On the Unique Solvability of Certain Nonlinear Singular Partial Differential Equations

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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.