Global existence and blow-up of solutions to some quasilinear wave equation in one space dimension

Abstract

We consider the global existence and blow up of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = \partial_x(c(u)^2 \partial_x u)$, which has richly physical backgrounds. Under the assumption that $c(u(0,x))\geq \delta$ for some $\delta>0$, we give sufficient conditions for the existence of global smooth solutions and the occurrence of two types of blow-up respectively. One of the two types is that $L^{\infty}$-norm of $\partial_t u$ or $\partial_x u$ goes up to the infinity. The other type is that $c(u)$ vanishes, that is, the equation degenerates.