On the Unique Solvability of Certain Nonlinear Singular Partial Differential Equations

Abstract

We study the singular nonlinear equation $t{u}_t=F(t,x,u,u_x)$, 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 $t\to 0$ uniformly in some neighborhood of $x=0$, then the equation has a unique solution $u(t,x)$ with the same growth order.