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 $({\left|x\right|}^{-\gamma }\ast v^2)v$ with $\gamma \in (0,n)$, where $v=v(x,t)$ is an unknown function on ${\mathbf{R}}^n\times [0,T)$. 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 ${\partial }_{t}v(x,0)=O\left({\left|x\right|}^{-(1+\nu )}\right)$ as $\left|x\right|\to \infty$. Here $\nu$ belongs to the scaling supercritical case $\nu <\frac{n-\gamma }{2}$. The proof of our main result is based on the combination of the arguments in the papers Takamura et al. (2010) and Takamura (1995). Especially, 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 ($n\ge 4$).

在本文中，我们研究了具有时变标度不变阻尼的波动方程的柯西问题，即 $\frac{2}{\mathrm{1个}+Ť}{\partial }_{Ť}v$ 和三次卷积 $（{\left|X\right|}^{-\gamma }\ast v^2）v$ 与 $\gamma \in （0，ñ）$，在哪里 $v=v（X，Ť）$ 是一个未知函数 ${\mathbf{[R}}^{ñ}\times [0，Ť）$。我们本文的目的是证明一个小的数据爆炸结果，并为缓慢衰减正初始数据显示该问题的寿命的较高估计。$（v（X，0），{\partial }_{Ť}v（X，0））$ 如 ${\partial }_{Ť}v（X，0）=Ø（{\left|X\right|}^{-（\mathrm{1个}+\nu ）}）$ 如 $\left|X\right|\to \infty$。这里$\nu$ 属于规模超临界情况 $\nu <\frac{ñ-\gamma }{2}$。我们主要结果的证明是基于Takamura等人论文的论点组合。（2010年）和高村（1995年）。特别是，我们的主要新贡献是在高空间维度上估算卷积项，即$ñ\ge 4$。本文是在高空间维度上用三次卷积处理波动方程的第一个爆炸结果（$ñ\ge 4$）。