arXiv 2003.103292020-03-24math.APmath.CA

具有不定常尺度不变阻尼和立方卷积的wave equation的临界指数

Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution

Masahiro Ikeda, Tomoyuki Tanaka, Kyouhei Wakasa

摘要：

本文研究了具有不定常尺度不变阻尼$\frac{2}{1+t}\partial_t v$和三维空间上$(|x|^{-γ}*v^2)v$与$γ\in \left(-\frac{1}{2},3\right)$的三次卷积的波动方程的Cauchy问题，初始数据为紧支集$\left(v(x,0),\partial_tv(x,0)\right)\in C^2(\mathbb{R}^3)\times C^1(\mathbb{R}^3)$，其中$v=v(x,t)$是$\mathbb{R}^3\times[0,T)$上的未知函数.这里，$T$表示$v$的最大存在时间。本文的第一个目的是证明该问题解的唯一整体存在性和解在超临界情形$γ\in (0,3)$下的渐近性态，并给出临界或亚临界情形$γ\in \left(-\frac{1}{2},0\right]$下寿命的较低估计.其证明的关键部分是在比无阻尼情况更弱的条件下得到一个更弱的估计，并通过耗散项的作用恢复这种弱点。本文的第二个目的是证明一个小的数据爆破和几乎尖锐的上估计的寿命的正数据与紧支持下的亚临界情况下$γ\in \left(-\frac{1}{2},0\right)$。证明的基本部分是在\cite{H20}中改进定理6.1的证明的论点，以获得寿命的上限估计。我们的两个结果确定了区分小解整体存在和爆破的临界指数$γ_c$是$0$，即$γ_c=0$。结果表明，由于尺度不变阻尼项的影响，临界指数从$2$移动到$0$。

英文摘要：

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.

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