On Global Existence of the Fractional Reaction-Diffusion System’s Solution

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-21-2023-11
Published: Feb 21, 2023
Cite as:
Iqbal M. Batiha, Nabila Barrouk, Adel Ouannas, Waseem G. Alshanti, On Global Existence of the Fractional Reaction-Diffusion System’s Solution, Int. J. Anal. Appl., 21 (2023), 11.

Main Article Content

Iqbal M. Batiha, Nabila Barrouk, Adel Ouannas, Waseem G. Alshanti

Abstract

The purpose of this paper is to prove the global existence of solution for one of most significant fractional partial differential system called the fractional reaction-diffusion system. This will be carried out by combining the compact semigroup methods with some L1-estimate methods. Our investigation can be applied to a wide class of fractional partial differential equations even if they contain nonlinear terms in their constructions.

Article Details

References

  1. M. Ilic, F. Liu, I. Turner, V. Anh, Numerical Approximation of a Fractional-In-Space Diffusion Equation (II) – with Nonhomogeneous Boundary Conditions, Fract. Calc. Appl. Anal. 9 (2006), 333-349. http://eudml.org/doc/11286.
  2. G. Karch, Nonlinear Evolution Equations With Anomalous Diffusion, In: Qualitative Properties of Solutions to Partial Differential Equations. Jindrich Necas Center for Mathematical Modelling Lecture Notes, vol. 5 (Matfyzpress, Prague, 2009), pp. 25–68.
  3. R.A. Fisher, the Wave of Advance of Advantageous Genes, Ann. Eugenics. 7 (1937), 355–369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x.
  4. S. Bonafede, D. Schmitt, "Triangular" Reaction–diffusion Systems With Integrable Initial Data, Nonlinear Anal.: Theory Methods Appl. 33 (1998), 785–801. https://doi.org/10.1016/s0362-546x(98)00042-x.
  5. T. Diagana, Some Remarks on Some Strongly Coupled Reaction-Diffusion Equations, (2003). https://doi.org/10.48550/ARXIV.MATH/0305152.
  6. S. Kouachi, A. Youkana, Global Existence for a Class of Reaction-Diffusion Systems, Bull. Polish. Acad. Sci. Math. 49 (2001), 1-6.
  7. T.E. Oussaeif, B. Antara, A. Ouannas, et al. Existence and Uniqueness of the Solution for an Inverse Problem of a Fractional Diffusion Equation with Integral Condition, J. Funct. Spaces. 2022 (2022), 7667370. https://doi.org/10.1155/2022/7667370.
  8. A. Ouannas, F. Mesdoui, S. Momani, et al. Synchronization of Fitzhugh-Nagumo Reaction-Diffusion Systems via One-Dimensional Linear Control Law, Arch. Control Sci. 31 (2021), 333–345. https://doi.org/10.24425/acs.2021.137421.
  9. I.M. Batiha, A. Ouannas, R. Albadarneh, et al. Existence and Uniqueness of Solutions for Generalized Sturm–liouville and Langevin Equations via Caputo–hadamard Fractional-Order Operator, Eng. Comput. 39 (2022), 2581–2603. https://doi.org/10.1108/ec-07-2021-0393.
  10. Z. Chebana, T.E. Oussaeif, A. Ouannas, et al. Solvability of Dirichlet Problem for a Fractional Partial Differential Equation by Using Energy Inequality and Faedo-Galerkin Method, Innov. J. Math. 1 (2022), 34–44. https://doi.org/10.55059/ijm.2022.1.1/4.
  11. N.D. Alikakos, Lp -Bounds of Solutions of Reaction-Diffusion Equations, Commun. Part. Differ. Equ. 4 (1979), 827–868. https://doi.org/10.1080/03605307908820113.
  12. K. Masuda, On the Global Existence and Asymptotic Behavior of Solutions of Reaction-Diffusion Equations, Hokkaido Math. J. 12 (1983), 360-370. https://doi.org/10.14492/hokmj/1470081012.
  13. A. Haraux, A. Youkana, On a Result of K. Masuda Concerning Reaction-Diffusion Equations, Tohoku Math. J. (2). 40 (1988), 159-163. https://doi.org/10.2748/tmj/1178228084.
  14. A. Barabanova, On the Global Existence of Solutions of a Reaction-Diffusion Equation With Exponential Nonlinearity, Proc. Amer. Math. Soc. 122 (1994), 827-831.
  15. T.E. Oussaeif, B. Antara, A. Ouannas, et al. Existence and Uniqueness of the Solution for an Inverse Problem of a Fractional Diffusion Equation with Integral Condition, J. Funct. Spaces. 2022 (2022), 7667370. https://doi.org/10.1155/2022/7667370.
  16. I.M. Batiha, Z. Chebana, T.E. Oussaeif, et al. On a Weak Solution of a Fractional-Order Temporal Equation, Math. Stat. 10 (2022), 1116–1120. https://doi.org/10.13189/ms.2022.100522.
  17. N. Anakira, Z. Chebana, T.-E. Oussaeif, I.M. Batiha, A. Ouannas, A Study of a Weak Solution of a Diffusion Problem for a Temporal Fractional Differential Equation, Nonlinear Funct. Anal. Appl. 27 (2022), 679–689. https://doi.org/10.22771/NFAA.2022.27.03.14.
  18. I.M. Batiha, Solvability of the Solution of Superlinear Hyperbolic Dirichlet Problem, Int. J. Anal. Appl. 20 (2022), 62. https://doi.org/10.28924/2291-8639-20-2022-62.
  19. M. Bezziou, Z. Dahmani, I. Jebril, et al. Solvability for a Differential System of Duffing Type via Caputo-Hadamard Approach, Appl. Math. Inform. Sci. 11 (2022), 341–352. https://doi.org/10.18576/amis/160222.
  20. A. Moumeni, N. Barrouk, Existence of Global Solutions for Systems of Reaction-Diffusion With Compact Result, Int. J. Pure Appl. Math. 102 (2015), 169-186. https://doi.org/10.12732/ijpam.v102i2.1.
  21. A. Moumeni, N. Barrouk, Triangular Reaction Diffusion Systems With Compact Result, Glob. J. Pure Appl. Math. 11 (2015), 4729-4747.
  22. A. Haraux, M. Kirane, Estimations C1 Pour des Problèmes Paraboliques Semi-Lineaires, Ann. Fac. Sci. Toulouse Math. 5 (1983), 265-280. http://www.numdam.org/item?id=AFST_1983_5_5_3-4_265_0.
  23. D. Hnaien, F. Kellil, R. Lassoued, Asymptotic Behavior of Global Solutions of an Anomalous Diffusion System, J. Math. Anal. Appl. 421 (2015), 1519–1530. https://doi.org/10.1016/j.jmaa.2014.07.083.
  24. O.A. Ladyzenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI (1968).
  25. N. Dunford, J.T. Schwartz, Linear Operators, 3 Volume Set, Interscience, (1972).
  26. F. Rothe, Global Existence of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, Berlin, (1984).