We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systemsuttx=aΔutx−guvm,vttx=dΔvtx+λtxguvm,
- reaction–diffusion systems
- local existence
- comparison principle
- global existence
In the sequel, we use the notations.
the set of natural numbers and
For the integer part of .
the set of integers.
For and :
the closed ball of center and radius
the boundary of
Let a subset. denote the boundary of
: the natural logaritm function.
the surface area of , where is the Gamma function.
the Banach space of bounded and uniformly continuous functions on with the supremum norm
which is a Banach space endowed with the norm
For ), we denote by
For two regular functions, and
Reaction-Diffuison equations are nonlinear parabolic partial differential equations arises in many fields of sciences like chemistry, physics, biology, ecology and even medicine. It appears usually as coupled systems.
The somewhat general form of these systems of two equations is
where with is an open set, is the Laplacian operator, , are two real positive constants called the coefficients of the diffusion. For a chemical reaction where two substances and , and represent their concentrations at time and position respectively, and and represent the rate of production of these substances in the given order. For more details see [1, 2].
In this chapter, we are concerned with the existence of global solutions to the reaction–diffusion system
with initial data
Whe assume that.
One of the essential questions for (1)–(3) is the existence of global solutions and possibly bounds uniform in time. Recently, Collet and Xin in their paper  published in 1996 have studied the system (1)–(3) but with , and In this particular case, this system describes the evolution of the mass fraction of reactant and that of the product for the autocatalytic chemical reaction of the form They proved the existence of global solutions and showed that the norm of can not grow faster than for any space dimension.
It is worth mentioning here the result of S. Badraoui  who studied the system
where , , , , is an even positive integer. He has proved that if , are nonnegative and are in that:
If on , then the solution is global and uniformly bounded.
If on then the solution is global.
Our work here is a continuation of the work of Collet and Xin . We treat the same question in a slightly general case. Inspired by the same ideas in  we prove that the system (1)–(3) under the assumptions (H1) to (H5) has a unique global nonnegative classical solution.
The chapter is organized as follows: In section 2, we treat the existence of local solution and reveal its positivity using the maximum principle.
In section 3, firstly, we prove by a simple comparison argument that if , the solution is uniformly bounded and we give an upper bound of it. Afterwards, we attack the hard case in which where we used the Lyapunov functional and the cut-off function We show that for sufficiently large and small enough we can control the -norms of on every unit spacial cub in from which we deduce the -norm of at any time
We emphazise here that I have engaged to calculate the constants encountered in all equations and inequalities exactly.
2. Existence of a local solution and its positivity
Here ; the operator is defined as
where . The function is defined as .
It is known that for the operator generates an analytic semigroup in the space :
Hence, the operator generates an analytic semigroup defined by
where is the semigroup generated by the operator , and is the semigroup generated by the operator .
We can write the first equation as
for some . Thanks to the assumption (H4)-ii we deduce that is bounded on Whence, by the theorem 9 on page 43 in , we obtain that
The second equation can be written as
By the same theorem we get
For the existence of a global solution, we use the contraposed of the characterization of the maximal existence time ( on page 193) as follows
3. Existence of a global solution
For this task we will use the fact that the solution is nonnegative.
Then, the solution is global and uniformly bounded on . More precisely, we have the estimates
The solution satisfies the integral equations
Here and are the semigroups generated by the operators and in the space respectively. As is nonnegative, then from (15) we get
Since , using the explicit expression of and , one can observe that (see )
This last inequality leads to the veracity of (14).
In the case where , it seems that the idea of comparison cannot be applied. Nevertheless, we can prove the existence of global classical solutions; but it appears that their boundedness is not assured.
We need some lemmas.
Then for any a smooth nonnegative function with exponential spacial decay at infinity, we have
Using integration by parts, we get
In fact, let then we have by the Geen theorem
where is the derivative of with respect to the unit outer normal to the boundary .
From (27) we obtain
By the same way we get
From (30) we find that
where are two arbitrary constants.
The inequality (33) is satisfied if
As a consequence of (33) we have
Calulate and estimate it
We pove that
To do this, it sufficies to compute the discriminant of the trinoma in
where is defined by (49).
Now, let us estimate We have ( on page 485)
In the following step we trie to control the second component of the solution on any unit spacial cube in the norms with
Let be an arbitrary fixed point and
By induction we prove that
Let and then we have by the imbedding theorem for spaces
Taking enough small such that . Combining this with (37)
with where . Then, there exists a positive constant
As , then from (67)
Also, it’s clear that , but every point is of the form
where and for all with at least one of the
It’s easy to prove that
We have obviously
Let an arbitrary point and be the family of pairwise disjoint measurable cubes of the form (64) covering such that the center of is .
Firstly, using the fact that and applying the left-hand inequality in (66)
By Hölder ineguality with and
and by (56) we have
On the other hand, we deduce from the right-hand inequality in (66) that
and finally we have for all
As the function in on the right-hand side of the estimate (83) is continuous on . As on and satisfied (83), we conclude from (11) that . Whence, the solution is global.
In this more general case, by examining the proof of the theorem 3.2; we see that under the same assumptions above, the system has also a global nonnegative classical solution.
4. Illustrative example
To illustrate the previous study about global existence, we give the following reaction–diffusion system
where are real positive constants and is a real nonnegative constant. If and are nonnegative; the system (84) admits a unique global nonnegative classical solution
5. Conclusion and perspectives
We have prouved in the case where that the solution is global, but it remains an interesting question that if it is uniformly bounded or not.
As perspectives, we will replace the function satisfying the hypothesis (H4) by the function with is a real constant and replace the term by with ; namely that reaction term is of exponential growth. The system was studied on bounded domain by J. I. Kanel and M. Mokhtar in .
J. Smoller, Shock Waves and Reaction-Diffusion Equations, New York-Heidelberg-Berlin, Springer-Verlag, 1983
J. David Logan, An introduction to nonlinear partial differential equations, Second edition, John Wiley and Sons, Inc., Publication, 2008
P. Collet & J. Xin, Global Existence and Large Time Asymptotic Bounds of Solutions of Thermal Diffusive Combustion Systems on , Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 23(1996), 625–642
J. D. Avrin, Qualitative theory for a model of laminar flames with arbitrary nonnegative initial data, J.D.E, 84(1990), 209–308
B. Larrouturou, The equations of one-dimentional unsteady flames propagation: existence and uniqueness, SIAM J. Math. Anal. 19, pp. 32–59, 1988
S. Badraoui; Existence of Global Solutions for Systems of Reaction-Diffusion Equations on Unbounded Domains, Electron. J. Diff. Eqns., Vol 2002, No. 74, pp. 1–10, 2002
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, New York, 1981
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983
A. Friedman, Partial di erential quations of parabolic type, Robert E. Krieger Publisching Company Malabar, Florida, 1983
R. H. Martin & M. Pierre, Nonlinear reaction-diffusion systems, in “Nonlinear Equations in the Applied Sciences” W. F. Ames and C. Rogers eds., Academic Press, Boston, 1992
S. D. Chatterji, Cours d’Analyse, 1 Analyse Vectorielle,Presse polytechniques et universitaires romands, 1997
J. I. Kanel and M. Mokhtar, Global solutions of Reaction-Diffusion systems with a Balance Law and Nonlinearities of Exponential Growth, J. Differential Equations 165, pp. 24–41, 2000