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

For the $2$-D semilinear wave equation with scale-invariant damping $\square u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\geq 1$, $\mu>0$ and $p>1$, it is conjectured that the global small data weak solution $u$ exists when $p>p_{s}(2+\mu) =\frac{\mu+3+\sqrt{\mu^2+14\mu+17}}{2(\mu+1)}$ for $0<\mu\leq 2$ and $p>p_f(2)=2$ for $\mu\geq 2$. In our previous papers, the global small solution $u$ has been obtained for $p>p_{s}(2+\mu)$ and $0<\mu<2$ but $\mu\not=1$. In the present paper, by the vector field method together with the delicate analysis on the Bessel functions, we will show the global existence of small solution $u$ for $p>2$ and $\mu>2$. In forthcoming paper, for $\mu=1$ and $p>p_{s}(2+\mu)=p_{s}(3)=1+\sqrt 2$, the global solution $u$ is also obtained. Therefore, collecting our series of conclusions together with partial results from others, this open question has been solved completely.