Abstract and Applied Analysis

Abstract and Applied Analysis / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 127394 | https://doi.org/10.1155/2008/127394

B. Aylaj, M. E. Achhab, M. Laabissi, "State Trajectories Analysis for a Class of Tubular Reactor Nonlinear Nonautonomous Models", Abstract and Applied Analysis, vol. 2008, Article ID 127394, 13 pages, 2008. https://doi.org/10.1155/2008/127394

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

Academic Editor: Nicholas Dimitrios Alikakos
Received04 Jul 2007
Accepted03 Sep 2007
Published25 Feb 2008

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 š¶0-semigroup.

1. Introduction

Several chemical and biochemical processes are typically described by nonlinear coupled partial differential equations ā€œPDEā€ 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 [1ā€“5]. 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 [0,+āˆž[. 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 š‘”. The considered class of models correspond to the following chemical reaction: š‘›š“+š‘ššµā†’š‘ƒ,(1.1) whose kinetic is given by š‘Ÿ=(āˆ’š‘˜1š¶š‘ššæš‘›,āˆ’š‘˜2š¶š‘ššæš‘›)š‘‡, where š¶ and šæ are the concentrations of the reactants š“ and šµ, respectively, š‘˜1 and š‘˜2 are the kinetic constants and š‘š,š‘› are the order of the reaction to š“ and šµ, respectively. More precisely, we study the global existence and uniqueness of the trajectories of the models which describe the evolution of two reactant concentrations š¶ and šæ: šœ•š¶šœ•š‘”=āˆ’šœˆšœ•š¶šœ•šœ‰+š·1šœ•2š¶šœ•šœ‰2āˆ’š‘˜1š¶š‘ššæš‘›,(1.2)šœ•šæšœ•š‘”=āˆ’šœˆšœ•šæšœ•šœ‰+š·2šœ•2šæšœ•šœ‰2āˆ’š‘˜2š¶š‘ššæš‘›,(1.3)for0x000a0šœ‰āˆˆ]0,š‘™[andš‘”>0š·1šœ•š¶šœ•šœ‰(0,š‘”)āˆ’šœˆš¶(0,š‘”)+šœˆš¶in(š‘”)=0=š·1šœ•š¶š·šœ•šœ‰(š‘™)āˆ€š‘”>0(1.4)2šœ•šæšœ•šœ‰(0,š‘”)āˆ’šœˆšæ(0,š‘”)+šœˆšæin(š‘”)=0=š·2šœ•šæšœ•šœ‰(š‘™,š‘”)āˆ€š‘”>0,(1.5)š¶(šœ‰,0)=š¶0(šœ‰),šæ(šœ‰,0)=šæ0(šœ‰)foršœ‰āˆˆ]0,š‘™[.(1.6), with the following boundary and initial conditions: š·1,š·2šœˆš‘”,šœ‰

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

In the above equations, š‘™ are the dispersion coefficients, š‘š is the superficial fluid velocity, š‘› denote the time and space independent variables, respectively, š¶in is the length of the reactor, šæin and šœˆ=0 are two positive integers, š‘›=š‘š=1 and šæāˆž 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, š‘š=1, [8] established the asymptotic behavior of solutions for the second-order reaction (i.e., 1<š‘›<3/2). N. Alikakos [9] established global existence and š‘š=1 bounds of positive solutions, when š‘›>1 and š‘š=š‘›=1,2,3. This latter result has been generalized by [10] for the case (š¶) and (šæ).
In practice, the special cases š‘›=š‘š have been used as an industrial pulp bleaching model, where the two reactants are chlorine dioxide š·1=š·2 and lignin š·1ā‰ š·2. In particular, [3] studied approximate solutions by using several methods (orthogonal collocation, finite elements, and finite difference methods), when š‘š=š‘›=1 and š·2=4š·1. The reader can find another model with š·2=16š·1 in [11], where the numerical analysis has been done for 0ā‰¤šœ‰ā‰¤š‘™ and š‘”ā‰„0, 0ā‰¤š¶ā‰¤š¶,0ā‰¤šæā‰¤š¶šæ,(1.7)in(š‘”)ā‰¤š¶,šæin(š‘”)ā‰¤šæ,(1.8) (see also [12]).
Recently, the existence of global solutions for problems such as (1.2)ā€“(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 š¶ and for all šæ: š‘‹ā€–ā‹…ā€–,š½=[š‘Ž,š‘[(š‘Ž<š‘ā‰¤+āˆž), where {š’Æ(š‘”);š‘”ā‰„0} and š¶0 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)ā€“(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)ā€“(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 [13ā€“16].

