Global existence of small data weak solutions to the semilinear wave equations with time-dependent scale-invariant damping

In this paper, we are concerned with the global existence of small data weak solutions to the $n-$dimensional semilinear wave equation ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u+\frac{\mu }{t}{\mathrm{\partial }}_{t}u=\mathrm{\mid }u{\mathrm{\mid }}^p$ with time-dependent scale-invariant damping, where $n\ge 2$, $t\ge 1$, $\mu \in (0,1)\cup (1,2]$ and $p>1$. This equation can be changed into the semilinear generalized Tricomi equation ${\mathrm{\partial }}_t^{2}u-t^m\mathrm{\Delta }u=t^{\alpha (m)}\mathrm{\mid }u{\mathrm{\mid }}^p$, where $m=m(\mu )>0$ and $\alpha (m)\in \mathbb{R}$ are two suitable constants. At first, for the more general semilinear Tricomi equation ${\mathrm{\partial }}_t^{2}v-t^m\mathrm{\Delta }v=t^{\alpha }\mathrm{\mid }v{\mathrm{\mid }}^p$ with any fixed constant $m>0$ and arbitrary parameter $\alpha \in \mathbb{R}$, we shall show that in the case of $\alpha \le -2$, $n\ge 3$ and $p>1$, the small data weak solution $v$ exists globally; in the case of $\alpha >-2$, through determining the conformal exponent $p_{conf}(n,m,\alpha )>1$, the global small data weak solution $v$ exists when some extra restrictions of $p\ge p_{conf}(n,m,\alpha )$ are given. Returning to the original equation ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u+\frac{\mu }{t}{\mathrm{\partial }}_{t}u=\mathrm{\mid }u{\mathrm{\mid }}^p$, the corresponding global existence results on the small data solution $u$ can be obtained.