On the critical decay for the wave equation with a cubic convolution in 3D
On the critical decay for the wave equation with a cubic convolution in 3D
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DOI：
10.3934/dcds.2021048
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发表时间：
2020-09
影响因子：
1.1
通讯作者：
Tomoyuki Tanaka;Kyouhei Wakasa
中科院分区：
文献类型：
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作者：
Tomoyuki Tanaka;Kyouhei Wakasa
我们考虑三维空间中具有立方卷积的波动方程$\Partial_t^2u-\Delta u=(|x|^{-\Gamma}*u^2)u$。这里，$0<；\Gamma<；3$和$*$代表空间变量的卷积。众所周知，如果初始数据是光滑的、小的和紧支撑的，则$\Gamma\ge2$确保解的唯一的全局存在性。另一方面，众所周知，对于衰减率不够快的初始数据，解在有限时间内爆破，即使是在$2\le\Gamma<；3$时。在本文中，我们考虑函数具有临界衰减率的时空加权$L空间中$2\le\r<；3$的柯西问题。当$\Gamma=2$时，我们给出了寿命的最优估计。这对Kubo猜想给出了肯定的回答(见Kubo(2004)中定理2.1之后的注解)。当$2<；\Gamma<；3$时，我们还证明了小数据解决方案在全球范围内的独一无二的存在。
We consider the wave equation with a cubic convolution $\partial_t^2 u-\Delta u=(|x|^{-\gamma}*u^2)u$ in three space dimensions. Here, $0<\gamma<3$ and $*$ stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then $\gamma\ge2$ assures unique global existence of solutions. On the other hand, it is also well known that solutions blow up in finite time for initial data whose decay rate is not rapid enough even when $2\le \gamma<3$. In this paper, we consider the Cauchy problem for $2\le \gamma<3$ in the space-time weighted $L^\infty$ space in which functions have critical decay rate. When $\gamma=2$, we give an optimal estimate of the lifespan. This gives an affirmative answer to the Kubo conjecture (see Remark right after Theorem 2.1 in Kubo(2004)). When $2<\gamma<3$, we also prove unique global existence of solutions for small data.