Solutions of parabolic equations  from families of Banach spaces depending on time

S. D. Ivasyshen

A version of Carleman's formula for functions holomorphic in a rectangle is proved. It is applied to the evaluation of the integral of $\zeta$-function logarithm with the summing factor $\exp(-t)$ along the critical line. This allowed to obtain a new statement equivalent to the riemann hypothesis.

1. Eidelman S.D. Fundamental matrices of solutions of general parabolic systems// Dokl. AN SSSR. – 1958. – V.120, №5. – P. 980–983. (in Russian)

2. Eidelman S.D. On fundamental solutions of parabolic systems. II// Mat. Sbornik. – 1961. – V.53, №1. – P. 73–136. (in Russian)

3. Eidelman S.D. Parabolic systems. – Moscow: Nauka, 1964. – 443 p. (in Russian) English edition: North- Holland, Amsterdam, 1969.

4. Ivasyshen S.D. Integral representation and initial values of solutions of $\overrightarrow{2b}$-parabolic systems// Ukr. Math. Zh. – 1990. – V.42, №4. – P. 500–506. (in Russian)

5. Eidelman S.D., Ivasyshen S.D. On solutions of parabolic equations from families of Banach spaces dependent on time// Operator Theory: Adv. and Appl. – 2000. – V.117. – P. 111–125.

6. Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and pseudodifferential equations of parabolic type// Operator Theory: Adv. and Appl. – 2004. – V.152. – 390 p.

7. Eidelman S.D. On a class of parabolic systems// Dokl. AN SSSR. – 1960. – V.133, №1. – P. 40–43. (in Russian)

8. Ivasyshen S.D., Eidelman S.D. $\overrightarrow{2b}$-parabolic systems// Trudy Sem. Funkt. Anal. – Kiev, Inst. Mat. AN Ukr. SSR. – 1968. – V.1. – P. 3–175, 271–273. (in Russian)

9. Ivasyshen S.D., Kondur O.S. Properties of some class of solutions for the homogeneous parabolic by Peyrovski system of arbitrary order// Dop. NAN Ukr. – 1996. – №11. – P. 12–15. (in Ukrainian)

10. Ivasyshen S.D., Kondur O.S. On the Green matrix of the Cauchy problem and the characterization of certain classes of solutions for $\overrightarrow{2b}$-parabolic systems of an arbitrary order// Mat. Stud. – 2000. – V.14, №1. – P. 73–84. (in Ukrainian)

11. Ivasyshen S.D., Dutchak T.V. Integral representations and initial values of solutions for $\overrightarrow{2b}$-parabolic systems from weighting Orlich spaces// Visnyk. Kyiv. Univer. Matematyka i mechanika. – 1989. – V.31. – P. 27–33. (in Ukrainian)

12. Ivasyshen S.D., Lavrenchuk V.P. On a integral representation of solutions for the parabolic system of linear equations with Bessel operator// Nelineinye granichnye zadachi: mezhved. sbornik nauch. tr. – 1992. – V.4. – P. 19–25. (in Russian)

13. Voznyak O.G., Ivasyshen S.D. The Cauchy problem for parabolic systems with degeneration on the initial hyperplane// Dop. AN Ukr. – 1994. – №6. – P. 7–11. (in Ukrainian)

14. Voznyak O.G. On integral representations of solutions for parabolic systems with the degeneration on the initial hyperplane// Materialy mizhnar. mat. conf., prysviach. pamiati Hansa Hana. – Chernivtsi: Ruta, 1995. – P. 42–60. (in Ukrainian)

15. Balabushenko T.M., Ivasyshen S. D., Lavrenchuk V.P., Melnychuk L.M. The fundamental solutions of the Cauchy problem for some parabolic equations with Bessel operator and increasing coefficients// Nauk. Visnyk Cherniver. Univer. – 2006. – V.288. – P. 5–11. (in Ukrainian)

16. Balabushenko T.M., Ivasyshen S. D., Lavrenchuk V.P., Melnychuk L.M. The Cauchy problem for some parabolic equations with Bessel operator and increasing coefficients// Nauk. Visnyk Cherniver. Univer. – 2006. – V. 314–315. – P. 7–16. (in Ukrainian)

17. Balabushenko T.M., Ivasyshen S. D., Lavrenchuk V.P., Melnychuk L.M. The integral of solutions some parabolic equations with Bessel operator and increasing coefficients// Nauk. Visnyk Cherniver. Univer. – 2007. – V.336–337. – P. 7–15. (in Ukrainian)

