Degeneracy in finite time of 1D quasilinear wave equations Ⅱ

Yuusuke Sugiyama

doi:10.3934/eect.2017031

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2017, Volume 6, Issue 4: 615-628. Doi: 10.3934/eect.2017031

This issue Previous Article Exact controllability results for a class of abstract nonlocal Cauchy problem with impulsive conditions Next Article Asymptotic profile of solutions to the linearized double dispersion equation on the half space $\mathbb{R}^{n}_{+}$

Degeneracy in finite time of 1D quasilinear wave equations Ⅱ

Yuusuke Sugiyama

Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjuku-ku, Tokyo 162-8601, Japan

Received: January 2017

Revised: August 2017

Published: December 2017

The first author is supported by Grant-in-Aid for Young Scientists Research (B), 16K17631.

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Abstract

We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2 _x u + λ c(u)c'(u)(\partial_x u)^2$ 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 $0≤q λ < 2$ degenerates in finite time.

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Mathematics Subject Classification: Primary: 35L15, 35A01; Secondary: 35B44, 35L65.

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Figure 1. the two characteristic curves on the $(x,t)$ plane

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