Uniqueness of the solution of nonlinear totally characteristic partial differential equations

Abstract

Let us consider the following nonlinear singular partial differential equation in the complex domain with two independent variables $(t,x)\in {\mathbf{C}}^2$. When the equation is of totally characteristic type, this equation was solved in [2] and [9] under certain Poincaré condition. In this paper, the author will prove the uniqueness of the solution under the assumption that $u(t,x)$ is holomorphic in for some $r>0$, $\theta >0$, $R>0$ and that it satisfies $u(t,x)=O\left(\right|t{|}^a)$ (as $t0$) uniformly in $x$ for some $a>0$. The result is applied to the problem of removable singularities of the solution.