On the singular solutions of nonlinear singular partial differential equations I

Abstract

Let us consider the following nonlinear singular partial differential equation: in the complex domain. Denote by ${\mathcal{S}}_{+}$ [resp. ${\mathcal{S}}_{log}$] the set of all the solutions $u(t,x)$ with asymptotics $u(t,x)=O\left(\right|t{|}^a)$ [resp. $u(t,x)=O(1/|\log t{|}^a\left)\right]$(as $t\to O$ uniformly in $x$) for some $a>0$. Clearly ${\mathcal{S}}_{log}\supset {\mathcal{S}}_{+}$. The paper gives a sufficient condition for ${\mathcal{S}}_{log}={\mathcal{S}}_{+}$ to be valid.