Unique solvability of some nonlinear partial differential equations with Fuchsian and irregular singularities

Abstract

The paper considers nonlinear partial differential equations of the form t(∂u/∂t) = F(t,x,u,∂u/∂x), with independent variables (t,x) ∈ ℝ × ℂ, and where F(t,x,u,v) is a function continuous in t and holomorphic in the other variables. It is shown that the equation has a unique solution in a sectorial domain centered at the origin under the condition that F(0,x,0,0) = 0, Re F_u(0,0,0,0) < 0, and F_v(0,x,0,0) = x^p+1γ(x), where γ(0) ≠ 0 and p is any positive integer. In this case, the equation has a Fuchsian singularity at t = 0 and an irregular singularity at x = 0.