On the Unique Solvability of Nonlinear Fuchsian Partial Differential Equations

摘要

We consider a singular nonlinear partial differential equation of the form (t partial derivative(t))(m)u = F(t,x, {(t partial derivative(t))(j)partial derivative(alpha)(x)u}((j,alpha)is an element of Im)) with arbitrary order m and I-m = {(j, alpha) is an element of N x N-n ; j + |alpha| = m, j m} under the condition that F(t, x, {Z(j,alpha)}((j,alpha)is an element of Im)) is continuous in t and holomorphic in the other variables, and it satisfies F(0, x, 0) equivalent to 0 and (partial derivative F/partial derivative z(j,alpha))(0, x, 0) equivalent to 0 for any (j, alpha) is an element of I-m boolean AND {|alpha| 0}. In this case, the equation is said to be a nonlinear Fuchsian partial differential equation. We show that if F(t, x, 0) vanishes at a certain order as t tends to 0 then the equation has a unique solution with the same decay order.