2. Preliminaries

Let š‘‹ be a real Banach space with norm š’œ and let ā„¬ be a linear contraction Ī©-semigroup on š‘‹ generated by Ī©. Let š½Ć—š‘‹ be a nonlinear continuous operator form š¼ into š•€, where š‘‹ is a subset of Ģ‡š‘„(š‘”)=š’œš‘„(š‘”)+ā„¬(š‘”,š‘„(š‘”)),šœ<š‘”<š‘,š‘„(šœ)=š‘„šœāˆˆĪ©(šœ),(2.1). Ī©(šœ) and Ī© denote, respectively, the identity operator of šœāˆˆš½ 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: Ī©(š‘”)={š‘„āˆˆš‘‹;(š‘”,š‘„)āˆˆĪ©}. where Ī©(š‘”)ā‰ āˆ… denote the section of š‘”āˆˆš½ at š‘‘(š‘„;š’Ÿ)=inf{ā€–š‘„āˆ’š‘¦ā€–,š‘¦āˆˆš’Ÿ},, given by š‘„āˆˆš‘‹ Assume that š’Ÿ for all š‘‹. Moreover, recall that Ī© for Ī©=š½Ć—š’Ÿ and Ī© is a subset of š‘„(š‘”)āˆˆĪ©(š‘”).

The semilinear nonautonomous evolution equations have been treated by a number of authors [14, 15, 17ā€“21]. However, one may find that in most cases š½ is cylindrical, that is, Ī© [14, 22]. More generally, the cylindrical case of ā„¬(š‘”,ā‹…) will not be convenient for the study of evolution system satisfying time-dependent constraints, that is, š‘™ā„¬ on Ī© (see our problem in Section 3). A noncylindrical (š‘”š‘›,š‘„š‘›)āˆˆĪ©,š‘”š‘›ā†‘š‘” 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 š½ is š‘„š‘›ā†’š‘„-dissipative [16].

Theorem 2.1 (see [16]). Suppose that the following conditions are fulfilled: (i)š‘‹ is closed from the left, that is, if š‘›ā†’āˆž in (š‘”,š‘„)āˆˆĪ©, and for0x000a0all0x000a0(š‘”,š‘„)āˆˆĪ©,liminfā„Žā†“0(1/ā„Ž)š‘‘(š’Æ(ā„Ž)š‘„+ā„Žā„¬(š‘”,š‘„),Ī©(š‘”+ā„Ž))=0 in ā„¬ as Ī©, then š‘™ā„¬āˆˆā„+;(ii)(ā„¬(š‘”,ā‹…)āˆ’š‘™ā„¬š¼);(iii)Ī©(š‘”) is continuous on š‘”āˆˆš½ and there exists Ī© such that the operator š½Ć—š‘‹ is dissipative on š‘”āˆˆš½,Ī©(š‘”)ā‰ āˆ… for all (šœ,š‘„šœ)āˆˆĪ©,. If š½ is a connected subset of Ī©(š‘”) such that for all š’Æ(š‘ ), then, for each š’Æ(š‘ )(Ī©(š‘”))āŠ‚Ī©(š‘”) (2.1) has a unique mild solution on š‘”,š‘ ā‰„0,.

Comment 2. It is shown in [16] that the ā€œsubtangential conditionā€ (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 ā„¬āˆ¶Ī©ā†’š‘‹ is Ī©-invariant, that is, Ī©(š‘”) for all š’Æ(š‘ ) we have the following lemma.

Lemma 2.2. Let š‘”,š‘ ā‰„0 be continuous and let limā„Žā†“01infā„Žš‘‘ī‚€ī‚š‘„+ā„Žā„¬(š‘”,š‘„);Ī©(š‘”+ā„Ž)=0āˆ€(š‘”,š‘„)āˆˆĪ©(2.2) be closed from the left. If limā„Žā†“01infā„Žš‘‘ī‚€ī‚š’Æ(ā„Ž)š‘„+ā„Žā„¬(š‘”,š‘„),Ī©(š‘”+ā„Ž)=0,āˆ€(š‘”,š‘„)āˆˆĪ©.(2.3) is (š‘”,š‘„)āˆˆĪ©,-invariant for all šœ–>0,, then the following subtangential condition ā„Žāˆˆ(0,šœ–] implies the condition š‘¦āˆˆĪ©(š‘”+ā„Ž)

