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 $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $μ>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,μ)=\max\{p_s(n+μ), p_f(n)\}$ with the Strauss exponent $p_{s}(n+μ)=\frac{n+μ+1+\sqrt{(n+μ)^2+10(n+μ)-7}}{2(n+μ-1)}$ and the Fujita exponent $p_f(n)=1+\frac{2}{n}$. The blowup of weak solution $u$ has been shown when $1<p\le p_{crit}(n,μ)$ 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 $μ\geq\frac{14}{5}$ (equivalent to $p_{crit}(3,μ)=p_f(3)=\frac53$) and $p>\max\{\frac53, 1+\frac{2}μ\}$.