Gevrey asymptotic theory for first-order linear and semi-linear partial differential equations of nilpotent type

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We studied two complex dimensional singular first-order linear partial differential equations of nilpotent type. Our main purpose was to give conditions for equations under which the divergent power series solution is Borel summable. As a consequence, we could conjecture conditions, in forms of global conditions (analytic continuation property, growth conditions or decreasing conditions) for coefficients of equations. Moreover, we could accomplish the proof of conjecture for some special, but have not been treated previously, equations.