Global existence of small data solutions to 3-D semilinear Euler-Poisson-Darboux equations

Abstract

There is an interesting open question: for $n$-D ($n\ge 1$) semilinear Euler-Poisson-Darboux equation ${\partial }_t^{2}u-\hat{I}”u+\frac{\hat{I}}{¼}t{\partial }_{t}u=|u{|}^p$, where $t\ge 1$, $p>1$ and $\hat{I}¼>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,\hat{I}¼)=\max \{p_s(n+\hat{I}¼),p_f(n)\}$ with the Strauss exponent $p_s(n+\hat{I}¼)=\frac{n+\hat{I}¼+1+\sqrt{(n+\hat{I}¼{)}^2+10(n+\hat{I}¼)-7}}{2(n+\hat{I}¼-1)}$ and the Fujita exponent $p_f(n)=1+\frac{2}{n}$. The blowup of weak solution $u$ has been shown when $1<p\le p_{crit}(n,\hat{I}¼)$ meanwhile this open question has been solved for $n=1,2$. In the present paper, we focus on this open question for $n=3$ and establish the global existence of small data solution $u$ for $\hat{I}¼\ge \frac{14}{5}$ (equivalent to $p_{crit}(3,\hat{I}¼)=p_f(3)=\frac{5}{3}$) and $p>\max \{\frac{5}{3},1+\frac{2}{\hat{I}}¼\}$.