Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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


Authors: Ding Bingbing, Ingo Witt and Yin Huicheng
Journal: Quart. Appl. Math. 73 (2015), 773-796
MSC (2010): Primary 35L65, 35J70, 35R35
DOI: https://doi.org/10.1090/qam/1410
Published electronically: September 11, 2015
MathSciNet review: 3432283
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Abstract: This paper is concerned with the lifespan of and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $ \partial _t^2u-\sum _{i=1}^2\partial _i(c_i^2(u)\partial _iu)$ $ =0$, where $ c_i(u)\in C^{\infty }(\mathbb{R}^n)$, $ c_i(0)\neq 0$, and $ (c_1'(0))^2+(c_2'(0))^2\neq 0$. This equation has an interesting physical background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $ (u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x))$ with $ u_0(x), u_1(x)\in C_0^{\infty }(\mathbb{R}^2)$, and $ \varepsilon >0$ is small, we will show that the classical solution $ u(t,x)$ stops to be smooth at some finite time $ T_{\varepsilon }$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives $ \nabla _{t,x}u(t,x)$, while $ u(t,x)$ itself is continuous up to the blowup time  $ T_{\varepsilon }$.


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Additional Information

Ding Bingbing
Affiliation: Department of Mathematics and IMS, Nanjing University, Nanjing 210093, People’s Republic of China
Email: 13851929236@163.com

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/qam/1410
Keywords: Nonlinear wave equation, blowup, lifespan, blowup system, Klainerman fields, Nash-Moser-H\"ormander iteration
Received by editor(s): April 21, 2014
Received by editor(s) in revised form: July 19, 2014
Published electronically: September 11, 2015
Additional Notes: The first and third authors were supported by the NSFC (No. 10931007, No. 11025105), by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and by the DFG via the Sino-German project “Analysis of PDEs and Applications.” Some parts of this work were done when the first and third authors were visiting the Mathematical Institute of the University of Göttingen. The third author is the corresponding author.
The second author was supported by the DFG via the Sino-German project “Analysis of PDEs and Applications”.
Article copyright: © Copyright 2015 Brown University

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