Dynamics of nonlinear hyperbolic equations of Kirchhoff type

Abstract

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$u_{tt}-\left(a \int_\Omega |\nabla u|^2 \dif x +b\right)\Delta u = \lambda u+ |u|^{p-1}u ,$$ where $a$, $b>0$, $p>1$, $\lambda \in \mathbb{R}$ and the initial energy is arbitrarily large. We prove several new theorems on the dynamics such as the boundedness or finite time blow-up of solution under the different range of $a$, $b$, $\lambda$ and the initial data for the following cases: (i) $1<p<3$, (ii) $p=3$ and $a>1/\Lambda$, (iii) $p=3$, $a \leq 1/\Lambda$ and $\lam <b\lam_1$, (iv) $p=3$, $a < 1/\Lambda$ and $\lam >b\lam_1$, (v) $p>3$ and $\lam\leq b\lam_1$, (vi) $p>3$ and $\lam> b\lam_1$, where $\lam_1 = \inf\left\{\|\nabla u\|^2_2 :~ u\in H^1_0(\Omega)\ {\rm and}\ \|u\|_2 =1\right\}$, and $\Lambda = \inf\left\{\|\nabla u\|^4_2 :~ u\in H^1_0(\Omega)\ {\rm and}\ \|u\|_4 =1\right\}$. Moreover, we prove the invariance of some stable and unstable sets of the solution for suitable $a$, $b$ and $\lam$, and give the sufficient conditions of initial data to generate a vacuum region of the solution. Due to the nonlocal effect caused by the nonlocal integro-differential term, we show many interesting differences between the blow-up phenomenon of the problem for $a>0$ and $a=0$.