8.4 Existence of expansions using Fuchsian methods, T3-case

In [54Jump To The Next Citation Point], Kichenassamy and Rendall proved the existence of expansions in the real analytic setting. In other words, they proved that, given real analytic va,ϕ, q,ψ with 0 < va < 1, there is a unique solution to Equations (16View Equation) – (17View Equation) with expansions of the form of Equations (37View Equation) – (38View Equation). Note that the number of functions that are freely specifiable in the expansions coincides with the number of functions that need to be specified in order to obtain a unique solution to the initial value problem corresponding to Equations (16View Equation) – (17View Equation). For reasons mentioned in Section 8.2, the expansions suffer from a potential consistency problem in the case of va ≥ 1 and q𝜃 ⁄= 0. However, in [54Jump To The Next Citation Point] it was proven that if q is constant, the condition on va can be relaxed to va > 0. The regularity condition of [54Jump To The Next Citation Point] was relaxed to smoothness in [68Jump To The Next Citation Point].
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