18. Ivasyshen S.D., Pasichnyk H.S. Cauchy problem for the Fokker–Planck–Kolmogorov equation of multidimensional normal Markov process// Mat. Metody Fiz.-Mech. Polya. – 2010. – V.53, №1. – P. 15–22. (in Ukrainian)

19. Ivasyshen S.D., Pasichnyk G.S. On the Cauchy problem for $\overrightarrow{2b}$-parabolic systems with increasing coefficients// Ukr. Mat. Zh. – 2000. – V.52, №11. – P. 1484–1496. (in Ukrainian)

20. Ivasyshen S.D., Androsova L.N. On integral representation and initial values of solutions of certain degenerate parabolic equations// Dokl. AN Ukr. SSR. Ser. A. – 1989. – №1. – P. 16–19. (in Russian)

21. Ivasyshen S.D., Androsova L.N. Integral representation of solutions of a class of degenerate parabolic Kolmogorov equations// Diff. Uravn. – 1991. – V.27, №3. – P. 479–487. (in Russian)

22. Ivasyshen S.D., Eidelman S.D. $\overrightarrow{2b}$-parabolic equations with degeneration in some of the variables// Dokl. AN. – 1998. – V.360, №3. – P. 303–305. (in Russian)

23. Ivasyshen S.D., Eidelman S.D. On the Cauchy problem for degenerate equations of Kolmogorov type with a $\overrightarrow{2b}$-parabolic part with respect to the main group of variables// Diff. Uravn. – 2000. – V.36, №4. – P. 527–536. (in Russian)

24. Ivasyshen S.D., Eidelman S.D. On the integral representation of solutions of Kolmogorov type degenerate equations with a $\overrightarrow{2b}$-parabolic part with respect to the main group of variables// Diff. Uravn. – 2000. – V.36, №5. – P. 647–655. (in Russian)

25. Dron’ V.S., Ivasyshen S.D. On correct solvability of the Cauchy problem degenerate parabolic equations of Kolmogorov type// Ukr. Mat. Visnyk. – 2004. – V.1, №1. – P. 61–68. (in Ukrainian)

26. Ivasyshen S.D., Lajuk V.V. The Cauchy problem for some degenerate parabolic Kolmogorov type equations// Mat. Metody Fiz.-Mech. Polya. – 2007. – V.50, №3. – P. 56–65. (in Ukrainian)

27. Ivasyshen S.D., Lajuk V.V. Characterization solutions for some class ultraparabolic equations of Kolmogorov type// Ukr. Mat. Visnyk. – 2010. – V.7, №1. – P. 1–38. (in Ukrainian)

28. Voznyak O.G. , Ivasyshen S.D. Fundamental solutions of the Cauchy problem for a class of parabolic equations, and their applications// Dop. NAN Ukr. – 1996. – №10. – P. 11–16. (in Ukrainian)

29. Ivasyshen S.D.,Voznyak O.G. On fundamental solutions of the Cauchy problem for a class of degenerate parabolic equations// Mat. Metody Fiz.-Mech. Polya. – 1998. – V.41, №2. – P. 13–19. (in Ukrainian)

30. Ivasyshen S.D., Kondur O.S. On integral representations of solutions for normal parabolic boundary problems// Integralni peretvorennia ta ich zastosuvannia do kraiovych zadach: zb. nauk. pr. – Keiv: Inst. Mat. AN Ukr. – 1993. – V.4. – P. 82–96. (in Ukrainian)

31. Kondur O.S. Normal parabolic boundary problems with discontinuous coefficients// Dop. AN Ukr. – 1994. – №12. – P. 18–22. (in Ukrainian)

32. Ivasyshen S.D. A priory estimates of solutions for the $\overrightarrow{2b}$-parabolic systems// Dop. AN Ukr. RSR. – 1965. – №9. – P. 1121–1125. (in Ukrainian)

33. Ivasyshen S.D., Lavrenchuk V.P. On solvobility for the Cauchy problem and boundary problems for general parabolic systems in a class of increasing functions// Dop. AN Ukr. RSR. Ser. A. – 1967. – №4. – P. 299–303. (in Ukrainian)

34. Ivasyshen S.D. Integral representations solutions for general parabolic boundary problems and correct solvability in spaces of increasing functions// Dop. AN Ukr. RSR. Ser. A. – 1973. – №7. – P. 596–599. (in Ukrainian)

