发表机构
Xinjiang University; Yili Normal University; Nanjing Normal University（新疆大学; 伊犁师范大学; 南京师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对三维半线性Euler-Poisson-Darboux方程的临界指数推测，利用超几何Riemann表示与精细逐点时空加权估计，证明了特定参数范围内小数据径向对称解的全局存在性，解决了对应条件下的开放问题。
AI 中文摘要
针对三维半线性Euler-Poisson-Darboux方程$\boldsymbol{\u25a1 u+\frac{\u03bc}{t}\u2202_tu=|u|^p}$（其中$t\boldsymbol{\u22651}$、$\boldsymbol{\u03bc>0}$且$\boldsymbol{p>1}$），学界推测存在临界指数$p_{crit}(3,\u03bc)=\u200b\boldsymbol{\u200bmax}\boldsymbol{\brace{p_s(3+\u03bc), p_f(3)}}$，其中Strauss指数$p_s(3+\u03bc)=\frac{\u03bc+4+\u221a{\u03bc^2+16\u03bc+32}}{2(\u03bc+2)}$，Fujita指数$p_f(3)=\frac{5}{3}$；该推测认为当$p>p_{crit}(3,\u03bc)$时小数据解$u$全局存在，当$1<p\u2264 p_{crit}(3,\u03bc)$时解$u$会在有限时间内爆破。目前已证实$1<p\u2264 p_{crit}(3,\u03bc)$时解$u$会爆破，但$p>p_{crit}(3,\u03bc)$时小解$u$的全局存在性仍为开放问题。注意当$0<\u03bc<\frac{14}{5}$时$p_{crit}(3,\u03bc)=p_s(3+\u03bc)$，当$\u03bc\u2265\frac{14}{5}$时$p_{crit}(3,\u03bc)=p_f(3)$。近期文献[16]的作者已得到$p>\u200b\boldsymbol{\u200bmax}\boldsymbol{\brace{\frac{5}{3},1+\frac{2}{\u03bc}}}$且$\u03bc\u2265\frac{14}{5}$条件下的全局小解$u$。本文利用超几何Riemann表示，通过建立若干精细的逐点时空加权估计，证明了剩余参数范围$\frac{5}{3}<p\u22641+\frac{2}{\u03bc}$且$\frac{14}{5}\u2264\u03bc<3$内小数据径向解$u$的全局存在性。由此，在径向对称情形且$\u03bc\u2265\frac{14}{5}$的条件下，$p>p_{crit}(3,\u03bc)=\frac{5}{3}$时的小数据解$u$全局存在性问题得到解决。
英文摘要
For the 3D semilinear Euler-Poisson-Darboux equation $\square u+\fracμ{t}\partial_tu=|u|^p$, where $t\geq1$, $μ>0$ and $p>1$, it is conjectured that there is a critical exponent $p_{crit}(3,μ)=\max\{p_s(3+μ), p_f(3)\}$ with the Strauss exponent $p_s(3+μ)=\frac{μ+4+\sqrt{μ^2+16μ+32}}{2(μ+2)}$ and the Fujita exponent $p_f(3)=\frac{5}{3}$ such that when $p>p_{crit}(3,μ)$, the small data solution $u$ exists globally, otherwise, when $1<p\le p_{crit}(3,μ)$, the solution $u$ can blow up in finite time. It is pointed out that for $1<p\le p_{crit}(3,μ)$, the blowup of solution $u$ has been shown. However, it is still open for the global existence of small solution $u$ when $p>p_{crit}(3,μ)$. Note that $p_{crit}(3,μ)=p_s(3+μ)$ for $0<μ<\frac{14}{5}$ and $p_{crit}(3,μ)=p_f(3)$ for $μ\geq\frac{14}{5}$. In the recent paper [16], the authors have obtained the global small solution $u$ for $p>\max\{\frac{5}{3},1+\frac{2}μ\}$ and $μ\geq\frac{14}{5}$. In this paper, by utilizing the hypergeometric Riemann representation and establishing some delicate pointwise spacetime weighted estimates, we prove the global existence of small data radial solution $u$ in the remaining range of $\frac{5}{3}<p\leq1+\frac{2}μ$ and $\frac{14}{5}\leqμ<3$. Therefore, for the radially symmetric case and $μ\geq\frac{14}{5}$, the global existence problem of small data solution $u$ is solved when $p>p_{crit}(3,μ)=\frac{5}{3}$.
