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