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 $\mbox{Re}\, \gamma(0)<0$.