The Fourier problem for weakly nonlinear integro-differential elliptic-parabolic
systems

M. M. Bokalo; I. V. Skira

doi:10.15330/ms.51.1.59-73

The Fourier problem or, in other words, the problem without initial conditions for weakly nonlinear elliptic-parabolic systems are considered in this paper. The existence and uniqueness solutions of the problem are proved. The estimates of these solutions are received.

1. R.E. Showalter, Singular nonlinear evolution equations, Rocky Mountain J. Math., 10 (1980), №3, 499– 507.

2. R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Amer. Math. Soc., 49 (1997), Providence.

3. M.M. Bokalo, O.M. Buhrii, R.A. Mashiyev, Unique solvability of initial boundary value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, J. Nonl. Evol. Equ. Appl., 2013 (2014), 67-87.

4. M.M. Bokalo, Almost periodic solutions of anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, Electronic Journal of Differential Equations, 169 (2014), 1-13.

5. M. Bokalo, I. Skira, Almost periodic solutions for nonlinear integro-differential elliptic-parabolic equations with variable exponents of nonlinearty, International Journal of Evolution Equations, 10 (2017), №3.4, 297-314.

6. M. Bokalo, I. Skira, Solutions for higher-order anisotropic elliptic-parabolic equations in time unbounded domains, New Trends in Mathematical Sciences, 6 (2018), №2, 29-42.

7. M.M. Bokalo, I.V. Skira, The Fouirer problem for nonlinear integro-differential elliptic-parabolic systems with variable exponents of nonlinearty, Visnyk Lviv. Universytetu. Serija meh.-mat., 83 (2017), 109-122. (in Ukrainian)

8. M. Bokalo, I. Skira, Well-posedness of the Fourier problem for higher-order weakly nonlinear integrodifferential elliptic-parabolic equations, Visnyk of the Lviv Univ. Series Mech. Math., 85 (2018), 91-106.

9. A. N. Tihonov, Uniqueness theorems for the heat equation, Matem. Sbornik, 2 (1935), 199-216.

10. O.A. Oleinik, G.A. Iosifyan, An analogue of Saint-Venants principle and the uniqueness of solutions of boundary value problems for parabolic equations in unbounded domains, Russian Mathematical Surveys, 31, 6 (1976), 153-178.

11. P.Ya. Pukach, On the problem without initial conditions for a nonlinear degenerating parabolic system, Ukrainian Mathematical Journal, 46 (1994), №4, 484-487.

12. S.P. Lavrenyuk, M.B. Ptashnik, Problem without initial conditions for a nonlinear pseudoparabolic system, Differential Equations, 36 (2000), №5, 739-748.

13. M. Bokalo, A. Lorenzi Linear evolution first-order problems without initial conditions, Milan J. Math., 77 (2009), 437-494.

14. N.P. Protsakh, A problem without initial conditions for a nonlinear ultraparabolic equation with degeneration, Journal of Mathematical Sciences, 168 (2010), №4, 505-522.

15. O. Buhrii, N. Buhrii, On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity, New Trends in Mathematical Sciences, 5 (2017), №3, 128-153.

16. O. Buhrii, N. Buhrii, Integro-differential systems with variable exponents of nonlinearity, Open Mathematics, 15 (2017), 859-883.

17. M. Loayza, Asymptotic behavior of solutions to parabolic problems with nonlinear nonlocal terms, Electronic Journal of Differential Equations, 2013 (2013), №228, 1-12.

18. R.E. Showalter, Hilbert space methods for partial differential equations, Monographs and Studies in Mathematics (Monographs in differential equations), V.1, Pitman, London-San Francisco, Calif.- Melbourne, 1977, xii+196 p.

19. H. Gayevskyy, K. Greger, K. Zaharias, Nonlinear operator equations and operator differential equations, Mir, Moscow, 1978.

