Problems nonlocal with respect to time for a class of
singular evolution equations

V. V. Horodetskyj; O. V. Martynyuk

We introduce and study new spaces of main and generalized functions which are natural objects for investigation of the Cauchy problem and nonlocal multipoint problems for wide class of pseudodifferential equations containing parabolic type evolution equations. To construct pseudodifferential operators we use new classes of function symbols not differentiable at zero, including a known class of symbols that possess a \parabolicity" condition. Some of the results establish properties of the fundamental solution of a nonlocal multipoint in time problem containing pseudodifferential operators in the evolution equation and boundary conditions. We prove the well solvability of the considered problem and find an expansion for the solution as the convolution of the fundamental solution by the boundary generalized function of Sobolev-Schwartz distribution type.

1. A.A. Dezin, Operators with first derivative with respect to “time” and non-local boundary conditions, Izv. Akad. Nauk SSSR Ser. Mat., 31 (1967), 61–86. (in Russian)

2. V.V. Gorodets’kii, O.M. Lenyuk, A two-point problem for a class of evolution equations, Mat. Stud., 28 (2007), .2, 175–182. (in Ukrainian)

3. V.V. Gorodets’kii, D.I. Spizhavka, A multipoint problem for evolution equations with pseudo-Bessel operators, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 12 (2009), 7–12.

4. V.V. Gorodets’kyj, I.S. Tupkalo, Multipoint problem for the evolutionary singular equations of the infinite order, Nauk. Visn. Chernivets’kogo Univ., Mat., 528 (2010), 27–35. (in Ukrainian)

5. M. Junusov, Operator equations with small parameter and nonlocal boundary conditions, Differentsial’nye Uravneniya, 17 (1981), .1, 172–181. (in Russian)

6. A.A. Kerefov, Nonlocal boundary value problems for parabolic equations, Differentsial’nye Uravneniya, 15 (1979), .1, 74–78. (in Russian)

7. G. Korn, T. Korn, Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov, [Mathematical handbook for scientists and engineers] Opredeleniya, teoremy, formuly, [Definitions, theorems, and formulas] Translated from the second English edition by I. G. Aramanovich, A.M. Berezman, I.A. Vainshtein, L.Z. Rumshiskiўi and L.Ya. Tslaf. Edited by Aramanovich. Fifth edition. “Nauka”, Moscow, 1984. 832 p. (in Russian)

8. B.M. Levitan, Expansion in Fourier series and integrals with Bessel functions, Uspehi Matem. Nauk (N.S.), 6 (1951), .2(42), 102–143. (in Russian)

9. A.H. Mamyan, General boundaty problems on layer, Dokl. Akad. Nauk SSSR, 267 (1982), .2, 292–293. (in Russian)

10. O.V. Martynyuk, The Cauchy problem for singular evolution equations in countably normed spaces of infinitely differentiable functions. I, Zb. Nauk. Prats’. Kamianets-Podilskogo Univ., Fiz.-Mat., 5 (2011), 179–192. (in Ukrainian)

11. O.V. Martynyuk, The Cauchy problem for singular evolution equations in countably normed spaces of infinitely differentiable functions. II, Zb. Nauk. Prats’. Kamianets-Podilskogo Univ., Fiz.-Mat., 6 (2012), 157–171. (in Ukrainian)

12. O.V. Martynyuk, The Cauchy problem for singular evolution equations in countably normed spaces of infinitely differentiable functions. III, Nauk. Visn. Chernivets’kogo Univ., Mat., 2 (2012), .1, 55–62. (in Ukrainian)

13. O.V. Martynyuk, V.V. Gorodets’kyj, The Cauchy problem for singular evolution equations with coefficients unbounded in time, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, 2 (2012), 19–24. (in Ukrainian)

14. M.I. Matiichuk, Parabolic singularly boundary poblem, Inst. Math. NANU, 1999, 176 p. (in Ukrainian)

15. A.M. Nakhushev, On nonlocal boundary value problems with displacement and their relationship with loaded equations, Differentsial’nye Uravneniya, 21 (1985), 92–101. (in Russian)

16. B.Yu. Ptashnyk, V.S. Il’kiv, I.Ya. Kmit’, V.M. Polischuk, Nonlocal boundary value problems with partial differential equations, Naukova Dumka, 2002, 416 p. (in Ukrainian)

17. V.K. Romanko, Boundary value problems for a certain class of differential operators, Differencial’nye Uravnenija, 10 (1974), 117–131. (in Russian)

18. V.K. Romanko, Boundary value problems for certain operator-differential equations, Dokl. Akad. Nauk SSSR, 227 (1976), .4, 812–815. (in Russian)

