On the Unique Solvability of Certain Nonlinear Singular Partial Differential Equations

Abstract

We study the singular nonlinear equation $tu_{t}=F(t,x,u,u_{x})$, where the function $F$ is assumed to be continuous in $t$ and holomorphic in the other variables. Under some growth conditions on the coefficients of the partial Taylor expansion of $F$, we show that if $F(t,x,0,0)$ is of order $O(\mu(t)^{\alpha})$ for some $\alpha\in[0,1]$ as $t\rightarrow0$ uniformly in some neighborhood of $x=0$, then the equation has a unique solution $u(t,x)$ with the same growth order.