Abstract and Applied Analysis

VolumeΒ 2008, Article IDΒ 127394, 13 pages

http://dx.doi.org/10.1155/2008/127394

## State Trajectories Analysis for a Class of Tubular Reactor Nonlinear Nonautonomous Models

^{1}Center de Recherche INRIA Futurs-Site de Bordeaux-IMB, 351 Cours de la LibΓ©ration, Talent Cedex, Bordeaux 33405, France^{2}DΓ©partement de MathΓ©matique et Informatique, FacultΓ© des Sciences, UniversitΓ© Chouaib Doukkali, El Jadida BP 20, Morocco

Received 4 July 2007; Accepted 3 September 2007

Academic Editor: Nicholas DimitriosΒ Alikakos

Copyright Β© 2008 B. Aylaj et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The existence and uniqueness of global mild solutions are proven for a class of semilinear nonautonomous evolution equations. Moreover, it is shown that the system, under considerations, has a unique steady state. This analysis uses, essentially, the dissipativity, a subtangential condition, and the positivity of the related -semigroup.

#### 1. Introduction

Several chemical and biochemical processes are typically described by nonlinear coupled partial differential equations βPDEβ and hence by distributed parameter models (see [1] and the references within). The source of nonlinearities is essentially the kinetics of the reactions involved in the process. For numerical simulation as well as for control design problems, many authors approximate those distributed parameter systems by lumped parameter models [1β5]. However, an important number of questions remained unsolved. In particular, to study the stability of the tubular reactor, the trajectory must exist on the whole real positive time interval . In our previous works [6, 7], we have proven the global state trajectories existence for a class of nonlinear systems arising from convection-dispersion-reaction systems, assuming that the inlet concentrations are independent of time. In this paper, we investigate the question in the case where the involved inlet concentrations are functions of time . The considered class of models correspond to the following chemical reaction: whose kinetic is given by , where and are the concentrations of the reactants and , respectively, and are the kinetic constants and are the order of the reaction to and , respectively. More precisely, we study the global existence and uniqueness of the trajectories of the models which describe the evolution of two reactant concentrations and : , with the following boundary and initial conditions:

Additionally, the existence and uniqueness of the corresponding equilibrium profile will be proven.

In the above equations, are the dispersion coefficients, is the superficial fluid velocity, denote the time and space independent variables, respectively, is the length of the reactor, and are two positive integers, and are the inlet concentration. For further discussion of parameters, we refer to [3].

*Comment 1. *(i) The nonlinear
models considered in this paper have been studied in a qualitative manner by
several authors. In the case, , [8]
established the asymptotic behavior of solutions for the second-order reaction
(i.e., ). N. Alikakos [9]
established global existence and bounds of
positive solutions, when and . This latter result has been generalized by [10] for
the case and .

In practice, the special cases have been used
as an industrial pulp bleaching model, where the two reactants are chlorine
dioxide and lignin . In particular, [3] studied approximate solutions by
using several methods (orthogonal collocation, finite elements, and finite
difference methods), when and . The reader can find another model with in [11], where
the numerical analysis has been done for and , (see also
[12]).

Recently, the existence of global solutions for
problems such as (1.2)β(1.6) has been
extensively studied in [6, 7] with constant inlet concentrations.

(ii) For technological limitations and economical considerations, the following
saturation conditions are usually fulfilled for all and for all : where and are positive
constants.

This paper is organized as follows. In Section 2, we will recall briefly some basic results and preliminary facts from semilinear nonautonomous evolution equations which will be used throughout Section 4. In Section 3, the problem (1.2)β(1.6) is converted through some transformations to a homogeneous form where the semigroup theory applies. In Section 4 we establish the main global existence result for system (1.2)β(1.6). We report the existence and uniqueness of equilibrium profiles results in Section 5. Finally, the main conclusions are outlined in Section 6. The background of our approach can be found in [13β16].

#### 2. Preliminaries

Let be a real Banach space with norm and let be a linear contraction -semigroup on generated by . Let be a nonlinear continuous operator form into , where is a subset of . and denote, respectively, the identity operator of and the function identically equal to 1.

