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 $\mathrm{\square }u+\frac{\mu }{t}{\mathrm{\partial }}_{t}u=\mathrm{\mid }u{\mathrm{\mid }}^p$, where $t\ge 1$, $\mu >0$ 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 $0<\mu \le 2$ and $p>p_f(2)=2$ for $\mu \ge 2$. In our previous papers, the global small solution $u$ has been obtained for $p>p_s(2+\mu )$ and $0<\mu <2$ but $\mu ≠1$. 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.