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

TL;DR

This paper investigates conditions under which solutions to a quasilinear wave equation in one dimension either exist globally or blow up, including scenarios where the wave speed degenerates, with implications for physical models.

Contribution

It provides new sufficient conditions for global existence and blow-up of solutions to a class of quasilinear wave equations, including degeneracy cases.

Findings

Conditions for global smooth solutions under positive wave speed.

Criteria for blow-up where derivatives become unbounded.

Identification of degeneracy leading to solution breakdown.

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.