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: ${\partial }_t^{2}u=c(u{)}^2{\partial }_x^{2}u+\lambda c(u)c^{\prime }(u)({\partial }_{x}u{)}^2$ 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 $0\le \lambda <2$ degenerates in finite time.

14 pages, 1 figure. Some typos are fixed. The proof in the case u_1 does not belong to L^1 added

References in corpus (1)

• Global Conservative Solutions to a Nonlinear Variational Wave Equation