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

Abstract

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