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arXiv · 1601.05191

Degeneracy in finite time of 1D quasilinear wave equations II

Yuusuke Sugiyama
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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.

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Yuusuke Sugiyama. 2016-01-20. Degeneracy in finite time of 1D quasilinear wave equations II. https://arxiv.org/abs/1601.05191

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