REPEATED KERNELS OF THE GREEN’S FUNCTION OF PARABOLIC SHILOV EQUATIONS WITH VARIABLE COEFFICIENTS AND NEGATIVE GENUS

Abstract

The concept of parabolicity by Shilov generalizes the concept of parabolicity by Petrovsky of equations with partial derivatives and leads to a significant expansion of the known Petrovsky class with those parabolic equations, the order of which may not coincide with the parabolicity index. Generally speaking, such an extension deprives of the parabolic stability сoncerning the change of the coefficients of parabolic Shilov equations, which is inherent to the Petrovsky class equations. As a result, significant difficulties arise in the study of the Cauchy problem for parabolic Shilov equations with variable coefficients. In the 60s of the last century, Y.I. Zhytomyrsky defined a special class of parabolic Shilov equations, which extends the Shilov class and at the same time is parabolically resistant to changes in the junior coefficients. For this class, by the method of successive approximations, he established the correct solvability of the Cauchy problem in the class of bounded initial functions of finite smoothness. However, to obtain more general results, it is important to know the Green’s function of the Cauchy problem. In this publication, for parabolic Shilov equations with bounded smooth variable coefficients and negative genus, estimates of repeated kernels of the Green’s function of the Cauchy problem are established, which allow us to investigate the properties of the density of volume potential of this function. These results are important for the development of the Cauchy problem theory for parabolic Shilov equations by classical means of the Green’s function.