Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations

Bingbing Ding; Ingo Witt; Huicheng Yin

doi:10.3934/cpaa.2017035

Article Contents

2017, Volume 16, Issue 3: 719-744. Doi: 10.3934/cpaa.2017035

This issue Previous Article Next Article On the decay and stability of global solutions to the 3D inhomogeneous MHD system

Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations

1.
School of Mathematical Sciences, Jiangsu Provincial Key Laboratory for Numerical, Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing, 210023, China
2.
Mathematical Institute, University of Göttingen, Bunsenstr. 3-5, Göttingen, D-37073, Germany

^* Corresponding author
^* Corresponding author

Received: January 2016

Revised: March 2016

Published online: February 2016

The first author and the third author were supported by the NSFC (No. 11571177) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The second author was supported by the DFG via the Sino-German project "Analysis of PDEs and application.".

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Abstract

For the 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (\nabla u)\partial _{ij}^2u = 0$, whose coefficients are independent of the solution $u$, the blowup result of small data solution has been established in ^[1,2] when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\sum\nolimits_{i,j = 0}^2 {{g_{ij}}} (u,\nabla u)\partial _{ij}^2u = 0$, whose coefficients depend on $u$ and $\nabla u$ simultaneously. When the first weak null condition is not fulfilled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution $u$ blows up in finite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.

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Mathematics Subject Classification: 35L05, 35L72.

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