Abstract
The main purpose of this paper is to extend the result of Barabanova (Proc. Am. Math. Soc. 122:827–831, 1994) on the global existence, uniqueness, uniform boundedness, and the asymptotic behavior of solutions for a weakly coupled class of reaction-diffusion systems on a growing domain with an isotropic growth. Numerical simulations are used to affirm and support the analytical findings.
Similar content being viewed by others
References
Crampin, E.J., Gaffney, E.A., Maini, P.K.: Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol. 61, 1093–1120 (1999). https://doi.org/10.1006/bulm.1999.0131
Alikakos, N.D.: \(L^{p}\)-bounds of solutions of reaction-diffusion equations. Commun. Partial Differ. Equ. 4, 827–868 (1979). https://doi.org/10.1080/03605307908820113
Haraux, A., Youkana, A.: On a result of K. Masuda concerning reaction-diffusion equations. Tohoku Math. J. 40, 159–163 (1988). https://doi.org/10.2748/tmj/1178228084
Hollis, S.L., Martin, R.H., Pierre, M.: Global existence and boundedness in reaction-diffusion systems. SIAM J. Math. Anal. 18, 744–761 (1987). https://doi.org/10.1137/0518057
Masuda, K.: On the global existence and asymptotic behavior of solutions of reaction-diffusion equations. Hokkaido Math. J. 12, 360–370 (1983). https://doi.org/10.14492/hokmj/1470081012
Barabanova, A.: On the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity. Proc. Am. Math. Soc. 122, 827–831 (1994). https://doi.org/10.1090/S0002-9939-1994-1207533-6
Pao, C.V.: Asymptotic stability of reaction-diffusion systems in chemical reactor and combustion theory. J. Math. Anal. Appl. 82, 503–526 (1981). https://doi.org/10.1016/0022-247X(81)90213-4
Kirane, M.: Global bounds and asymptotics for a system of reaction-diffusion equations. J. Math. Anal. Appl. 138, 328–342 (1989). https://doi.org/10.1016/0022-247X(89)90293-X
Kouachi, S., Youkana, A.: Global existence for a class of reaction-diffusion systems. Bull. Pol. Acad. Sci., Math. 49, 303–308 (2001)
Rebiai, B., Benachour, S.: Global classical solutions for reaction-diffusion systems with nonlinearities of exponential growth. J. Evol. Equ. 10, 511–527 (2010). https://doi.org/10.1007/s00028-010-0059-x
Abdelmalek, S., Youkana, A.: Global existence of solutions for some coupled systems of reaction-diffusion equations. Int. J. Math. Anal. 5, 425–432 (2011)
Rebiai, B.: Global classical solutions for reaction-diffusion systems with a triangular matrix of diffusion coefficients. Electron. J. Differ. Equ. 2011, 99 (2011)
Abdelmalek, S., Kirane, M., Youkana, A.: A Lyapunov functional for a triangular reaction-diffusion system with nonlinearities of exponential growth. Math. Methods Appl. Sci. 36, 80–85 (2013). https://doi.org/10.1002/mma.2572
Djebara, L., Abdelmalek, S., Bendoukha, S.: Global existence and asymptotic behavior of solutions for some coupled systems via a Lyapunov functional. Acta Math. Sci. 39, 1538–1550 (2019). https://doi.org/10.1007/s10473-019-0606-7
Kelkel, J., Surulescu, C.: A weak solution approach to a reaction-diffusion system modeling pattern formation on seashells. Math. Methods Appl. Sci. 32, 2267–2286 (2009). https://doi.org/10.1002/mma.1133
Venkataraman, C., Lakkis, O., Madzvamuse, A.: Global existence for semilinear reaction-diffusion systems on evolving domains. J. Math. Biol. 64, 41–67 (2012). https://doi.org/10.1007/s00285-011-0404-x
Morgan, J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20, 1128–1144 (1989). https://doi.org/10.1137/0520075
Labadie, M.: Reaction-Diffusion equations and some applications to Biology, Theses, Université Pierre et Marie Curie – Paris VI (2011)
Henry, D.: Geometric Theory of semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)
Rothe, F.: Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Mathematics, vol. 1072. Springer, Berlin (1984)
Haraux, A., Kirane, M.: Estimations \(C^{1}\) pour des problèmes paraboliques semi-linéaires. Ann. Fac. Sci. Toulouse Math. 5, 265–280 (1983)
Madzvamuse, A.: Stability analysis of reaction-diffusion systems with constant coefficients on growing domains. Int. J. Dyn. Syst. Differ. Equ. 1, 250–262 (2008). https://doi.org/10.1504/IJDSDE.2008.023002
Ladyz̆enskaja, O., Solonnikov, V., Ural’ceva, N.: Linear and Quasi-Linear Equations of Parabolic Type. Am. Math. Soc., Providence (1968)
Lieberman, G.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)
Wang, M.: A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment. J. Funct. Anal. 270, 483–508 (2016)
Wang, M.: Note on the Lyapunov functional method. Appl. Math. Lett. 75, 102–107 (2018)
Pierre, M.: Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78, 417–455 (2010). https://doi.org/10.1007/s00032-010-0133-4
Acknowledgements
We are grateful to the anonymous referees for their accurate revision of the manuscript and their valuable and constructive comments. The authors also would like to thank Professors Amar Youkana, Mokhtar Kirane and Zhigui Lin for their discussion from which we have benefited immensely. This work would not have been possible without the financial support of the Directorate-General for Scientific Research and Technological Development (DGRSDT) of Algeria.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Douaifia, R., Abdelmalek, S. & Bendoukha, S. Global Existence and Asymptotic Stability for a Class of Coupled Reaction-Diffusion Systems on Growing Domains. Acta Appl Math 171, 17 (2021). https://doi.org/10.1007/s10440-021-00385-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10440-021-00385-7