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

Abstract

For the 3D semilinear Euler-Poisson-Darboux equation $\square u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\geq1$, $\mu>0$ 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 $1 p_{crit}(3,\mu)$. 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 $\mu\geq\frac{14}{5}$. In the recent paper [16], the authors have obtained the global small solution $u$ for $p>\max\{\frac{5}{3},1+\frac{2}{\mu}\}$ and $\mu\geq\frac{14}{5}$. 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 $\frac{5}{3} p_{crit}(3,\mu)=\frac{5}{3}$.