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Invariant regions and global existence of solutions for reaction-diffusion systems with a general full matrix
Journal of Inequalities and Applications volume 2014, Article number: 24 (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.
We are mainly interested in the global existence in time of solutions to a reaction-diffusion system of the form
with the following boundary conditions:
and the initial data
where Ω is an open bounded domain in with boundary ∂ Ω of class , denotes the outward normal derivative on ∂ Ω, Δ denotes the Laplacian operator with respect to the x variable, a, b, c, d, σ are positive constants satisfying the condition , which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion is positive definite, . The eigenvalues and () of the matrix are positive. We assume that
and the initial data are assumed to be in the following region:
For more details, one may consult .
The function f is a nonnegative continuously differentiable function on Σ such that
In addition we suppose that
where and φ is a nonnegative function of class such that
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 .
Furthermore, if , then
Therefore, if there exists a positive constant C such that
Multiplying (1.2) first through by and adding (1.1) and then by and subtracting (1.1), we get
with the boundary conditions
and the initial data
for any in and
To prove that Σ is an invariant region for system (1.1)-(1.4) it suffices to prove that the region
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 .
From (1.6), its clear that the region is invariant for system (2.1)-(2.4) and from (2.5) we have
Remark 2 We note that if , then and .
3 Existence of global solutions
A simple application of the comparison theorem [, Theorem 10.1] to system (2.1)-(2.4) implies that for any initial conditions and , we have
To prove the global existence of the solutions of problem (1.1)-(1.4), one needs to prove it for problem (2.1)-(2.4). As regards this subject, it is well known that it suffices to derive a uniform estimate of for some , i.e.
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 .
For , we put
We firstly introduce the following lemmas, which are useful in our main results.
Lemma 1 Let be a solution of (2.1)-(2.4). Then
Proof We integrate both sides of (2.1), satisfied by w, which is positive and then we obtain
Lemma 2 Assume that and let
where is the solution of (2.1)-(2.4) on . Then under the assumptions (1.7)-(1.8) there exist two positive constants and such that
Proof The proof is similar to that in Melkemi et al. .
Differentiating with respect to t and using the Green formula one obtains
We observe that H is given by
is a quadratic form with respect to ∇w and ∇z, which is nonnegative if
and we have chosen such that
It is easy to see that the left-hand side of (3.13) can be written as
the inequality (3.13) holds, , and consequently
and the second term S can be estimated as
On the other hand
Taking into account the fact that , and using (3.19), we observe that
Then for , and for , , we have
On the other hand, we deduce that the function
is bounded on the compact interval ; then there exists such that
From (3.17) and (3.20), we deduce immediately the following inequality:
by (3.3), we have
and from (3.7), it follows that
and from (3.7) and (3.9), we conclude that
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 .
Proof Multiplying the inequality (3.24) by and then integrating, we deduce that there exists a positive constant independent of t, such that
From (3.6), we observe that
and it follows, for all , that
and as we select we can proceed to bound .
and is such that
using (1.7), we deduce
From (2.7), we have
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 3 Because and , we deduce that
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.
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.
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 24
Rebai B: Global classical solutions for reaction-diffusion systems with a triangular matrix of diffusion coefficients. Electron. J. Differ. Equ. 2011., 2011: Article ID 99
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Henry D Lecture Notes in Mathematics 840. In Geometric Theory of Semi-Linear Parabolic Equations. Springer, New York; 1984.
Pazy A Applied Mathematical Sciences 44. In Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York; 1983.
Smoller J: Shock Waves and Reaction-Diffusion Equations. Springer, New York; 1983.
The author would like to thank the anonymous referee for his/her valuable suggestions.
The author declares that they have no competing interests.
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Boukerrioua, K. Invariant regions and global existence of solutions for reaction-diffusion systems with a general full matrix. J Inequal Appl 2014, 24 (2014). https://doi.org/10.1186/1029-242X-2014-24
- global existence
- reaction-diffusion systems
- Lyapunov functional