# American Institute of Mathematical Sciences

May  2017, 16(3): 719-744. doi: 10.3934/cpaa.2017035

## 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

Received  January 2016 Revised  March 2016 Published  February 2016

Fund Project: 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.".

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.

Citation: Bingbing Ding, Ingo Witt, Huicheng Yin. Blowup time and blowup mechanism of small data solutions to general 2-D quasilinear wave equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 719-744. doi: 10.3934/cpaa.2017035
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