Large time behavior of solutions to 1D quasilinear wave equations (Regularity and Singularity for Partial Differential Equations with Conservation Laws)

Abstract

This note is a survey of [17]. We consider the large time behavior of solutions of the Cauchy problem of the quasilinear wave equation: ∂2 t u = ∂x((1 + u)2a∂xu), which describes shearing-motion in elastic-plastic rod. If 1+u(0, x) is bounded away from a positive constant, we can construct a local solution for smooth initial data. When 1+u(t, x) is going to 0 in finite time, the equation degenerates. We give a sufficient condition that the equation degenerates in finite time.