INHOMOGENEOUS DIFFERENTIAL EQUATIONS OF VECTOR ORDER WITH DISSIPATIVE PARABOLICITY AND POSITIVE GENUS

Abstract

Parabolicity in the sense of both Petrosky and Shilov has a scalar character. It is not able to take into account the specificity of the heterogeneity of the environment. In this regard, in the early 70-s, S.D. Eidelman proposed the so-called $\vec{2b}$-parabolicity, which is a natural generalization of the Petrovsky parabolicity for the case of an anisotropic medium. A detailed study of the Cauchy problem for equations with such parabolicity was carried out in the works of S.D. Eidelman, S.D. Ivasishena, M.I. Matiichuk and their students. An extension of parabolicity according to Shilov for the case of anisotropic media is $\{\vec{p},\vec h\}$-parabolicity. The class of equations with such parabolicity is quite broad, it includes the classes of Eidelman, Petrovskii, and Shilov and allows unifying the classical theory of the Cauchy problem for parabolic equations. In this work, for inhomogeneous $\{\vec{p},\vec h\}$-parabolic equations with vector positive genus, the conditions under which the Cauchy problem in the class of generalized initial functions of the type of Gelfand and Shilov distributions will be correctly solvable are investigated. At the same time, the inhomogeneities of the equations are continuous functions of finite smoothness with respect to the set of variables, which decrease with respect to the spatial variable, and are unbounded with the integrable feature with respect to the time variable.