Degeneracy in finite time of 1D quasilinear wave equations II

TL;DR

This paper investigates the finite-time degeneracy of solutions to a class of 1D quasilinear wave equations, showing conditions under which the solutions degenerate due to the vanishing of the wave speed function.

Contribution

It provides a sufficient condition for finite-time degeneracy in solutions of a nonlinear wave equation with variable wave speed, extending understanding of wave behavior with degeneracy.

Findings

Solutions degenerate in finite time when the wave speed function approaches zero.

Degeneracy occurs for equations with parameter λ in [0, 2) under certain conditions.

The equation remains well-posed locally if the wave speed stays away from zero.

Abstract

We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2_x u + \lambda c(u)c'(u)(\partial_x u)^2$ with the parameter $\lambda \in [0,2]$. If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(\cdot )$ 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 that the equation with $0\leq \lambda < 2$ degenerates in finite time.