Schwarzschild流形上大数据半线性wave equation的时空积分估计
A space-time integral estimate for a large data semi-linear wave equation on the Schwarzschild manifold
摘要：
本文考虑R × R × S ^2中的波动方程（- \dt^2 + \dr^2 -V-V_L（-Δ S^2}））u = fF '（|u| ^2）u，其中（t，ρ，θ，θ）.具有单个闭测地线曲面的球对称流形上或史瓦西流形外部的波动方程可以简化为这种形式。利用不需要球谐分解的光滑Morawetz估计，我们证明了在能量类中，即使初始数据很大，L^2_{loc}也会衰减.这需要对电势V、V_L和f有一定的条件。我们证明了在平滑Morawetz估计的权重的一个关键条件可以减少到一个ODE条件，这是数值验证。
英文摘要：
We consider the wave equation (-\dt^2+\dr^2 -V -V_L(-Δ_{S^2})) u = fF'(|u| ^2) u with (t,ρ,θ,ϕ) in R x R x S^2. The wave equation on a spherically symmetric manifold with a single closed geodesic surface or on the exterior of the Schwarzschild manifold can be reduced to this form. Using a smoothed Morawetz estimate which does not require a spherical harmonic decomposition, we show that there is decay in L^2_{loc} for initial data in the energy class, even if the initial data is large. This requires certain conditions on the potentials V, V_L, and f. We show that a key condition on the weight in the smoothed Morawetz estimate can be reduced to an ODE condition, which is verified numerically.
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