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

Abstract

In this paper, we are concerned with the global existence of small data weak solutions to the $n-$dimensional semilinear wave equation $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$ with time-dependent scale-invariant damping, where $n\geq 2$, $t\geq 1$, $\mu\in(0,1)\cup(1,2]$ and $p>1$. This equation can be changed into the semilinear generalized Tricomi equation $\partial_t^2u-t^m\Delta u=t^{\alpha(m)}|u|^p$, where $m=m(\mu)>0$ and $\alpha(m)\in\Bbb R$ are two suitable constants. At first, for the more general semilinear Tricomi equation $\partial_t^2v-t^m\Delta v=t^{\alpha}|v|^p$ with any fixed constant $m>0$ and arbitrary parameter $\alpha\in\Bbb R$, we shall show that in the case of $\alpha\leq -2$, $n\geq 3$ 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 $p\geq p_{conf}(n,m,\alpha)$ are given. Returning to the original equation $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, the corresponding global existence results on the small data solution $u$ can be obtained.