Global small data radial symmetric solutions of 3D semilinear Euler-Poisson-Darboux equations

Abstract

For the 3D semilinear Euler-Poisson-Darboux equation $\square u+\frac{\mu }{t}{\partial }_{t}u=|u{|}^p$, where $t\ge 1$, $\mu >0$ and $p>1$, it is conjectured that there is a critical exponent $p_{crit}(3,\mu )=\max \{p_s(3+\mu ),p_f(3)\}$ with the Strauss exponent $p_s(3+\mu )=\frac{\mu +4+\sqrt{{\mu }^2+16\mu +32}}{2(\mu +2)}$ and the Fujita exponent $p_f(3)=\frac{5}{3}$ such that when $p>p_{crit}(3,\mu )$, the small data solution $u$ exists globally, otherwise, when $1<p\le p_{crit}(3,\mu )$, the solution $u$ can blow up in finite time. It is pointed out that for $1<p\le p_{crit}(3,\mu )$, the blowup of solution $u$ has been shown. However, it is still open for the global existence of small solution $u$ when $p>p_{crit}(3,\mu )$. Note that $p_{crit}(3,\mu )=p_s(3+\mu )$ for $0<\mu <\frac{14}{5}$ and $p_{crit}(3,\mu )=p_f(3)$ for $\mu \ge \frac{14}{5}$. In the recent paper [16], the authors have obtained the global small solution $u$ for $p>\max \{\frac{5}{3},1+\frac{2}{\mu }\}$ and $\mu \ge \frac{14}{5}$. 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 $\frac{5}{3}<p\le 1+\frac{2}{\mu }$ and $\frac{14}{5}\le \mu <3$. Therefore, for the radially symmetric case and $\mu \ge \frac{14}{5}$, the global existence problem of small data solution $u$ is solved when $p>p_{crit}(3,\mu )=\frac{5}{3}$.