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 $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, 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 $\mu\geq 2$. 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 $p\geq p_{conf}(2,\mu)$ 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 $\mu=1, p>p_s(\mu+2)=1+\sqrt 2$.