## Differential and Integral Equations

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

Yuusuke Sugiyama

#### 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