论文标题
在3D中用立方卷积的波方程的临界衰减
On the critical decay for the wave equation with a cubic convolution in 3D
论文作者
论文摘要
我们在三个空间维度中考虑了带有立方卷积$ \ partial_t^2 u-ΔU=（| x |^{ - γ}*u^2）U $的波动方程。在这里，$ 0 <γ<3 $和$*$在太空变量中的卷积代表。众所周知，如果初始数据平滑，小且紧凑，则$γ\ ge2 $确保了解决方案的独特全球存在。另一方面，也众所周知，在有限的时间内，解决方案的衰减率也不够快，即使$ 2 \leγ<3 $也不够快。在本文中，我们认为cauchy问题的价格为$ 2 \leγ<3 $在时空加权$ l^\ infty $空间中，功能具有关键的衰减率。当$γ= 2 $时，我们给出了寿命的最佳估计。这给出了库拜猜想的肯定答案（请参阅Kubo（2004）的定理2.1之后的备注）。当$ 2 <γ<3 $时，我们还证明了用于小数据的独特全球解决方案。
We consider the wave equation with a cubic convolution $\partial_t^2 u-Δu=(|x|^{-γ}*u^2)u$ in three space dimensions. Here, $0<γ<3$ and $*$ stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then $γ\ge2$ 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 $2\le γ<3$. In this paper, we consider the Cauchy problem for $2\le γ<3$ in the space-time weighted $L^\infty$ space in which functions have critical decay rate. When $γ=2$, 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 $2<γ<3$, we also prove unique global existence of solutions for small data.
