论文标题
波动方程的关键指数，具有时间依赖的比例不变性阻尼和立方卷积
Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution
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，该问题与时间有关的规模不变性$ \ frac {2} {1+t} {1+t} \ partial_t v $和一个立方卷积$（| x | x |^{ - γ}*v^2）v $ with $ frac左右（$ frac）初始数据$ \ left的空间维度（v（x，0），\ partial_tv（x，0）\右）\ in C^2（\ Mathbb {r}^3）\ times c^1（\ times c^1（\ times c^1） $ \ mathbb {r}^3 \ times [0，t）$。这里$ t $表示$ v $的最大存在时间。 本文的第一个目的是证明解决方案解决方案的独特全球存在和解决方案在（0,3）$中$γ\ in（0,3）$中的渐近行为，并显示出较低的寿命估计值，在临界或亚临界情况$γ\ in \ in \ in \ in \ eft \ eft \ frac frac {1}} {2} {2} {2} {2} {2}，priection priections priections priections prift for in IS ISSIFERS prift for IS ISSISTICES prections primits。弱情况下的情况较弱，没有阻尼，并通过耗散期限的效果来恢复弱点。 本文的第二个目的是证明较小的数据爆炸，并且在\ lest（ - \ frac {1} {2} {2}，0 \ right）$中，在亚临界情况下具有紧凑的支持的正数据，对寿命的几乎清晰的上限进行了估计。证明的必要部分是完善\ cite {H20}定理6.1证明的论点，以获得寿命的上限。 我们的两个结果确定，将全球存在和小型解决方案爆炸的关键指数$γ_C$为$ 0 $，即$γ_c= 0 $。结果，我们可以看到，由于比例尺不变式缩放术语的影响，关键指数从$ 2 $转移到$ 0 $。
In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping $\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-γ}*v^2)v$ with $γ\in \left(-\frac{1}{2},3\right)$ in three spatial dimension for initial data $\left(v(x,0),\partial_tv(x,0)\right)\in C^2(\mathbb{R}^3)\times C^1(\mathbb{R}^3)$ with a compact support, where $v=v(x,t)$ is an unknown function to the problem on $\mathbb{R}^3\times[0,T)$. Here $T$ denotes a maximal existence time of $v$. The first aim of the present paper is to prove unique global existence of the solution to the problem and asymptotic behavior of the solution in the supercritical case $γ\in (0,3)$, and show a lower estimate of the lifespan in the critical or subcritical case $γ\in \left(-\frac{1}{2},0\right]$. The essential part for their proofs is to derive a weaker estimate under the weaker condition than the case without damping and to recover the weakness by the effect of the dissipative term. The second aim of the present paper is to prove a small data blow-up and the almost sharp upper estimate of the lifespan for positive data with a compact support in the subcritical case $γ\in \left(-\frac{1}{2},0\right)$. The essential part for the proof is to refine the argument for the proof of Theorem 6.1 in \cite{H20} to obtain the upper estimate of the lifespan. Our two results determine that a critical exponent $γ_c$ which divides global existence and blow-up for small solutions is $0$, namely $γ_c=0$. As the result, we can see that the critical exponent shift from $2$ to $0$ due to the effect of the scale invariant damping term.
