Uniqueness of the solution of nonlinear totally characteristic partial differential equations

摘要

Let us consider the following nonlinear singular partial differential equation (t partial deriv/partial deriv t)~m u = F(t,x, {(t partial deriv/partial deriv t)~j(partial deriv/partial deriv x)~α u}_(j+α ≤ m, j < m)) in the complex domain with two independent variables (t,x) ∈ C~2. When the equation is of totally characteristic type, this equation was solved in [2] and [9] under certain Poincare 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) ∈ C~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 → 0) uniformly in x for some a > 0. The result is applied to the problem of removable singularities of the solution.