Existence and uniqueness theorem for a class of singular nonlinear partial differential equations
Summary: This paper deals with singular nonlinear partial differential equations of the form \(t\partial u/\partial t =F\left(t,x,u,\partial u/\partial x\right)\), with independent variables \((t,x)\in \mathbb{R}\times\mathbb{C}\), and where \(F(t,x,u,v)\) is a function continuous in \(t\) and holomorphic in the other variables. Using the Banach fixed point theorem (also known as the contraction mapping principle), we show that a unique solution \(u(t,x)\) exists under the condition that \(F(0,x,0,0)=0, F_u(0,x,0,0)=0\) and \(F_v(0,x,0,0)=x \gamma(x)\) with \(\text{Re}\, \gamma(0)<0\).
scientific article; zbMATH DE number 6126566