Global Existence of Solutions for A Gierer-Meinhardt System with Two Activators and Two Inhibitors

Document Type: Original Article


1 Department of mathematics, University of Tebessa 12002 Algeria.

2 Larbi Tebessi University

3 Department of Electrical Engineering, College of Engineering at Yanbu, Taibah University, Saudi Arabia


This paper deals with a Gierer-Meinhardt model with 2 activators and 2 inhibitors described by a reaction-diffusion system with fractional reactions. The purpose of this paper is to prove the existence of a global solution. Our technique is based on a suitable Lyapunov functional.


[1] S. Abdelmalek, A.Gouadria and A. Youkana, Global solutions for an m-component system of activator-inhibitor type, Abstr. Appl. Anal., Volume 2013 (2013), Article ID 939405, 9 pages.
[2] S. Abdelmalek, H. Loua , and A. Youkana, Global existence of solutions for Gierer{Meinhardet system with three equations, Elec. J. Diff. Eqs., Vol. 2012(2012), No. 55, pp. 1-8.
[3] S. Abdelmalek and S. Kouachi, A simple proof of Sylvester's (determinants) identity, App. Math. Scie. Vol. 2, No. 32,2008, pp. 1571-1580. 
[4] C. Cooper, Chaotic behavior in coupled Gierer-Meinhardt equations, Computers & Graphics, Vol. 25 (2001), pp. 159-170.
[5] S. Henine, S. Abdelmalek, A. Youkana, Boundedness and large time behavior of solutions for a Gierer -Meinhardt system of three equations, Elec. J. Diff. Eqs., Vol. 2015 (2015), No. 94, pp. 1-11. 
[6] A. Trembley, Memoires pour servir a l'histoire d'un genre de polypes d'eau douce, a bras en forme de cornes, 1744. 
[7] A. Haraux and M. Kirane, Estimations C1 pour des problemes paraboliques semi-lineaires, Ann. Fac. Sci. Toulouse 5 (1983), pp. 265-280. 
[8] A. Gierer and H. A. Meinhardt, Theory of biological pattern formation, Kybernetik, Vol. 12, 1972, pp. 30-39.
[9] D. Henry, Geometric Theory of semi-linear parabolic equations, Lecture Notes in Mathematics 840, Springer-Verlag, New-York, 1984.
[10] H. Jiang, Global existence of Solution of an Activator-Inhibitor System, Disc. Cont. Dyn. Sys., Vol. 14, No. 4, 2006, pp.737-751.
[11] K. Masuda and K. Takahashi, Reaction-diffusion systems in the Gierer-Meinhardt theory of biological pattern formation, Japan J. Appl. Math., Vol. 4, No. 1, 1987, pp. 47-58.
[12] H. Meinhardt, A. Koch, and G. Bernasconi, Models of pattern formation applied to plant development, Symmetry in Plants (D. Barabe and R. V. Jean, Eds), World Scientific Publishing, Singapore, pp. 723-758.
[13] L. Mingde, C. Shaohua and Q. Yuchun, Boundedness and blow up for the general activator-inhibitor model, Acta Mathematicae Applicatae Sinica, Vol. 11, No. 1, 1995. 
[14] F. Rothe, Global solutions of reaction-diffusion equations, Lecture Notes in Mathematics 1072, Springer-Verlag, Berlin,1984. 
[15] A. M. Turing, The chemical basis of morphogenesis, Philosophical Trans. Roy. Soc. (B), Vol. 237, 1952, pp. 37-72. 
[16] J. Wu. and Y. Li, Global classical solution for the activator-inhibitor model, Acta Mathematicae Applicatae Sinica (in Chinese), Vol. 13, 1990, pp. 501-505.