论文标题
紧凑支撑的能量波方程的辐射
Radiation of the energy-critical wave equation with compact support
论文作者
论文摘要
我们证明了（非自由度）解决方案的外部能量下限，可在空间尺寸中的能源至关重要的非线性波方程$ 3 \ le d \ le 5 $，并具有紧凑的初始数据。特别是，这表明具有紧凑空间支持的非平凡全局解决方案必须具有辐射性，因为至少以下一个是正确的：（1）$ \ int_ {| x |> |> |> |> |> |> | \ partial_tial_t u |^2 + |^2 + | \ nabla U |^2 \ right） \ Mathrm {for} \ \ Mathrm {all} \ T \ ge 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t \ t \ t \ le 0，$（$（2）$ \ int_ {| t） \ right）\ mathrm {d} x \geη_2（\ varepsilon，u）> 0，\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ thallm {all} \ t \ in \ t \ in \ mathbb {r}，r}，\ varepsilon> 0。作为应用程序，我们获得了有关紧凑型属性解决方案的刚性猜想的部分结果，包括有关这种解决方案的全球存在的新证明。
We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions $3 \le d \le 5$, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in the sense that at least one of the following is true: (1) $\int_{|x|> |t|} \left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x \ge η_1(u) > 0, \ \mathrm{for} \ \mathrm{all} \ t \ge 0 \ \mathrm{or} \ \mathrm{all} \ t \le 0,$ (2) $\int_{|x|> -\varepsilon +|t|} \left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x \ge η_2(\varepsilon, u) > 0, \ \mathrm{for} \ \mathrm{all} \ t \in \mathbb{R}, \varepsilon > 0.$ In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.
