Formal solutions with Gevrey type estimates of nonlinear partial differential equations. (English) Zbl 0810.35006

Summary: Let \(L(u)= L(z, \partial^ \alpha u\); \(|\alpha |\leq m)\) be a nonlinear partial differential operator defined in a neighbourhood \(\Omega\) of \(z=0\) in \(\mathbb{C}^{n+1}\), where \(z= (z_ 0, z')\in \mathbb{C}\times \mathbb{C}^ n\). \(L(u)\) is a polynomial of the unknown and its derivatives \(\{\partial^ \alpha u\); \(|\alpha |\leq m\}\) with degree \(M\). The main purpose of this paper is to find a formal solution \(u(z)\) of \(L(u)= g(z)\) in the form \[ u(z)= z_ 0^ q \Biggl( \sum_{n=0}^{+\infty} u_ n (z') z_ 0^{q_ n} \Biggr), \qquad u_ 0(z') \not\equiv 0, \] where \(q\in \mathbb{R}\) and \(0= q_ 0< q_ 1<\dots< q_ n<\dots\to +\infty\), and to obtain estimates of coefficients \(\{u_ n (z')\); \(n\geq 0\}\). It is shown under some conditions that we can construct formal solutions with \[ | u_ n (z')|\leq AB^{q_ n} \Gamma \biggl( {{q_ n} \over {\gamma_ *}} +1\biggr), \qquad 0< \gamma_ * \leq\infty, \] which we often call the Gevrey type estimate.


35C10 Series solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35G20 Nonlinear higher-order PDEs