35. Ivasyshen S.D., Lavrenchuk V.P. On correct solvability general boundary problems for parabolic system with increasing coefficients// Ukr. Mat. Zh. – 1978. – V.30, №1. – P. 100–106. (in Russian)

36. Ivasyshen S.D. On correct solvability parabolic boundary problems in spaces of increasing functions// Ukr. Mat. Zh. – 1982. – V.34, №1. – P. 25–30. (in Russian)

37. Ivasyshen S.D. Green matrices of parabolic boundary value problems. – Kiev: Vyshcha shkola, 1990. – 200 p. (in Russian)

38. Ivasyshen S.D., Medynsky I.P. Cauchy problem for $\overrightarrow{2b}$-parabolic systems with degeneration on the initial hyperplane// Mat. Metody Fiz.-Mech. Polya. – 2003. – V.46, №3. – P. 15–24. (in Ukrainian)

39. Ivasyshen S.D.,Medynsky I.P. A priory estimates of solutions for $\overrightarrow{2b}$-parabolic system with the degeneration on the initial hyperplane// Nelin. analiz: Pr. Ukr. Mat. Kongres-2001, Kyiv: Ins. Mat. NAN Ukr. – 2006. – P. 28–41. (in Ukrainian)

40. Ivasyshen S.D., Ivasyuk H.P. Solonnikov parabolic systems with quasihomogeneous structure// Ukr. Mat. Zh. – 2006. – V.58, №11. – P. 1501–1510. (in Ukrainian)

41. Ivasyshen S.D., Ivasyuk H.P. Initial value problems for Solonnikov–Eidelman parabolic systems// Dop. NAN. Ukr. – 2007. – №9. – P. 7–11. (in Ukrainian)

42. Ivasyshen S.D., Ivasyuk H.P. Correct solvability Solonnikov–Eidelman initial value problems// Ukr. Mat. Zh. – 2009. – V.61, №5. – P. 650–671. (in Ukrainian)

