On The Formal Solutions Of A Type Of Singular Nonlinear Partial Differential Equations

Recently,people find that formal solutions to a type of doubly singular ordinary differential equations(systems)were summable in a monomial of two variables,and that formal solutions to many singular partial differential equations were multisummable.So summability theory of formal power series in several variables is of great importance to the research of formal solutions to partial differential equations.Especially the constrction(of monomial summability theory for formal power series in two variable)makes it more and more convenient for people to study the summability of formal solutions to partial differential equations.In this paper,we give a type of partial differential equation and show monomial summability of formal solutions to them.This enriches the achievements in the research of formal solutions to differential equations.It is a application of monomial summability theory of formal power series.The following is the main research work of this paper.Firstly,we give a type of partial differential equation an make several hypotheses on them to make some sure that they have formal power series solutions in special form.We give a concrete example of partial differential equation,and work out the formal solution to it and point out the Gevrey order in a monomial of it.This means that this type of partial differential equation have indeed formal solutions which are monomial summable.Secondly,by formal transformation,we translate the partial differential equation into two sequence of ordinary differential equations.We select,according to their solutions,special domains such that all solutions are well defined.Using the fixed point theorem,we prove that the partial differential equation exists unique analytic bounded solution on such domains.Finally,we prove monomial summability of the formal solution to the partial differential equation by applying an important result of summability theory.