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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.

##### MSC:
 35C10 Series solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs 35G20 Nonlinear higher-order PDEs
##### Keywords:
formal solution; Gevrey type estimate