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

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 $.