Differential and Integral Equations

Global existence of solutions to some quasilinear wave equation in one space dimension

Yuusuke Sugiyama

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Under some conditions on the initial data, we show the global existence of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = c(u)^2 \partial^2 _x u+\lambda c(u)c'(u)(\partial_x u)^2$ for $0 \leq \lambda\leq 2$, which has richly physical background. In [16], P. Zhang and Y. Zheng show the global existence of solutions to this equation with $\lambda =1$. It is difficult to apply their method to this global existence problem of this equation with $\lambda =0$ directly. As an application of the theorem of the global existence of solutions to $\partial_{t}^2 u = c(u)^2 \partial^2 _x u$, we construct solutions to $\partial_{t}^2 u = (u+1)^2 \partial^2 _x u$ which is going to $-1$ in finite time with $u(0,x)+1\geq \delta $ for some $\delta>0 $.

Article information

Source
Differential Integral Equations, Volume 26, Number 5/6 (2013), 487-504.

Dates
First available in Project Euclid: 14 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1363266076

Mathematical Reviews number (MathSciNet)
MR3086397

Zentralblatt MATH identifier
1299.35196

Subjects
Primary: 35L15: Initial value problems for second-order hyperbolic equations 35A01: Existence problems: global existence, local existence, non-existence

Citation

Sugiyama, Yuusuke. Global existence of solutions to some quasilinear wave equation in one space dimension. Differential Integral Equations 26 (2013), no. 5/6, 487--504. https://projecteuclid.org/euclid.die/1363266076


Export citation