Uniqueness of the solution of nonlinear totally characteristic partial differential equations (Q2583056)

Let us consider the following nonlinear singular partial differential equation \[ (t\partial/\partial t)^mu=F(t,x,{(t\partial/\partial t)^j(\partial/\partial x)^\alpha u}_{j+\alpha\leq m,\,j<m}) \] in the complex domain with two independent variables \((t,x)\in\mathbb{C}^2\). When the equation is of totally characteristic type, this equation was solved in [\textit{H. Chen} and \textit{H. Tahara}, Publ. Res. Inst. Math. Sci. 35, No. 4, 621--636 (1999; Zbl 0961.35002)] and [J. Math. Soc. Japan 55, No. 4, 1095--1113 (2003; Zbl 1061.35009)] under certain Poincaré conditions. In this paper, the author will prove the uniqueness of the solution under the assumption that \(u(t,x)\) is holomorphic in \(\{(t,x)\in\mathbb{C}^2; ~0<|t|<r,|\text{arg}t|<\vartheta,|x|<R\}\) for some \(r>0\), \(\vartheta>0\), \(R>0\) and that it satisfies \(u(t,x)=O(|t|a)\) (as \(t\to 0\)) uniformly in \(x\) for some \(a>0\). The result is applied to the problem of removable singularities of the solution.