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

For the $2$-D semilinear wave equation with scale-invariant damping ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u+\frac{\mu }{t}{\mathrm{\partial }}_{t}u=\mathrm{\mid }u{\mathrm{\mid }}^p$, where $t\ge 1$ and $p>1$, in the paper [T. Imai, M. Kato, H. Takamura, K. Wakasa, The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions, J. Differential Equations 269 (2020), no. 10, 8387-8424], 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 $\mu \in (0,2)$ and $p>p_f(2)=2$ for $\mu \ge 2$. In our previous paper, the global small solution $u$ has been obtained for $p_s(2+\mu )<p<p_{conf}(2,\mu )=\frac{\mu +5}{\mu +1}$ and $\mu \in (0,1)\cup (1,2)$. In the present paper, we will show the global existence of small solution $u$ for $p\ge p_{conf}(2,\mu )$ and $\mu \in (0,1)\cup (1,2)$. In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $\mu >2,p>2$ or $\mu =1,p>p_s(\mu +2)=1+\sqrt{2}$.