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

Abstract

Abstract. We treat a nonlinear partial di¤erential equation F ðx; u; u x Þ ¼ 0 in a neighborhood of x ¼ 0 A R d , where F ðx; u; pÞ 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 F pi ð0; 0; 0Þ 0 0 for some i. In this paper we study the existence of a classical solution under the condition that However if F ðx; u; pÞ is not analytic, the methods used for analytical case are not available. It seems that general existence results under condition (0.2) are not known. It is the purpose of this paper to construct under the condition