Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping

In this paper we prove a sharp global existence result for semilinear wave equations with time-dependent scale-invariant damping terms if the initial data is small. More specifically, we consider Cauchy problem of ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u+\frac{\mu }{t}{\mathrm{\partial }}_{t}u=\mathrm{\mid }u{\mathrm{\mid }}^p$, where $n\ge 3$, $t\ge 1$ and $\mu \in (0,1)\cup (1,2)$. For critical exponent $p_{crit}(n,\mu )$ which is the positive root of $(n+\mu -1)p^2-(n+\mu +1)p-2=0$ and conformal exponent $p_{conf}(n,\mu )=\frac{n+\mu +3}{n+\mu -1}$, we establish global existence for $n\ge 3$ and $p_{crit}(n,\mu )<p\le p_{conf}(n,\mu )$. The proof is based on changing the wave equation 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, then we investigate more general semilinear Tricomi equation ${\mathrm{\partial }}_t^{2}v-t^m\mathrm{\Delta }v=t^{\alpha }\mathrm{\mid }v{\mathrm{\mid }}^p$ and establish related weighted Strichartz estimates. Returning to the original wave equation, the corresponding global existence results on the small data solution $u$ can be obtained.