wave equations在辐射区的解
Solutions of wave equations in the radiation regime
摘要：
我们研究Minkowski时空上线性和半线性波动方程的“双曲柯西问题”，初始数据在加权索伯列夫空间中，允许在边界处有奇异行为，或者初始数据为多齐次。具体来说，我们考虑非线性对称双曲系统的形式，其中包括标量场与$λϕ^p$非线性，以及波映射，与初始数据上的双曲面;几个结果证明适用于一般时空承认共形完成零无穷大，以及一大类方程具有类似的非线性结构。我们证明了具有受控渐近行为的解的存在性，以及当初始数据具有这种展开式时解的渐近展开。特别是，我们证明了多齐次初始数据（满足相容性条件）导致的解决方案是多齐次的共形边界Scri的闵可夫斯基时空。
英文摘要：
We study the ``hyperboloidal Cauchy problem'' for linear and semi-linear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behaviour at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a $λϕ^p$ nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behaviour, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary Scri of the Minkowski space-time.
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