This section is devoted to investigate sufficient conditions for the existence and uniqueness of global mild solutions to the following abstract Cauchy problem: where denote the section of at , given by Assume that for all . Moreover, recall that for and is a subset of .

The semilinear nonautonomous evolution equations have been treated by a number of authors [14, 15, 17β21]. However, one may find that in most cases is cylindrical, that is, [14, 22]. More generally, the cylindrical case of will not be convenient for the study of evolution system satisfying time-dependent constraints, that is, on (see our problem in Section 3). A noncylindrical case was studied in [16, 19].

The following result gives sufficient conditions for the existence and uniqueness of global mild solutions to the semilinear equations of type (2.1). It is a particular version of [16, Theorem 8.1], when the nonlinear is -dissipative [16].

Theorem 2.1 (see [16]). *Suppose that the following conditions are fulfilled:
*(i)* is closed from
the left, that is, if in , and in as , then ;*(ii)*;*(iii)* is continuous on and there exists such that the operator is dissipative on for all .**
If is a connected
subset of such that for
all , then, for each (2.1) has a unique mild solution on .*

*Comment 2. *It is shown in [16] that the
βsubtangential conditionβ (ii) is a necessary
condition for the existence of the mild solutions of (2.1). For more
details on the conditions of Theorem 2.1, we refer to [16].

In the particular case when is -invariant, that is, for all we have the following lemma.

Lemma 2.2. *Let be continuous and let be closed from the left. If is -invariant for all , then the
following subtangential condition ** implies the
condition *

*Proof. *Let given from condition (2.2) it follows, by [23, Lemma 3] (see also [24, Lemma 1]), that there is and such that . Let now and . We get such that . By the
invariance properties of , we have . Consequently, By using the continuity of -semigroup the desired result (2.3) is obtained.

Theorem 2.1 with Lemma 2.2 obviously imply the following.

Corollary 2.3. *Suppose that the following conditions are
fulfilled:
*(i)* is closed from
the left, that is, if in , and in as then ;*(ii)* is -invariant, for
all ;*(iii)*;*(iv)* is continuous on and there exists such that the operator is dissipative on , for all .**
If is a connected
subset of such that for
all , then, for each
(2.1) has a unique mild solution on .*

#### 3. Abstract Semigroup Formulation

Throughout the sequel, we assume , the Hilbert space with the usual inner product and the induced norm for all and in .

Clearly, the Hilbert space is a real Banach lattice, where for all given , Recall that for every pair , the set is called the order interval between and . Clearly, is nonempty if (for more details, see, e.g., [25]). A bounded linear operator on is said to be positive if for all Similarly, a family of bounded linear operators of is said to be a positive on if is a on and is a positive operator for all .

In the following, we will assume that and are positive -functions. Let us consider the following state transformation: Then, we obtain the new equivalent system for all and : with where

This PDEs describing the reactor dynamics may be formally written in the abstract form as where denote the section of at , which is given in view of (1.7) by

The linear operator is defined by The nonlinear operator is defined on by It is shown in [7] that the linear operator given by (3.14) is the infinitesimal generator of contraction semigroup on where and are the generated, respectively, by and .

#### 4. Global Existence

This section is concerned with the existence and the uniqueness of mild solution for our problem given by (3.6)β(3.9) In order to be able to apply Corollary 2.3, we need the following lemmas.

Lemma 4.1. *For each *

*Proof. *Let Observe that is given by , where Denote we have, for Let be sufficiently
small such that

Let, now, then Hence By using the regularity of the inlet function , we get where as Whence By similar
considerations as above, taking into account the regularity of the function we also get Observe, now, that
combining the latter with (4.8)-(4.9) we get the desired result (4.1).

The following lemma is useful to establish the dissipativity property.

Lemma 4.2. *There exists such that the
operator is dissipative
on for each .*

*Proof. *Let and let be in Denote and let also Observe that, for each for Hence, by
applying the mean value theorem, for , we get Finally, Consequently, is an -dissipative
operator on [14, page 245], where

Finally, we state the invariance properties of the state trajectories of the model given by (3.6)β(3.9).

