论文标题
二维波 - 具有半线性非线性的klein-gordon方程
Two dimensional wave--Klein-Gordon equations with semilinear nonlinearities
论文作者
论文摘要
从Lindblad和Rodnianski的弱效率状况的工作中，众所周知，“糟糕的”二次采购术语可以在三个空间维度的耦合半线性波方程中出现，前提是这样的术语作为“好”变量的来源出现，并且良好的变量可以通过良好的变量来回到系统中。在这些思想的推动下，我们在本文中研究了两个空间维度的耦合波 - klein-gordon方程的解决方案的解决方案的较小数据和尖锐的衰减。特别是，我们考虑了针对klein-gordon方程的波方程和低于关键的半线性非线性的关键半线性非线性。我们两个系统的一个有趣特征是，如果要交换我们的PDE的非线性，则波动方程中的非线性项将导致有限的时间爆炸。
From the work on the weak-null condition by Lindblad and Rodnianski, it is well-known that `bad' quadratic sourcing terms are allowed to appear in coupled semilinear wave equations in three spatial dimensions, provided that such terms appear as sources for `good' variables and that the good variables feed back into the system via `good' sourcing terms. Motivated by these ideas, in this paper we investigate the small data global existence and pointwise decay of solutions to two systems of coupled wave--Klein-Gordon equations in two spatial dimensions. In particular, we consider critical semilinear nonlinearities for the wave equation and below-critical semilinear nonlinearities for the Klein-Gordon equation. An interesting feature of our two systems is that if the nonlinearities of our PDEs were to be swapped, the nonlinear term in the wave equation would lead to finite-time blow-up.