	Solutions of parabolic equations from families of Banach spaces depending on time(in Ukrainian)
Author	S. D. Ivasyshen ivasyshen_-sd@mail.ru National Technical University of Ukraine, ``Kyiv Polytechnic Institute''
Abstract	A version of Carleman's formula for functions holomorphic in a rectangle is proved. It is applied to the evaluation of the integral of $\zeta$-function logarithm with the summing factor $\exp(-t)$ along the critical line. This allowed to obtain a new statement equivalent to the riemann hypothesis.
Keywords	parabolic equation; degenerate parabolic equation; Cauchy problem; boundary-value problem; weight space of functions and generalized measures; families of Banach spaces depending on time; correct solvability; integral representation; set of initial values
Reference	1. Eidelman S.D. Fundamental matrices of solutions of general parabolic systems// Dokl. AN SSSR. – 1958. – V.120, №5. – P. 980–983. (in Russian) 2. Eidelman S.D. On fundamental solutions of parabolic systems. II// Mat. Sbornik. – 1961. – V.53, №1. – P. 73–136. (in Russian) 3. Eidelman S.D. Parabolic systems. – Moscow: Nauka, 1964. – 443 p. (in Russian) English edition: North- Holland, Amsterdam, 1969. 4. Ivasyshen S.D. Integral representation and initial values of solutions of $\overrightarrow{2b}$-parabolic systems// Ukr. Math. Zh. – 1990. – V.42, №4. – P. 500–506. (in Russian) 5. Eidelman S.D., Ivasyshen S.D. On solutions of parabolic equations from families of Banach spaces dependent on time// Operator Theory: Adv. and Appl. – 2000. – V.117. – P. 111–125. 6. Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and pseudodifferential equations of parabolic type// Operator Theory: Adv. and Appl. – 2004. – V.152. – 390 p. 7. Eidelman S.D. On a class of parabolic systems// Dokl. AN SSSR. – 1960. – V.133, №1. – P. 40–43. (in Russian) 8. Ivasyshen S.D., Eidelman S.D. $\overrightarrow{2b}$-parabolic systems// Trudy Sem. Funkt. Anal. – Kiev, Inst. Mat. AN Ukr. SSR. – 1968. – V.1. – P. 3–175, 271–273. (in Russian) 9. Ivasyshen S.D., Kondur O.S. Properties of some class of solutions for the homogeneous parabolic by Peyrovski system of arbitrary order// Dop. NAN Ukr. – 1996. – №11. – P. 12–15. (in Ukrainian) 10. Ivasyshen S.D., Kondur O.S. On the Green matrix of the Cauchy problem and the characterization of certain classes of solutions for $\overrightarrow{2b}$-parabolic systems of an arbitrary order// Mat. Stud. – 2000. – V.14, №1. – P. 73–84. (in Ukrainian) 11. Ivasyshen S.D., Dutchak T.V. Integral representations and initial values of solutions for $\overrightarrow{2b}$-parabolic systems from weighting Orlich spaces// Visnyk. Kyiv. Univer. Matematyka i mechanika. – 1989. – V.31. – P. 27–33. (in Ukrainian) 12. Ivasyshen S.D., Lavrenchuk V.P. On a integral representation of solutions for the parabolic system of linear equations with Bessel operator// Nelineinye granichnye zadachi: mezhved. sbornik nauch. tr. – 1992. – V.4. – P. 19–25. (in Russian) 13. Voznyak O.G., Ivasyshen S.D. The Cauchy problem for parabolic systems with degeneration on the initial hyperplane// Dop. AN Ukr. – 1994. – №6. – P. 7–11. (in Ukrainian) 14. Voznyak O.G. On integral representations of solutions for parabolic systems with the degeneration on the initial hyperplane// Materialy mizhnar. mat. conf., prysviach. pamiati Hansa Hana. – Chernivtsi: Ruta, 1995. – P. 42–60. (in Ukrainian) 15. Balabushenko T.M., Ivasyshen S. D., Lavrenchuk V.P., Melnychuk L.M. The fundamental solutions of the Cauchy problem for some parabolic equations with Bessel operator and increasing coefficients// Nauk. Visnyk Cherniver. Univer. – 2006. – V.288. – P. 5–11. (in Ukrainian) 16. Balabushenko T.M., Ivasyshen S. D., Lavrenchuk V.P., Melnychuk L.M. The Cauchy problem for some parabolic equations with Bessel operator and increasing coefficients// Nauk. Visnyk Cherniver. Univer. – 2006. – V. 314–315. – P. 7–16. (in Ukrainian) 17. Balabushenko T.M., Ivasyshen S. D., Lavrenchuk V.P., Melnychuk L.M. The integral of solutions some parabolic equations with Bessel operator and increasing coefficients// Nauk. Visnyk Cherniver. Univer. – 2007. – V.336–337. – P. 7–15. (in Ukrainian) 18. Ivasyshen S.D., Pasichnyk H.S. Cauchy problem for the Fokker–Planck–Kolmogorov equation of multidimensional normal Markov process// Mat. Metody Fiz.