Proposition 4.3. *One has that *

*Proof. *Let and . We have Hence, by using the positivity of [26], we have Since, (see [26]) and
by using the inequalities
(1.8)
(i.e., and ), the invariance
of holds for all Thus, for all

Now, we are in a position to state and prove our global existence result for problem (3.6)β(3.9).

Theorem 4.4. *Let and be positive -functions.
Then, for every the problem (3.6)β(3.9) has a unique
global mild solution.*

*Proof. *Since is continuous function in by Corollary 2.3,
it is sufficient to prove the condition (i) in Corollary 2.3
and to check that the subset is connected

(a)*Let us first show that is closed from
the left.*Let and with then there
exists a subsequence of which is also
denoted by such that , that is, on which implies,
by continuity of and , that is, on , hence for each .(b)*Let us, now, check that is connected in :*Let and define such that for
all Since and are continuous
functions in it follows that in is also a
continuous function. Observe that is surjective;
since is connected in , we get that is also
connected in

Thus the proof of the theorem is complete.

The next section deals with the existence and uniqueness results of equilibrium profile solutions for a nonlinear model given by (3.6)β(3.9).

#### 5. Equilibrium Profiles

In the steady-state solution analysis, the inlet functions and are independent of time which implies that the domain is also independent of . If we denote by and the values of and which correspond to the steady-state solutions, the corresponding steady-state system to the models (3.6)β(3.9) is given by the following equations: with The following existence result can be proven as in the case where and are independent of time.

Theorem 5.1 (see [7, 27]). *The tubular reactor modelled by the nonlinear coupled partial differential equations given
by (3.6)β(3.9) has at least
one equilibrium profile in .*

The sequel of this paper will deal with the uniqueness analysis of steady states in the important case where

First, since , we denote with for .

Now, we derive a positivity lemma, which will play a fundamental role in the proof of the uniqueness result of steady states.

Lemma 5.2. *Let be a bounded nonnegative
function defined in . If satisfies the equations ** then in .*

*Proof. *Let be the solution of problem (5.5), then
We have, Since is nonnegative
function in then by (5.8) and taking into account (5.6)
Which implies,
in view of (5.6)-(5.7), that Then, we get Clearly, by
using the Sobolev imbedding theorem, Therefore, since

Theorem 5.3. *For the steady-state problem given by (5.1)β(5.3) has a unique
solution in *

*Proof. *Let be solutions to (5.1)β(5.3) on To obtain the desired result, we will be showing that Let Then Hence, by
applying the mean value theorem, we get

where are some
intermediate values between and

By similar considerations as above, we also get for the same and

Now, we have the following system: where, for

Multiplying
(5.16) by and (5.17) by we get by addition of both equations that where By Lemma 5.2,
this system has a unique solution in Now, let
and
substituting the expression yields where Observe that,
for ,

Let , then we have By Lemma 5.2 we get Thus it follows, by (5.21), that which ensures
the desired result, that is,

#### 6. Conclusion

In this paper, we have studied the existence and uniqueness of the global mild solution for a class of tubular reactor nonlinear nonautonomous models. It has also been proven that the trajectories are satisfying time-dependent constraints, that is, Moreover, the set of physically meaningful admissible states is invariant under the dynamics of the reactions. In addition, the existence and uniqueness results of equilibrium profiles are reported.

An important open question is the stability analysis of equilibrium profile for system (1.2)β(1.6). This question is under investigation.

#### Acknowledgments

This paper presents research results of the Moroccan βProgramme ThΓ©matique d'Appui Γ la Recherche Scientifiqueβ *PROTARS III*, initiated by the Moroccan βCentre National de la Recherche Scientifique et Techniqueβ (**CNRST**). The scientific responsibility rests with its authors. The work
has been partially carried out within the framework of a collaboration
agreement between CESAME (UniversitΓ© Catholique de Louvain, Belgium) and LINMA
of the Faculty of sciences (UnivesitΓ© Chouaib Doukkali, Morocco), funded by the
Belgian Secretary of the State for Development Cooperation and by the CIUF
(Conseil Interuniversitaire de la CommunautΓ© FranΓ§aise, Belgium). The work of B. Aylaj is supported by a research grant
from the Agence Universitaire de la Francophonie.

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