Maillet’s type theorems for non linear singular partial differential equations without linear part

Abstract

In chapter 6, we have studied formal solutions of a class (6.3.1) of non linear singular partial differential equations of order l and we have proved that each formal power series solution is in some formal Gevrey class ε^{s} for s = s _l . The number s _l can be computed explicitly by looking at the coefficients of (6.3.1). Moreover, each formal solution û of the equation (6.3.1) is in some formal Gevrey class ε^{s} for s = s _l (û) ≤ s _l and s _l (û) can be calculated explicitly. We showed also in [19] (see also section 6.4 of chapter 6) that for some particular equations the number s _l (û) is the exact formal Gevrey index of û. The main assumption in chapter 6 was that the equation (6.3.1) has a linear part.