Invariant regions and global existence of solutions for reaction-diffusion systems with a general full matrix
© Boukerrioua; licensee Springer. 2014
Received: 30 July 2013
Accepted: 26 December 2013
Published: 23 January 2014
The aim of this study is to prove the global existence in time of solutions for reaction-diffusion systems. We make use of the appropriate techniques which are based on invariant regions and Lyapunov functional methods. We consider a full matrix of diffusion coefficients and we show the global existence of the solutions.
Keywordsglobal existence reaction-diffusion systems Lyapunov functional
For more details, one may consult .
Melkemi et al.  established the existence of global solutions (eventually uniformly bounded in time) using a novel approach that involved the use of a Lyapunov function for system (1.1)-(1.4) when . Along the same lines, Rebai  has proved the global existence of solutions for system (1.1)-(1.4), in the case , (triangular matrix). The present investigation is a continuation of results obtained in . Here, we follow the same reasoning as in , in the study of system (1.1)-(1.4), when , , that is, for a model that involves a general full matrix.
The components and represent either chemical concentrations or biological population densities and system (1.1)-(1.2) is a mathematical model describing various chemical and biological phenomena (see, e.g., Cussler ).
Remark 1 If , then we have . We note that the condition of parabolicity implies that , where A is the matrix of diffusion.
2 Local existence and invariant regions
Throughout the text we shall denote by the norm in , and by the norm in or .
For any initial data in or , , local existence and uniqueness of solutions to the initial value problem (1.1)-(1.4) follow from the basic existence theory for abstract semilinear differential equations (see Henry  and Pazy ). The solutions are classical on , where denotes the eventual blowing-up time in .
is invariant for system (2.1)-(2.4).
Now, to prove that the region is invariant for system (2.1)-(2.4), it suffices to show that for , and , for , see .
Remark 2 We note that if , then and .
3 Existence of global solutions
where C is a nonnegative constant independent of t.
From the assumptions (1.7) and (1.8), we are led to establish the uniform boundedness of the on in order to get that of on .
We firstly introduce the following lemmas, which are useful in our main results.
Proof The proof is similar to that in Melkemi et al. .
Now we can establish the global existence and uniform boundedness of the solutions of (2.1)-(2.4).
Theorem 1 Under the assumptions (1.7) and (1.8), the solutions of (2.1)-(2.4) are global and uniformly bounded on .
and as we select we can proceed to bound .
Using the regularity results for the solutions of the parabolic equations in , we conclude that the solutions of problem (2.1)-(2.4) are uniformly bounded on . □
By (2.7), it is easy to see that the solutions of problem (1.1)-(1.4) are also uniformly bounded on .
Remark 4 We note that and , because and .
We conclude by noting that the study of the global existence of strongly coupled systems has been a major development, and several articles are devoted to this subject. In our opinion, many other systems with non-constant diffusion matrix which are in the actual scope of the results previously given, should be taken in consideration and studied with more interest.
The author would like to thank the anonymous referee for his/her valuable suggestions.
- Kouachi S: Invariant regions and global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients and nonhomogeneous boundary conditions. Georgian Math. J. 2004,11(2):349-359.MathSciNetGoogle Scholar
- Melkemi L, Mokrane AZ, Youkana A: On the uniform boundedness of the solutions of systems of reaction-diffusion equations. Electron. J. Qual. Theory Differ. Equ. 2005., 2005: Article ID 24Google Scholar
- Rebai B: Global classical solutions for reaction-diffusion systems with a triangular matrix of diffusion coefficients. Electron. J. Differ. Equ. 2011., 2011: Article ID 99Google Scholar
- Cussler EL Chemical Engineering Monographs 3. In Multicomponent Diffusion. Elsevier, Amsterdam; 1976.Google Scholar
- Henry D Lecture Notes in Mathematics 840. In Geometric Theory of Semi-Linear Parabolic Equations. Springer, New York; 1984.Google Scholar
- Pazy A Applied Mathematical Sciences 44. In Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York; 1983.View ArticleGoogle Scholar
- Smoller J: Shock Waves and Reaction-Diffusion Equations. Springer, New York; 1983.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.