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 $c(u(0,x))\ge \delta$ 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.

arXiv comment11 pages

References in corpus (1)

• On the blowup and lifespan of smooth solutions to a class of 2-D nonlinear wave equations with small initial data