Dynamics of nonlinear hyperbolic equations of Kirchhoff type

In this paper, we study the initial boundary value problem of important hyperbolic Kirchhoff equation $$\begin{aligned} u_{tt}-\left( a \int _\text{\O}mega |\nabla u|^2 \mathrm {d}x +b\right) \Delta u = \lambda u+ |u|^{p-1}u , \end{aligned}$$ where a, $$b>0$$ $$p>1$$ $$\lambda \in {\mathbb {R}}$$ and energy is arbitrarily large. We prove several new theorems on dynamics such as boundedness or finite time blow-up solution under different range b, $$ data for following cases: (i) $$11/\Lambda (iii) $$a \le 1/\Lambda b\lambda (v) $$p>3$$ b\lambda (vi) > _1 \inf \left\{ \Vert \nabla u\Vert ^2_2 :~ u\in H^1_0(\text{\O}mega )\ \mathrm{and}\ _2 =1\right\} $$\Lambda ^4_2 _4 . Moreover, invariance some stable unstable sets suitable b give sufficient conditions to generate vacuum region solution. Due nonlocal effect caused by integro-differential term, show many interesting differences between phenomenon $$a>0$$ $$a=0$$