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 $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, 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 $p>p_{s}(2+μ) =\frac{μ+3+\sqrt{μ^2+14μ+17}}{2(μ+1)}$ for $μ\in (0, 2)$ and $p>p_f(2)=2$ for $μ\geq 2$. In our previous paper, the global small solution $u$ has been obtained for $p_{s}(2+μ)<p<p_{conf}(2,μ)=\frac{μ+5}{μ+1}$ and $μ\in(0,1)\cup(1,2)$. In the present paper, we will show the global existence of small solution $u$ for $p\geq p_{conf}(2,μ)$ and $μ\in(0,1)\cup(1,2)$. In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $μ>2, p>2$ or $μ=1, p>p_s(μ+2)=1+\sqrt 2$.