Existence and Uniqueness Theorem for a Class of Singular Nonlinear Partial Differential Equations

Abstract

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 \mathrm{Re}\, \gamma(0)<0 .