On the unique solvability of certain nonlinear singular partial differential equations (Q713225)

Summary: 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 \(t\rightarrow 0\) uniformly in some neighborhood of \(x=0\), then the equation has a unique solution \(u(t,x)\) with the same growth-order.