	Problems nonlocal with respect to time for a class of singular evolution equations
Author	V. V. Horodetskyj, O. V. Martynyuk alfaolga1@gmail.com Department Algebra and Information Studies, Chernivtsi National University
Abstract	We introduce and study new spaces of main and generalized functions which are natural objects for investigation of the Cauchy problem and nonlocal multipoint problems for wide class of pseudodifferential equations containing parabolic type evolution equations. To construct pseudodifferential operators we use new classes of function symbols not differentiable at zero, including a known class of symbols that possess a \parabolicity" condition. Some of the results establish properties of the fundamental solution of a nonlocal multipoint in time problem containing pseudodifferential operators in the evolution equation and boundary conditions. We prove the well solvability of the considered problem and find an expansion for the solution as the convolution of the fundamental solution by the boundary generalized function of Sobolev-Schwartz distribution type.
Keywords	nonlocal multipoint problem; pseudodifferential equations; fundamental solution; Bessel’s transform
Reference	1. A.A. Dezin, Operators with first derivative with respect to “time” and non-local boundary conditions, Izv. Akad. Nauk SSSR Ser. Mat., 31 (1967), 61–86. (in Russian) 2. V.V. Gorodets’kii, O.M. Lenyuk, A two-point problem for a class of evolution equations, Mat. Stud., 28 (2007), .2, 175–182. (in Ukrainian) 3. V.V. Gorodets’kii, D.I. Spizhavka, A multipoint problem for evolution equations with pseudo-Bessel operators, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 12 (2009), 7–12. 4. V.V. Gorodets’kyj, I.S. Tupkalo, Multipoint problem for the evolutionary singular equations of the infinite order, Nauk. Visn. Chernivets’kogo Univ., Mat., 528 (2010), 27–35. (in Ukrainian) 5. M. Junusov, Operator equations with small parameter and nonlocal boundary conditions, Differentsial’nye Uravneniya, 17 (1981), .1, 172–181. (in Russian) 6. A.A. Kerefov, Nonlocal boundary value problems for parabolic equations, Differentsial’nye Uravneniya, 15 (1979), .1, 74–78. (in Russian) 7. G. Korn, T. Korn, Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov, [Mathematical handbook for scientists and engineers] Opredeleniya, teoremy, formuly, [Definitions, theorems, and formulas] Translated from the second English edition by I. G. Aramanovich, A.M. Berezman, I.A. Vainshtein, L.Z. Rumshiskiўi and L.Ya. Tslaf. Edited by Aramanovich. Fifth edition. “Nauka”, Moscow, 1984. 832 p. (in Russian) 8. B.M. Levitan, Expansion in Fourier series and integrals with Bessel functions, Uspehi Matem. Nauk (N.S.), 6 (1951), .2(42), 102–143. (in Russian) 9. A.H. Mamyan, General boundaty problems on layer, Dokl. Akad. Nauk SSSR, 267 (1982), .2, 292–293. (in Russian) 10. O.V. Martynyuk, The Cauchy problem for singular evolution equations in countably normed spaces of infinitely differentiable functions. I, Zb. Nauk. Prats’. Kamianets-Podilskogo Univ., Fiz.-Mat., 5 (2011), 179–192. (in Ukrainian) 11. O.V. Martynyuk, The Cauchy problem for singular evolution equations in countably normed spaces of infinitely differentiable functions. II, Zb. Nauk. Prats’. Kamianets-Podilskogo Univ., Fiz.-Mat., 6 (2012), 157–171. (in Ukrainian) 12. O.V. Martynyuk, The Cauchy problem for singular evolution equations in countably normed spaces of infinitely differentiable functions. III, Nauk. Visn. Chernivets’kogo Univ., Mat., 2 (2012), .1, 55–62. (in Ukrainian) 13. O.V. Martynyuk, V.V. Gorodets’kyj, The Cauchy problem for singular evolution equations with coefficients unbounded in time, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, 2 (2012), 19–24. (in Ukrainian) 14. M.I. Matiichuk, Parabolic singularly boundary poblem, Inst. Math. NANU, 1999, 176 p. (in Ukrainian) 15. A.M. Nakhushev, On nonlocal boundary value problems with displacement and their relationship with loaded equations, Differentsial’nye Uravneniya, 21 (1985), 92–101. (in Russian) 16. B.Yu. Ptashnyk, V.S. Il’kiv, I.Ya. Kmit’, V.M. Polischuk, Nonlocal boundary value problems with partial differential equations, Naukova Dumka, 2002, 416 p. (in Ukrainian) 17. V.K. Romanko, Boundary value problems for a certain class of differential operators, Differencial’nye Uravnenija, 10 (1974), 117–131. (in Russian) 18. V.K. Romanko, Boundary value problems for certain operator-differential equations, Dokl. Akad. Nauk SSSR, 227 (1976), .4, 812–815. (in Russian)
Pages	165-180
Volume	42
Issue	2
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Problems nonlocal with respect to time for a class of singular evolution equations