Uniqueness of the solution of nonlinear totally characteristic partial differential equations

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Let us consider the following nonlinear singular partial differential equation (t ∂/∂ t)^m u =F ( t, x, {(t ∂/∂ t)^j (∂/∂ x)α u }j+α ≤ m, j<m ) in the complex domain with two independent variables (t, x) ∈ \BC^2. When the equation is of totally characteristic type, this equation was solved in \cite{ct2} and \cite{resont} 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 {(t, x) ∈ \BC^2 ; 0 < |t|<r, | \arg t| < θ, |x|<R} for some r>0, θ >0, R>0 and that it satisfies u(t, x) =O(|t|^a ) (as t \longrightarrow 0) uniformly in x for some a>0. The result is applied to the problem of removable singularities of the solution.


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