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

There is an interesting open question: for the $n$-D ($n\ge 1$) semilinear wave equation with scale-invariant damping $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $μ>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,μ)=\max\{p_s(n+μ), p_f(n)\}$ with $p_{s}(n+μ)=\frac{n+μ+1+\sqrt{(n+μ)^2+10(n+μ)-7}}{2(n+μ-1)}$ and $p_f(n)=1+\frac{2}{n}$. It is noticed that the weak solution $u$ can blow up in finite time when $1<p\le p_{crit}(n,μ)$. In addition, for $n=1$, this open question has been solved recently. We now systematically solve this open problem for $n=2$. As the first part, in the present paper, the global small solution $u$ is established for $p_{s}(2+μ)<p<p_{conf}(2,μ)=\frac{μ+5}{μ+1}$ and $μ\in(0,1)\cup(1,2)$. Our main ingredients are to find the suitable conformal power $p_{conf}(2,μ)$ and derive some new kinds of spacetime-weighted $L^{q}_tL^{q}_x([1, \infty)\times \mathbb{R}^2)$ or $L^q_tL^ν_rL^2_θ([1, \infty)\times [0, \infty)\times [0, 2π])$ Strichartz estimates for the solutions of linear generalized Tricomi equation $\partial_t^2v-t^mΔv=F(t,x)$ ($m>0$). In forthcoming papers, we shall show the global existence of small solution $u$ for the remaining cases of $p>1$ and $μ>0$.