On the singular solutions of nonlinear singular partial differential equations I

Abstract

Let us consider the following nonlinear singular partial differential equation: (t∂_t)^mu=F(t, x, {(t∂_t)^j∂_x^αu}_{j+|α|≤ m, j<m}) in the complex domain. Denote by \mathscr{S}₊[resp. \mathscr{S}_log] the set of all the solutions u(t, x) with asymptotics u(t, x)=O(|t|^a) [resp. u(t, x)=O(1/|log t|^a)] (as t→ 0 uniformly in x) for some a>0. Clearly \mathscr{S}_log⊃ \mathscr{S}₊. The paper gives a sufficient condition for \mathscr{S}_log=\mathscr{S}₊$ to be valid.