Genuine solutions and formal solutions with Gevrey type estimates of nonlinear partial differential equations. (English) Zbl 0860.35018

The author proves a theorem of Ramis-Sibuya type for a nonlinear partial differential equation \(L(U)= g(z)\). He uses the characteristic polygons techniques in order to settle the combinatorial conditions he needs to state the theorems. Let \[ \widehat u(z_0, z') =z^q_0 \left(\sum^\infty_{n=0} u_n(z') z_0^{q_n} \right) \in{\mathcal O}_{\mathbb{C}^n,0} [[z_0]] \] be a formal solution of Gevrey’s type, that is, for \(n\in\mathbb{N}\) we have the inequalities \(|u_n(z') |\leq AB^{q_n} \Gamma ((q_n/ \gamma) + 1)\), where \(\gamma\) is a positive real number associated to the characteristic polygon \(\Sigma\) of \(L\) (see the paper for details). If the operator \(L(\cdot + \widehat u) - L(\widehat u)\) is linearly nondegenerated and \(\Sigma\) verifies some combinatorial properties, then the author shows the existence of a solution \(u\) such that: (1) It is a holomorphic function in \(S \times D_1 \times \cdots \times D_n \subset \mathbb{C} \times \mathbb{C}^n\), where \(S\) is an open sector, and the \(D_j\) are open discs in \(\mathbb{C}\) centered at \(0 \in \mathbb{C}\). (2) The formal series \(\widehat u\) is the asymptotic expansion of \(u\) in \(S \times D_1 \times \cdots \times D_n \subset \mathbb{C} \times \mathbb{C}^n\).
The asymptotic developable functions introduced by the author are different from those of Majima, Haraoka and Zurro. Certainly it is an important work and it will induce the development of the subject.


35C20 Asymptotic expansions of solutions to PDEs
35A20 Analyticity in context of PDEs
35C10 Series solutions to PDEs