Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Existence and Behavior Results for a Nonlocal Nonlinear Parabolic Equation with Variable Exponent

  • Sert, Ugur (Department of Mathematics, Hacettepe University) ;
  • Ozturk, Eylem (Department of Mathematics, Hacettepe University)
  • Received : 2018.10.12
  • Accepted : 2019.11.18
  • Published : 2020.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this article, we study the solvability of the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations with nonstandard growth and nonlocal terms. We prove the existence of weak solutions of the considered problem under more general conditions. In addition, we investigate the behavior of the solution when the problem is homogeneous.

Keywords

References

  1. R. A. Adams, Sobolev spaces, Academic Press, New York, 1975.
  2. N. V. Afanas'eva and A. F. Tedeev, Theorems on the existence and nonexistence of solutions to the Cauchy problem for degenerate parabolic equations with a nonlocal source, Ukrain. Mat. Zh., 57(11)(2005), 1443-1464.
  3. Y. Alkhutov, S. N. Antontsev and V. Zhikov, Parabolic equations with variable order of nonlinearity, Collection of works of Institute of Mathematics NAS of Ukraine, 6(2009), 23-50.
  4. J. R. Anderson and K. Deng, Global existence for degenerate parabolic equations with a non-local forcing, Math. Methods Appl. Sci., 20(1997), 1069-1087. https://doi.org/10.1002/(SICI)1099-1476(19970910)20:13<1069::AID-MMA867>3.0.CO;2-Y
  5. S. N. Antontsev and S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 60(2005), 515-545. https://doi.org/10.1016/j.na.2004.09.026
  6. S. N. Antontsev and S. I. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity, J. Math. Sci., 150(5)(2008), 2289-2301. https://doi.org/10.1007/s10958-008-0129-6
  7. J. Bebernes and A. Bressan, Thermal behaviour for a confined reactive gas, J. Diff. Equations, 44(1992), 118-133. https://doi.org/10.1016/0022-0396(82)90028-6
  8. J. Bebernes and D. Eberly, Mathematical problems from combustion theory, Springer, New York 1989.
  9. M. Chipot and B. Lovat, On the asymptotic behavior of some nonlocal problems, Positivity, 3(1999), 65-81. https://doi.org/10.1023/A:1009706118910
  10. M. Chipot and B. Lovat, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 8(1)(2001), 35-51.
  11. M. Chipot, V. Valente and G. Vergara Cafarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Mat. Univ. Padova, 110(2003), 199-220.
  12. W. B. Deng, Z. W. Duan and C. H. Xie, The blow-up rate for a degenerate parabolic equation with a non-local source, J. Math. Anal. Appl., 264(2001), 577-597. https://doi.org/10.1006/jmaa.2001.7696
  13. K. Deng, M. K. Kwong and H. A. Levine, The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers' equation, Quart. Appl. Math., 50(1992), 173-200. https://doi.org/10.1090/qam/1146631
  14. L. Diening, P. Harjulehto, P. Hasto and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011.
  15. Yu. A. Dubinskii, Weak convergence for nonlinear elliptic and parabolic equations, (Russian) Mat. Sb. (N.S.), 67(109)(1965), 609-642.
  16. F. G. Duzgun and K. Soltanov, Existence, uniqueness and behaviour of solutions for a nonlinear diffusion equation with third type boundary value condition, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26(4)(2015), 445-463. https://doi.org/10.4171/RLM/715
  17. J. Furter and M. Grinfeld, Local vs. nonlocal interactions in population dynamics, J. Math. Biol., 27(1989), 65-80. https://doi.org/10.1007/BF00276081
  18. V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34(1998), 1005-1027. https://doi.org/10.1016/S0362-546X(97)00716-5
  19. A. S. Kalashnikov, Some problems of qualitative theory of nonlinear degenerate second-order parabolic equations, Russian Math. Surveys, 42(1987), 169-222. https://doi.org/10.1070/RM1987v042n02ABEH001309
  20. O. Kovacik and J. Rakosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(1991), 592-618. https://doi.org/10.21136/CMJ.1991.102493
  21. O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and quasilinear equations of parabolic type, Mathematical Monographs 23, American Mathematical Society, Providence, RI, 1969.
  22. J. L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod; Gauthier-Villars, Paris, 1969.
  23. C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.
  24. C. V. Pao, Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory, J. Math. Anal. Appl., 166(2)(1992), 591-600. https://doi.org/10.1016/0022-247X(92)90318-8
  25. V. Radulescu and D. Repovs, Partial differential equations with variable exponents: variational methods and quantitative analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015.
  26. M. M. Rao, Measure theory and integration, John Wiley & Sons, New York, 1987.
  27. P. Rouchon, Universial bounds for global solutions of a diffusion equation with a nonlocal reaction term, J. Differential Equations, 193(2003), 75-94. https://doi.org/10.1016/S0022-0396(03)00039-1
  28. U. Sert and K. N. Soltanov, On solvability of a class of nonlinear elliptic type equation with variable exponent , J. Appl. Anal. Comput., 7(3)(2017), 1139-1160.
  29. U. Sert and K. N. Soltanov, Solvability of nonlinear elliptic type equation with two unrelated nonstandard growths, J. Korean Math. Soc., 55(6)(2018), 1337-1358. https://doi.org/10.4134/JKMS.j170691
  30. K. N. Soltanov, Some embedding theorems and their applications to nonlinear equations, Differensial'nie Uravnenia, 20(12)(1984), 2181-2184.
  31. K. N. Soltanov, On some modification Navier-Stokes equations, Nonlinear Anal., 52(3)(2003), 769-793. https://doi.org/10.1016/S0362-546X(02)00132-3
  32. K. N. Soltanov, Some nonlinear equations of the nonstable filtration type and embedding theorems, Nonlinear Anal., 65(2006), 2103-2134. https://doi.org/10.1016/j.na.2005.11.053
  33. K. N. Soltanov and J. Sprekels, Nonlinear equations in non-reflexive Banach spaces and strongly nonlinear differential equations, Adv. Math. Sci. Appl., 9(2)(1999), 939-972.
  34. P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations, 153(1999), 374-406. https://doi.org/10.1006/jdeq.1998.3535
  35. M. X. Wang and Y. M. Wang, Blow-up properties of positive solutions for nonlocal reaction diffusion problems, Math. Methods Appl. Sci., 19(1996), 1141-1156. https://doi.org/10.1002/(SICI)1099-1476(19960925)19:14<1141::AID-MMA811>3.0.CO;2-9