-Mech. Polya. – 2010. – V.53, №1. – P. 15–22. (in Ukrainian) 19. Ivasyshen S.D., Pasichnyk G.S. On the Cauchy problem for $\overrightarrow{2b}$-parabolic systems with increasing coefficients// Ukr. Mat. Zh. – 2000. – V.52, №11. – P. 1484–1496. (in Ukrainian) 20. Ivasyshen S.D., Androsova L.N. On integral representation and initial values of solutions of certain degenerate parabolic equations// Dokl. AN Ukr. SSR. Ser. A. – 1989. – №1. – P. 16–19. (in Russian) 21. Ivasyshen S.D., Androsova L.N. Integral representation of solutions of a class of degenerate parabolic Kolmogorov equations// Diff. Uravn. – 1991. – V.27, №3. – P. 479–487. (in Russian) 22. Ivasyshen S.D., Eidelman S.D. $\overrightarrow{2b}$-parabolic equations with degeneration in some of the variables// Dokl. AN. – 1998. – V.360, №3. – P. 303–305. (in Russian) 23. Ivasyshen S.D., Eidelman S.D. On the Cauchy problem for degenerate equations of Kolmogorov type with a $\overrightarrow{2b}$-parabolic part with respect to the main group of variables// Diff. Uravn. – 2000. – V.36, №4. – P. 527–536. (in Russian) 24. Ivasyshen S.D., Eidelman S.D. On the integral representation of solutions of Kolmogorov type degenerate equations with a $\overrightarrow{2b}$-parabolic part with respect to the main group of variables// Diff. Uravn. – 2000. – V.36, №5. – P. 647–655. (in Russian) 25. Dron’ V.S., Ivasyshen S.D. On correct solvability of the Cauchy problem degenerate parabolic equations of Kolmogorov type// Ukr. Mat. Visnyk. – 2004. – V.1, №1. – P. 61–68. (in Ukrainian) 26. Ivasyshen S.D., Lajuk V.V. The Cauchy problem for some degenerate parabolic Kolmogorov type equations// Mat. Metody Fiz.-Mech. Polya. – 2007. – V.50, №3. – P. 56–65. (in Ukrainian) 27. Ivasyshen S.D., Lajuk V.V. Characterization solutions for some class ultraparabolic equations of Kolmogorov type// Ukr. Mat. Visnyk. – 2010. – V.7, №1. – P. 1–38. (in Ukrainian) 28. Voznyak O.G. , Ivasyshen S.D. Fundamental solutions of the Cauchy problem for a class of parabolic equations, and their applications// Dop. NAN Ukr. – 1996. – №10. – P. 11–16. (in Ukrainian) 29. Ivasyshen S.D.,Voznyak O.G. On fundamental solutions of the Cauchy problem for a class of degenerate parabolic equations// Mat. Metody Fiz.-Mech. Polya. – 1998. – V.41, №2. – P. 13–19. (in Ukrainian) 30. Ivasyshen S.D., Kondur O.S. On integral representations of solutions for normal parabolic boundary problems// Integralni peretvorennia ta ich zastosuvannia do kraiovych zadach: zb. nauk. pr. – Keiv: Inst. Mat. AN Ukr. – 1993. – V.4. – P. 82–96. (in Ukrainian) 31. Kondur O.S. Normal parabolic boundary problems with discontinuous coefficients// Dop. AN Ukr. – 1994. – №12. – P. 18–22. (in Ukrainian) 32. Ivasyshen S.D. A priory estimates of solutions for the $\overrightarrow{2b}$-parabolic systems// Dop. AN Ukr. RSR. – 1965. – №9. – P. 1121–1125. (in Ukrainian) 33. Ivasyshen S.D., Lavrenchuk V.P. On solvobility for the Cauchy problem and boundary problems for general parabolic systems in a class of increasing functions// Dop. AN Ukr. RSR. Ser. A. – 1967. – №4. – P. 299–303. (in Ukrainian) 34. Ivasyshen S.D. Integral representations solutions for general parabolic boundary problems and correct solvability in spaces of increasing functions// Dop. AN Ukr. RSR. Ser. A. – 1973. – №7. – P. 596–599. (in Ukrainian) 35. Ivasyshen S.D., Lavrenchuk V.P. On correct solvability general boundary problems for parabolic system with increasing coefficients// Ukr. Mat. Zh. – 1978. – V.30, №1. – P. 100–106. (in Russian) 36. Ivasyshen S.D. On correct solvability parabolic boundary problems in spaces of increasing functions// Ukr. Mat. Zh. – 1982. – V.34, №1. – P. 25–30. (in Russian) 37. Ivasyshen S.D. Green matrices of parabolic boundary value problems. – Kiev: Vyshcha shkola, 1990. – 200 p. (in Russian) 38. Ivasyshen S.D., Medynsky I.P. Cauchy problem for $\overrightarrow{2b}$-parabolic systems with degeneration on the initial hyperplane// Mat. Metody Fiz.-Mech. Polya. – 2003. – V.46, №3. – P. 15–24. (in Ukrainian) 39. Ivasyshen S.D.,Medynsky I.P. A priory estimates of solutions for $\overrightarrow{2b}$-parabolic system with the degeneration on the initial hyperplane// Nelin. analiz: Pr. Ukr. Mat. Kongres-2001, Kyiv: Ins. Mat. NAN Ukr. – 2006. – P. 28–41. (in Ukrainian) 40. Ivasyshen S.D., Ivasyuk H.P. Solonnikov parabolic systems with quasihomogeneous structure// Ukr. Mat. Zh. – 2006. – V.58, №11. – P. 1501–1510. (in Ukrainian) 41. Ivasyshen S.D., Ivasyuk H.P. Initial value problems for Solonnikov–Eidelman parabolic systems// Dop. NAN. Ukr. – 2007. – №9. – P. 7–11. (in Ukrainian) 42. Ivasyshen S.D., Ivasyuk H.P. Correct solvability Solonnikov–Eidelman initial value problems// Ukr. Mat. Zh. – 2009. – V.61, №5. – P. 650–671. (in Ukrainian)
Pages	172-181
Volume	40
Issue	2
Year	2013
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Solutions of parabolic equations from families of Banach spaces depending on time(in Ukrainian)