Existence of Classical Solutions near Characteristic Points of First Order Nonlinear Partial Differential Equations

Abstract

We treat a nonlinear partial differential equation F(x,u,ux) = 0 in a neighborhood of x = 0 ∈ Rd, where F(x,u,p) is a real-valued smooth function. It is well-known that solutions are constructed by solving noncharacteristic Cauchy problem with the method of characteristics, provided Fpi(0,0,0) ≠ 0 for some i. In this paper we study the existence of a classical solution under the condition that Fpi(0,0,0) = 0 for all 1 ≤ i ≤ d.