论文标题
波动方程的小数据爆炸，具有时间依赖的量表不变性阻尼和一个立方卷积，用于缓慢衰减初始数据
Small data blow-up for the wave equation with a time-dependent scale invariant damping and a cubic convolution for slowly decaying initial data
论文作者
论文摘要
在本文中，我们研究了波动方程的库奇问题，并具有时间相关的规模不变阻尼，即$ \ frac {2} {1+t} {1+t} \ partial_t v $和立方卷积$（| x | x | x |^{ - γ}*v^2）v $ in（0，n Newlennal in $ c in（nes） $ \ mathbb {r}^n \ times [0，t）$。我们本文的目的是证明一个小数据爆炸结果，并显示出对问题的寿命的上限，以缓慢衰减阳性的初始数据$（v（x，0），\ partial_t v（x，0））$，例如$ \ partial_t v（x，0）这里$ν$属于缩放超临界情况$ν<\ frac {n-γ} {2} $。我们的主要贡献是在高空间维度（即$ n \ ge 4 $）中估算卷积期限。本文是在高空间维度（$ n \ ge 4 $）中使用立方卷积处理波动方程的第一个爆炸结果。
In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e.$\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-γ}*v^2)v$ with $γ\in (0,n)$, where $v=v(x,t)$ is an unknown function on $\mathbb{R}^n\times[0,T)$. Our aim of the present paper is to prove a small data blow-up result and show an upper estimate of lifespan of the problem for slowly decaying positive initial data $(v(x,0),\partial_t v(x,0))$ such as $\partial_t v(x,0)=O(|x|^{-(1+ν)})$ as $|x|\rightarrow\infty$. Here $ν$ belongs to the scaling supercritical case $ν<\frac{n-γ}{2}$. Our main new contribution is to estimate the convolution term in high spatial dimensions, i.e. $n\ge 4$. This paper is the first blow-up result to treat wave equations with the cubic convolution in high spatial dimensions ($n\ge 4$).