20. J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires. Dunod Gauthier- Villars, Paris, 1969.

	The Fourier problem for weakly nonlinear integro-differential elliptic-parabolic systems
Author	M. M. Bokalo¹, I. V. Skira² mm.bokalo@gmail.com¹, irusichka.skira@gmail.com² Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	The Fourier problem or, in other words, the problem without initial conditions for weakly nonlinear elliptic-parabolic systems are considered in this paper. The existence and uniqueness solutions of the problem are proved. The estimates of these solutions are received.
Keywords	Fourier problem; elliptic-parabolic system; integro-differential equation; functional-differential system
DOI	doi:10.15330/ms.51.1.59-73
Reference	1. R.E. Showalter, Singular nonlinear evolution equations, Rocky Mountain J. Math., 10 (1980), №3, 499– 507. 2. R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Amer. Math. Soc., 49 (1997), Providence. 3. M.M. Bokalo, O.M. Buhrii, R.A. Mashiyev, Unique solvability of initial boundary value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, J. Nonl. Evol. Equ. Appl., 2013 (2014), 67-87. 4. M.M. Bokalo, Almost periodic solutions of anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, Electronic Journal of Differential Equations, 169 (2014), 1-13. 5. M. Bokalo, I. Skira, Almost periodic solutions for nonlinear integro-differential elliptic-parabolic equations with variable exponents of nonlinearty, International Journal of Evolution Equations, 10 (2017), №3.4, 297-314. 6. M. Bokalo, I. Skira, Solutions for higher-order anisotropic elliptic-parabolic equations in time unbounded domains, New Trends in Mathematical Sciences, 6 (2018), №2, 29-42. 7. M.M. Bokalo, I.V. Skira, The Fouirer problem for nonlinear integro-differential elliptic-parabolic systems with variable exponents of nonlinearty, Visnyk Lviv. Universytetu. Serija meh.-mat., 83 (2017), 109-122. (in Ukrainian) 8. M. Bokalo, I. Skira, Well-posedness of the Fourier problem for higher-order weakly nonlinear integrodifferential elliptic-parabolic equations, Visnyk of the Lviv Univ. Series Mech. Math., 85 (2018), 91-106. 9. A. N. Tihonov, Uniqueness theorems for the heat equation, Matem. Sbornik, 2 (1935), 199-216. 10. O.A. Oleinik, G.A. Iosifyan, An analogue of Saint-Venants principle and the uniqueness of solutions of boundary value problems for parabolic equations in unbounded domains, Russian Mathematical Surveys, 31, 6 (1976), 153-178. 11. P.Ya. Pukach, On the problem without initial conditions for a nonlinear degenerating parabolic system, Ukrainian Mathematical Journal, 46 (1994), №4, 484-487. 12. S.P. Lavrenyuk, M.B. Ptashnik, Problem without initial conditions for a nonlinear pseudoparabolic system, Differential Equations, 36 (2000), №5, 739-748. 13. M. Bokalo, A. Lorenzi Linear evolution first-order problems without initial conditions, Milan J. Math., 77 (2009), 437-494. 14. N.P. Protsakh, A problem without initial conditions for a nonlinear ultraparabolic equation with degeneration, Journal of Mathematical Sciences, 168 (2010), №4, 505-522. 15. O. Buhrii, N. Buhrii, On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity, New Trends in Mathematical Sciences, 5 (2017), №3, 128-153. 16. O. Buhrii, N. Buhrii, Integro-differential systems with variable exponents of nonlinearity, Open Mathematics, 15 (2017), 859-883. 17. M. Loayza, Asymptotic behavior of solutions to parabolic problems with nonlinear nonlocal terms, Electronic Journal of Differential Equations, 2013 (2013), №228, 1-12. 18. R.E. Showalter, Hilbert space methods for partial differential equations, Monographs and Studies in Mathematics (Monographs in differential equations), V.1, Pitman, London-San Francisco, Calif.- Melbourne, 1977, xii+196 p. 19. H. Gayevskyy, K. Greger, K. Zaharias, Nonlinear operator equations and operator differential equations, Mir, Moscow, 1978. 20. J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires. Dunod Gauthier- Villars, Paris, 1969.
Pages	59-73
Volume	51
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

The Fourier problem for weakly nonlinear integro-differential elliptic-parabolic systems