The small data solutions of general 3-D quasilinear wave equations. II
The small data solutions of general 3-D quasilinear wave equations. II
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DOI：
10.1016/j.jde.2016.04.002
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发表时间：
2016-07
影响因子：
2.4
通讯作者：
Bingbing Ding;I. Witt;Huicheng Yin
中科院分区：
文献类型：
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作者：
Bingbing Ding;I. Witt;Huicheng Yin
本文是文献[8]的继续，在文[8]中，作者证明了一般三维拟线性波动方程∑i，j=0 3gi j(u，∂u)∂i j 2 u=0在弱零条件下的整体光滑小数据解的存在性.本文证明了当弱零条件不成立且初值满足一般非退化条件时，方程∑i，j=0 3gi j(u，∂u)∂i j 2 u=0的光滑小数据解将在有限时间内爆破，并且完全确定了精确爆破时间.因此，结合文[8]中的主要结果，我们对三维拟线性波动方程∑i，j=0 3gi j(u，∂u)∂i j 2 u=0的爆破或整体解的存在性进行了基本完整的研究。
This paper is a continuation of the work in [8], where the authors established the global existence of smooth small data solutions to the general 3-D quasilinear wave equation∑ i, j= 0 3 g i j (u,∂ u)∂ i j 2 u= 0 when the weak null condition holds. In the present paper, we show that the smooth small data solutions of equation∑ i, j= 0 3 g i j (u,∂ u)∂ i j 2 u= 0 will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied, moreover, a precise blowup time is completely determined. Therefore, collecting the main results in this paper and [8], we have given a basically complete study on the blowup or global existence of small data solutions to the 3-D quasilinear wave equation∑ i, j= 0 3 g i j (u,∂ u)∂ i j 2 u= 0.