Global small data weak solutions of 2-D semilinear wave equations with scale-invariant damping, I

Abstract

There is an interesting open question: for the $n$-D ($n\ge 1$) semilinear wave equation with scale-invariant damping $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, 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 $p_{s}(n+\mu)=\frac{n+\mu+1+\sqrt{(n+\mu)^2+10(n+\mu)-7}}{2(n+\mu-1)}$ and $p_f(n)=1+\frac{2}{n}$. It is noticed that the weak solution $u$ can blow up in finite time when $1 0$). In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $p>1$ and $\mu>0$.