标题： 具有时间相关尺度不变阻尼的小振幅半线性波动方程的整体存在性 显示英文标题

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

摘要： 本文证明了当初始数据较小时，具有时间相关尺度不变阻尼项的半线性波动方程的严格整体存在性结果。更具体地，我们考虑了柯西问题：$\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$，其中 $n\ge 3$，$t\ge 1$ 和 $\mu\in(0,1)\cup(1,2)$。对于临界指数 $p_{crit}(n,\mu)$，它是 $(n+\mu-1)p^2-(n+\mu+1)p-2=0$ 的正根，以及共形指数 $p_{conf}(n,\mu)=\frac{n+\mu+3}{n+\mu-1}$，我们建立了 $n\geq3$ 和 $p_{crit}(n,\mu) < p\leq p_{conf}(n,\mu)$ 的全局存在性。证明基于 将波动方程转化为半线性广义Tricomi方程 $\partial_t^2u-t^m\Delta u=t^{\alpha(m)}|u|^p$，其中 $m=m(\mu)>0$ 且 $\alpha(m)\in\Bbb R$ 为两个合适的常数。然后，我们研究更一般的半线性Tricomi方程 $\partial_t^2v-t^m\Delta v=t^{\alpha}|v|^p$，并建立相关的加权Strichartz估计。回到原始波动方程，可以得到相应的小数据解 $u$ 的全局存在性结果。 显示英文摘要

摘要： 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 $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, 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 $n\geq3$ and $p_{crit}(n,\mu)<p\leq p_{conf}(n,\mu)$. The proof is based on changing the wave equation 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, then we investigate more general semilinear Tricomi equation $\partial_t^2v-t^m\Delta v=t^{\alpha}|v|^p$ 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.