{p}dx\right) \Delta _{p}u+\left\vert u_{t}\right\vert ^{m-2}u_{t}=\left\vert u\right\vert ^{r-2}u.$$ Under suitable assumptions and positive initial energy, we prove the global existence of solution by using the potential energy and Nehari's functionals. Finally, the stability of equation is established based on Komornik's integral inequality."