Degeneracy in finite time of 1D quasilinear wave equations Ⅱ

Abstract

We consider the large time behavior of solutions to the following nonlinear wave equation: ∂ 2 t u = c(u) 2 ∂ 2 x u + λc(u)c ′ (u)(∂xu) 2 with the parameter λ ∈ [0, 2]. If c(u(0, x)) is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if c(•) has a zero point, then c(u(t, x)) can be going to zero in finite time. When c(u(t, x)) is going to 0, the equation degenerates. We give a sufficient condition so that the equation with 0 ≤ λ < 2 degenerates in finite time.