论文标题
径向二次波方程的孤子分辨率在六个空间尺寸
Soliton resolution for the radial quadratic wave equation in six space dimensions
论文作者
论文摘要
我们考虑六个维度的二次半线性波方程。这个能量关键问题承认了基态解决方案，该解决方案是唯一的（缩放）阳性固定解决方案。我们证明，在能量标准中保持界限的任何球形对称溶液都渐近地发展为脱钩的调制基态，以及辐射项。作为该方法的副产品，我们证明了不发射任何辐射的多层溶液的不存在。证明遵循最后三位作者为大奇数启动的方法，将问题降低到排除这种非辐射多层的存在，是通过从合理的调制参数的普通微分方程的有限维度系统中得出矛盾的。相比之下，六个维度的难度是某些能量估计通道的失败和线性共振的相关存在。主要新颖性是使用新的能量估计渠道，以及用少量能量的非辐射溶液分类。
We consider the quadratic semilinear wave equation in six dimensions. This energy critical problem admits a ground state solution, which is the unique (up to scaling) positive stationary solution. We prove that any spherically symmetric solution, that remains bounded in the energy norm, evolves asymptotically to a sum of decoupled modulated ground states, plus a radiation term. As a by-product of the approach we prove the non-existence of multisoliton solutions that do not emit any radiation. The proof follows the method initiated for large odd dimensions by the last three authors, reducing the problem to ruling out the existence of such non-radiative multisolitons, by deriving a contradiction from a finite dimensional system of ordinary differential equations governing their modulation parameters. In comparison, the difficulty in six dimensions is the failure of certain channel of energy estimates and the related existence of a linear resonance. The main novelties are the use of new channel of energy estimates, as well as the classification of non-radiative solutions with small energy.
