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 $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $n\ge 3$, $t\ge 1$ and $μ\in(0,1)\cup(1,2)$. For critical exponent $p_{crit}(n,μ)$ which is the positive root of $(n+μ-1)p^2-(n+μ+1)p-2=0$ and conformal exponent $p_{conf}(n,μ)=\frac{n+μ+3}{n+μ-1}$, we establish global existence for $n\geq3$ and $p_{crit}(n,μ) 0$ and $α(m)\in\Bbb R$ are two suitable constants, then we investigate more general semilinear Tricomi equation $\partial_t^2v-t^mΔv=t^α|v|^p$ 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.