# Fourier problems for parabolic equations with variable exponents of nonlinearity and time delay

Author
Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract
The Fourier problem for nonlinear parabolic equations with variable exponents of nonlinearity and time delay is considered. The existence and uniqueness of weak solutions of the problem are investigated. Also, its a priori estimates are obtained.
Keywords
nonlinear parabolic equation; equation with time delay; Fourier problem; anisotropic parabolic equation; equation with variable exponents of nonlinearity
DOI
doi:10.15330/ms.47.1.47-58
Reference
1. Alkhutov Y., Antontsev S., Zhikov V., Parabolic equations with variable order of nonlinearity, Collection of works of Institute of Mathematics NAS of Ukraine, 6 (2009), 23-50.

2. Antontsev S., Shmarev S., Extinction of solutions of parabolic equations with variable anisotropic nonlinearities, Proceedings of the Steklov Institute of Mathematics, 261 (2008), 11-21.

3. Antontsev S., Shmarev S., Evolution PDEs with nonstandard growth conditions. Existence, uniqueness, localization, blow-up, Atlantis Studies in Differential Equations, 4, Atlantis Press, Paris, 2015, xviii+409 p.

4. Batkai A., Piazzera S., Semigroups for delay equations, in Resarch Notes in Mathematics, 10, A.K. Peters: Wellesley MA, 2005.

5. Batkai A., Schnaubelt R., Asymptotic behaviour of parabolic problems with delays in the highest order derivatives, Semigroup Forum, 69 (2004), ¹3, 369-399.

6. Bokalo M.M., The unique solvability of a problem without initial conditions for linear and nonlinear elliptic-parabolic equations, Ukr. Mat. Visn., 8 (2011), ¹1, 55-86, (in Ukrainian); Engl. Transl.: Journ. of Math. Sciences, 178 (2011), ¹1, 41-64.

7. Bokalo M., Dynamical problems without initial conditions for elliptic-parabolic equations in spatial unbounded domains, Electronic Journal of Differential Equations, 178 (2010), 1-24.

8. Bokalo N.M., Problem without initial conditions for some classes of nonlinear parabolic equations, J. Sov. Math., 51 (1990), ¹3, 2291-2322.

9. Bokalo M.M., Buhrii O.M., Mashiyev R.A., Unique solvability of initial boundary value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity, Journal of nonlinear evolution equations and applications, 2013 (2014), ¹6, 67-87.

10. Bokalo M., Dmytriv V., On a Fourier problem for coupled evolution system of equations with integral time delays, Visnyk of the Lviv Univ. Series Mech. Math., 60 (2002), 32-49.

11. Bokalo M., Ilnytska O., Problems for parabolic equations with variable exponents of nonlinearity and time delay, Applicable Analysis, 2016, http://dx.doi.org/10.1080/00036811.2016.1183771.

12. Bokalo M., Lorenzi A., Linear evolution first-order problems without initial conditions, Milan Journal of Mathematics. 77 (2009), 437-494.

13. Bokalo M.M., Pauchok I.B., On the well-posedness of the Fourier problem for higher-order nonlinear parabolic equations with variable exponents of nonlinearity, Mat. Stud., 26 (2006), ¹1, 25-48.

14. Bokalo M.M., Sikorskyy V.M., About properties of solutions of problem without initial conditions for equations generalized politropic filtration equation, Visnyk of the Lviv University. Serija meh.-mat., 51 (1998), 85-98.

15. Chueshov I., Rezounenko A., Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Commun. Pure Appl. Anal., 14 (2015), 1685-1704.

16. Coddington E.A., Levinson N., Theory of ordinary differential equations, New York, Toronto, London: McGraw-Hill book company, 1955.

17. Di Blasio G., Kunisch K., Sinestrari E., L2-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives, Journal of Mathematical Analysis and Applications, 102 (1984), 38-57.

18. Diening L., Harjulehto P., Hasto P., Ruzicka M., Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011, x+509 pp.

19. Dmytriv V.M., On a Fourier problem for coupled evolution system of equations with time delays, Mat. Stud., 16 (2001), 141-156.

20. Elsholts L.E., Norkin S.B., Introduction to the theory of differential equations with deviating argument, Moscow (RF), Nauka, 1971.

21. Ezzinbi K., Liu J.H., Periodic solutions of non-densely defined delay evolution equations, Journal of Applied Mathematics and Stochastic Analysis, 15 (2002), ¹2, 105-114.

22. Fan X., Zhao D., On the space $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$ Journal of Mathematical Analysis and Applications, 263 (2001), 424-446.

23. Fu Y., Pan N., Existence of solutions for nonlinear parabolic problem with p(x)-growth, Journal of Mathematical Analysis and Applications, 362 (2010), 313-326.

24. Jin Ch., Yin J., Traveling wavefronts for a time delayed non-Newtonian filtration equation, Physica D., 241 (2012), 1789-1803.

25. Khusainov D., Pokojovy M., Racke R., Strong and mild extrapolated L2-solutions to the heat equation with constant delay, SIAM Journal on Mathematical Analysis, 47 (2015), ¹1, 427-454.

26. Kova.cik O., Rakosnic J., On spaces Lp(x) and Wk; p(x), Czechoslovak Mathematical Journal, 41 (1991), ¹116, 592-618.

27. Mashiyev R.A., Buhrii O.M. Existence of solutions of the parabolic variational inequality with variable exponent of nonlinearity, Journal of Mathematical Analysis and Applications, 377 (2011), 450-463.

28. Musielak J., Orlicz spaces and modular spaces. Lecture Notes in Mathematics, 1034, Berlin-Heidelberg, Springer Verlag, 1983.

29. Mihailescu M., Radulescu V., Tersian S., Homoclinic solutions of difference equations with variable exponents, Topological Methods in Nonlinear Analysis, 38 (2011), 277-289.

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

31. Oleinik O.A., Iosifjan G.A., Analog of Saint-Venants principle and uniqueness of solutions of the boundary problems in unbounded domain for parabolic equations, Usp. Mat. Nauk., 31 (1976), ¹6, 142-166.

32. Orlicz W., Uber konjugierte Exponentenfolgen, Studia Mathematica (Lwow), 3 (1931), 200-211.

33. Pankov A.A., Bounded and almost periodic solutions of nonlinear operator differential equations, Kluwer, Dordrecht, 1990.

34. Rezounenko A.V., Wu J. A non-local PDE model for population dynamics with state-selective delay: Local theory and global attractors, Journal of Computational and Applied Mathematics, 190 (2006), 99-113.

35. Ruzicka M., Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, 1748, Berlin, Springer-Verlag, 2000.

36. Tihonov A.N., Uniqueness theorems for the heat equation, Matem. Sbornik, 2 (1935), 510-516.

37. Showalter R.E., Singular nonlinear evolution equations, Rocky Mountain J. Math., 10 (1980), ¹3, 499-507.

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

Pages
47-58
Volume
47
Issue
1
Year
2017
Journal
Matematychni Studii
Full text of paper
Table of content of issue