ON THE UNIQUENESS OF GENERALIZED SOLUTIONS OF SYSTEMS OF DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS // BSU bulletin. Mathematics, Informatics. - 2021. №1. . - С. 24-33.

ON THE UNIQUENESS OF GENERALIZED SOLUTIONS OF SYSTEMS OF DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

Работа выполнена при финансовой поддержке ЛФ ПНИПУ. Автор выражает благодарность руководству вуза — Кочневу Виктору Анатольевичу.

This paper studies the problem of uniqueness of extension of generalized solutions of systems of partial differential equations with constant coefficients.

Holmgren, I. M. Gelfand, G. E. Shilov, V. P. Palamodov, and other mathemati- cians dealt with the problem of extending the uniqueness of solutions of such systems. The problem of uniqueness of the Cauchy problem for evolutionary type with constant coefficients is also studied in I. M. Gelfand and G. E Shilov’s book. V. P. Palamodov

investigated the uniqueness problem and established more precise theorems on the possibility of extending generalized solutions given in a neighborhood of the boundary of the domain in the most important situations. Uniqueness problems similar to the Goursat problem were investigated by A. M. Berdimuratov. This paper is devoted to the following problem: under what conditions is any generalized solution of infinite order of a system of partial differential equations with constant coefficients defined in a neighborhood of three adjacent faces of a parallelepiped in, can be uniquely extended to some of its neighborhood.

Holmgren, I. M. Gelfand, G. E. Shilov, V. P. Palamodov, and other mathemati- cians dealt with the problem of extending the uniqueness of solutions of such systems. The problem of uniqueness of the Cauchy problem for evolutionary type with constant coefficients is also studied in I. M. Gelfand and G. E Shilov’s book. V. P. Palamodov

investigated the uniqueness problem and established more precise theorems on the possibility of extending generalized solutions given in a neighborhood of the boundary of the domain in the most important situations. Uniqueness problems similar to the Goursat problem were investigated by A. M. Berdimuratov. This paper is devoted to the following problem: under what conditions is any generalized solution of infinite order of a system of partial differential equations with constant coefficients defined in a neighborhood of three adjacent faces of a parallelepiped in, can be uniquely extended to some of its neighborhood.

algebraic variety; finite function; algebraic cone; improper point; Palamodov — Noether operator; entire analytic function; Fourier transform.

Palamodov V. P. Linear Differential Operators with Constant Coefficients. Mos- cow: Nauka Publ., 1967. 488 p.

Akhmedov Sh. A. Analog of Malgrange's Theorem. News of the Academy of Sci- ences of the Tajik SSR, Department of Physical, Mathematical and Geological and Chemical Sciences. 1983. Vol. 88, no. 2. Pp. 15–20.

Akhmedov Sh. A., Berdimuratov A. Analog of the Second Theorem Malgrange. News of the Academy of Sciences of the Tajik SSR, Department of Physical, Mathe- matical and Geological and Chemical Sciences. 1985. Vol. 96, no. 2. Pp. 3–7.

Berdimuratov A. M. An Analog of the Darboux — Goursat — Baudot Problem in Classes of Generalized Functions for Systems of Partial Differential Equations with Constant Coefficients. Dissertation. Bishkek, 1992. 110 p.

Berdimuratov A. M. Palamodov's Exponential Representation Method and Its Applications to Some Analogs of Classical Problems in Spaces of Generalized Func- tions. Bishkek, 2017. 134 p.

Akhmedov Sh. A. Analog of Malgrange's Theorem. News of the Academy of Sci- ences of the Tajik SSR, Department of Physical, Mathematical and Geological and Chemical Sciences. 1983. Vol. 88, no. 2. Pp. 15–20.

Akhmedov Sh. A., Berdimuratov A. Analog of the Second Theorem Malgrange. News of the Academy of Sciences of the Tajik SSR, Department of Physical, Mathe- matical and Geological and Chemical Sciences. 1985. Vol. 96, no. 2. Pp. 3–7.

Berdimuratov A. M. An Analog of the Darboux — Goursat — Baudot Problem in Classes of Generalized Functions for Systems of Partial Differential Equations with Constant Coefficients. Dissertation. Bishkek, 1992. 110 p.

Berdimuratov A. M. Palamodov's Exponential Representation Method and Its Applications to Some Analogs of Classical Problems in Spaces of Generalized Func- tions. Bishkek, 2017. 134 p.