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Advances in Mathematical Physics
Volume 2017 (2017), Article ID 1620417, 18 pages
https://doi.org/10.1155/2017/1620417
Research Article

Global Well-Posedness for a Class of Kirchhoff-Type Wave System

Department of Mathematics, Bohai University, Jinzhou, Liaoning 121013, China

Correspondence should be addressed to Xiaoli Jiang

Received 6 June 2017; Revised 9 August 2017; Accepted 8 November 2017; Published 7 December 2017

Academic Editor: Rehana Naz

Copyright © 2017 Xiaoli Jiang and Xiaofeng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In recent years, the small initial boundary value problem of the Kirchhoff-type wave system attracts many scholars’ attention. However, the big initial boundary value problem is also a topic of theoretical significance. In this paper, we devote oneself to the well-posedness of the Kirchhoff-type wave system under the big initial boundary conditions. Combining the potential well method with an improved convex method, we establish a criterion for the well-posedness of the system with nonlinear source and dissipative and viscoelastic terms. Based on the criteria, the energy of the system is divided into different levels. For the subcritical case, we prove that there exist the global solutions when the initial value belongs to the stable set, while the finite time blow-up occurs when the initial value belongs to the unstable set. For the supercritical case, we show that the corresponding solution blows up in a finite time if the initial value satisfies some given conditions.