Singular solutions and prolongation of holomorphic solutions to nonlinear differential equations (Q5931084)

The author considers nonlinear differential equations with holomorphic coefficients in the complex domain NEWLINE\[NEWLINE\partial^m_t u(t, x)= f(t,x,(\partial^j_t \partial^\alpha_x u(t,x))_{j\leq m-1,j+|\alpha|\leq m}),\quad t\in\mathbb{C},\;x\in\mathbb{C}^n.NEWLINE\]NEWLINE Under some exponent condition he constructs a solution as follows: NEWLINE\[NEWLINEu(t,x)= t^\sigma \sum^\infty_{j=0} u_j(t) t^{j/n},NEWLINE\]NEWLINE where the \(u_j\)'s are holomorphic in a common neighborhood of the origin in \(\mathbb{C}^n\) and \(u_0\neq 0\), \(\sigma\) a positive rational number, and all formal solutions of the above form converge near the origin in \(\mathbb{C}_t\times \mathbb{C}^n_x\).