Green’s Function of the Cauchy Problem for Equations with Dissipative Parabolicity, Negative Genus, and Variable Coefficients

Abstract

Green’s function of the Cauchy problem is constructed by the method of successive approximations, and its main properties are studied for a new class of linear differential equations with dissipative parabolicity and negative genus, whose coefficients are bounded, continuous in time, and infinitely differentiable by the spatial variable of the function. This class covers Shilov parabolic equations, as well as the class of the Zhitomirskii parabolic Shilov-type equations with variable coefficients and negative genus, and successfully complements the Petrovsky class of parabolic equations.