Maillet’s type theorems for non linear singular partial differential equations

Abstract

Near an irregular singular point of an ordinary linear differential equation there may exist formal power series solutions $\sum\limits_{k \ge 0} {{u_k}{t^k}}$ that diverge everywhere, but it is relatively easy to show that such solutions are in some formal Gevrey class i.e. the coefficients uk grow at most like a fixed power of k! as k tends to infinity.