Parabolic Operators Associated with Newton’s Polygon

Abstract

This chapter is devoted to studying Cauchy’s problem for differential operators in several variables. When studying Cauchy’s problem, there naturally arises separation of variables into a temporal variable and spatial variables. Taking into account separately the order of differentiation with respect to time and the total order of differentiation with respect to the spatial variables we associate with a differential operator a set of integral points in the plane that are used to construct Newton’s polygon and the related set of principal quasi-homogeneous parts of the operator. In terms of these quasi-homogeneous parts we describe a new class of parabolic operators which relates to the classical operators parabolic in Petrovskiĭ’s sense in the same way as N quasi-elliptic operators relate to quasi-elliptic operators.