Dynamics of nonlinear hyperbolic equations of Kirchhoff type

TL;DR

This paper investigates the dynamics of nonlinear hyperbolic Kirchhoff equations, establishing conditions for solution boundedness or blow-up across various parameter regimes, highlighting the effects of nonlocal terms.

Contribution

It provides new theorems on solution behavior, invariance of solution sets, and conditions for vacuum regions, emphasizing differences caused by nonlocal effects.

Findings

Conditions for finite time blow-up and boundedness of solutions.

Invariance of stable and unstable solution sets.

Differences in blow-up phenomena due to nonlocal effects.

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 $a=0$.