May/June 2013 Global existence of solutions to some quasilinear wave equation in one space dimension
Yuusuke Sugiyama
Differential Integral Equations 26(5/6): 487-504 (May/June 2013). DOI: 10.57262/die/1363266076

Abstract

Under some conditions on the initial data, we show the global existence of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = c(u)^2 \partial^2 _x u+\lambda c(u)c'(u)(\partial_x u)^2$ for $0 \leq \lambda\leq 2$, which has richly physical background. In [16], P. Zhang and Y. Zheng show the global existence of solutions to this equation with $\lambda =1$. It is difficult to apply their method to this global existence problem of this equation with $\lambda =0$ directly. As an application of the theorem of the global existence of solutions to $\partial_{t}^2 u = c(u)^2 \partial^2 _x u$, we construct solutions to $\partial_{t}^2 u = (u+1)^2 \partial^2 _x u$ which is going to $-1$ in finite time with $u(0,x)+1\geq \delta $ for some $\delta>0 $.

Citation

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Yuusuke Sugiyama. "Global existence of solutions to some quasilinear wave equation in one space dimension." Differential Integral Equations 26 (5/6) 487 - 504, May/June 2013. https://doi.org/10.57262/die/1363266076

Information

Published: May/June 2013
First available in Project Euclid: 14 March 2013

zbMATH: 1299.35196
MathSciNet: MR3086397
Digital Object Identifier: 10.57262/die/1363266076

Subjects:
Primary: 35A01 , 35L15

Rights: Copyright © 2013 Khayyam Publishing, Inc.

Vol.26 • No. 5/6 • May/June 2013
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