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I'm sure you can extract it from the proof, but does anyone know of a reference where the radius of convergence (in terms of radius of convergence of the initial data and PDE) of the solution given by the Cauchy-Kovalevskaya Theorem is written down?

On a related, but more speculative note, I'm curious if there are any results along the following lines: Suppose one is given a problem solvable by the Cauchy-Kovalevskaya Theorem along with a real analytic solution with uniform lower bound on its radius of convergence (alternatively which exists in some uniform strip about the hypersurface on which initial data is prescribed). If one approximates the data of the given solution (the right notion is part of the question) by some sequence of real analytic data, can one say anything about the radius of convergence of the corresponding solutions (or size of strip the solutions exist on).

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  • $\begingroup$ The reference to Hormander provided by user80744 is a good one. For your second question, if I understand correctly, the answer is "no", unless you also assume some lower bound on the radius of convergence of the data in the sequence (and then you get an estimate on the size of the domain of the corresponding sequence of solutions not from the assumption about what the data approximate but rather from applying C-K at each step of the sequence). If you clarify your question by indicating whether you want to assume lower bound on radius of convergence of each data function, I can elaborate. $\endgroup$
    – Erik Lundberg
    Commented Sep 22, 2023 at 15:50

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Theorem 9.4.5 in the first volume of Hörmander's four-volume treatise "The Analysis of Linear Partial Differential Operators" is a quantitative version of the Cauchy-Kovalevsky theorem. Condition (9.4.7) of that theorem gives bounds on the radii of convergence. The theorem is stated with zero initial data. To apply it with non-zero initial data, first move these into the right-hand side of the equation by applying the differential operator to the Taylor polynomial (in $z_n$) of the initial data.

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