Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On the blowup and lifespan of smooth solutions to a class of 2-D nonlinear wave equations with small initial data


Authors: Li Jun, Ingo Witt and Yin Huicheng
Journal: Quart. Appl. Math. 73 (2015), 219-251
MSC (2010): Primary 35L15, 35L65, 35L70
DOI: https://doi.org/10.1090/S0033-569X-2015-01374-2
Published electronically: January 29, 2015
MathSciNet review: 3357493
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Abstract: We are concerned with a class of two-dimensional nonlinear wave equations $ \partial _t^2u-$$ \text {div}(c^2(u)\nabla u)=0$ or $ \partial _t^2u-c(u)\mathrm {div}(c(u)\nabla u)=0$ with small initial data $ (u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x))$, where $ c(u)$ is a smooth function, $ c(0)\not =0$, $ x\in \mathbb{R}^2$, $ u_0(x), u_1(x)\in C_0^{\infty }(\mathbb{R}^2)$ depend only on $ r=\sqrt {x_1^2+x_2^2}$, and $ \varepsilon >0$ is sufficiently small. Such equations arise in a pressure-gradient model of fluid dynamics, as well as in a liquid crystal model or other variational wave equations. When $ c'(0)\not = 0$ or $ c'(0)=0$, $ c''(0)\not = 0$, we establish blowup and determine the lifespan of smooth solutions.


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Additional Information

Li Jun
Affiliation: Department of Mathematics and IMS, Nanjing University, Nanjing 210093, People’s Republic of China
Email: lijun@nju.edu.cn

Ingo Witt
Affiliation: Mathematical Institute, University of Göttingen, Bunsenstr. 3-5, D-37073 Göttingen, Germany
Email: iwitt@uni-math.gwdg.de

Yin Huicheng
Affiliation: Department of Mathematics and IMS, Nanjing University, Nanjing 210093, People’s Republic of China
Email: huicheng@nju.edu.cn

DOI: https://doi.org/10.1090/S0033-569X-2015-01374-2
Keywords: Nonlinear wave equation, blowup, lifespan, Klainerman-Sobolev inequality
Received by editor(s): January 30, 2013
Published electronically: January 29, 2015
Additional Notes: The first and third authors were supported by the NSFC (No. 10931007, No. 11025105, No. 11001122), by the Doctoral Program Foundation of the Ministry of Education of China (No. 20090091110005), and by the DFG via the joint Sino-German project “Analysis of PDEs and application”. This work was done when the first and third authors were visiting the Mathematical Institute of the University of Göttingen. The second author was partially supported by the DFG via the Sino-German project “Analysis of PDEs and application”. The second author is the corresponding author.
Article copyright: © Copyright 2015 Brown University


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