Proof. Let ā€–š‘¦āˆ’š‘„āˆ’ā„Žā„¬(š‘”,š‘„)ā€–ā‰¤ā„Žšœ– given š‘¢=š‘¦āˆ’š‘„āˆ’ā„Žā„¬(š‘”,š‘„) from condition (2.2) it follows, by [23, Lemma 3] (see also [24, Lemma 1]), that there is š‘£=(1/ā„Ž)š‘¢ and ā€–š‘£ā€–ā‰¤šœ– such that š‘¦=š‘„+ā„Ž(šµ(š‘”,š‘„)+š‘£)āˆˆĪ©(š‘”+ā„Ž). Let now Ī©(š‘”) and š’Æ(ā„Ž)š‘¦āˆˆĪ©(š‘”+ā„Ž). We get š‘‘ī‚€ī‚ā‰¤ā€–ā€–ā€–ā€–,ā‰¤ā€–ā€–ā€–ā€–,ā€–ā€–ā€–ā€–ā€–ā€–ā€–ā€–,ā€–ā€–ā€–ā€–š’Æ(ā„Ž)š‘„+ā„Žā„¬(š‘”,š‘„);Ī©(š‘”+ā„Ž)š’Æ(ā„Ž)š‘„+ā„Žā„¬(š‘”,š‘„)āˆ’š’Æ(ā„Ž)š‘¦ā„Žā„¬(š‘”,š‘„)āˆ’ā„Žš’Æ(ā„Ž)ā„¬(š‘”,š‘„)āˆ’ā„Žš’Æ(ā„Ž)š‘£ā‰¤ā„Žš’Æ(ā„Ž)ā„¬(š‘”,š‘„)āˆ’ā„¬(š‘”,š‘„)+ā„Žš’Æ(ā„Ž)š‘£ā‰¤ā„Žš’Æ(ā„Ž)ā„¬(š‘”,š‘„)āˆ’ā„¬(š‘”,š‘„)+ā„Žšœ–.(2.4) such that š¶0. By the invariance properties of (š’Æ(š‘”))š‘”ā‰„0,, we have Ī©. Consequently, (š‘”š‘›,š‘„š‘›)āˆˆĪ©,š‘”š‘›ā†‘š‘” By using the continuity of š½-semigroup š‘„š‘›ā†’š‘„ 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)š‘‹ is closed from the left, that is, if š‘›ā†’āˆž, in (š‘”,š‘„)āˆˆĪ©, and Ī©(š‘”) in š’Æ(š‘ ) as š‘”,š‘ ā‰„0 then for0x000a0all(š‘”,š‘„)āˆˆĪ©,liminfā„Žā†“0(1/ā„Ž)š‘‘(š‘„+ā„Žā„¬(š‘”,š‘„),Ī©(š‘”+ā„Ž))=0;(ii)ā„¬ is Ī©-invariant, for all š‘™ā„¬āˆˆā„+;(iii)(ā„¬(š‘”,ā‹…)āˆ’š‘™ā„¬š¼);(iv)Ī©(š‘”) is continuous on š‘”āˆˆš½ and there exists Ī© such that the operator š½Ć—š‘‹ is dissipative on š‘”āˆˆš½,Ī©(š‘”)ā‰ āˆ…, for all (šœ,š‘„šœ)āˆˆĪ©,. If š½ is a connected subset of š»=šæ2[0,1]āŠ•šæ2[0,1] such that for all š‘„ī‚¬ī‚€1,š‘„2ī‚,ī‚€š‘¦1,š‘¦2=ī‚¬š‘„ī‚ī‚­1,š‘¦1ī‚­šæ2+ī‚¬š‘„2,š‘¦2ī‚­šæ2(3.1), then, for each ā€–ā€–ī‚€š‘„1,š‘„2ī‚ā€–ā€–ā€–ā€–š‘„=(1ā€–ā€–2šæ2+ā€–ā€–š‘„2ā€–ā€–2šæ2)1/2(3.2) (2.1) has a unique mild solution on (š‘„1,š‘„2).

3. Abstract Semigroup Formulation

Throughout the sequel, we assume (š‘¦1,š‘¦2), the Hilbert space with the usual inner productš» and the induced norm š» for all š‘„=(š‘„1,š‘„2)āˆˆš»,š‘¦=(š‘¦1,š‘¦2)āˆˆš» and š‘„ā‰¤š‘¦iļ¬€0x000a0š‘„1(š‘§)ā‰¤š‘¦1(š‘§),š‘„2(š‘§)ā‰¤š‘¦2(š‘§)fora.e.š‘§āˆˆ[0,1].(3.3) in š‘„,š‘¦āˆˆš».

Clearly, the Hilbert space ī‚†[š‘„,š‘¦]=š‘¤āˆˆš»āˆ¶š‘„1ā‰¤š‘¤1ā‰¤š‘¦1,š‘„2ā‰¤š‘¤2ā‰¤š‘¦2ī‚‡=ī‚ƒš‘„1,š‘¦1ī‚„Ć—ī‚ƒš‘„2,š‘¦2ī‚„(3.4) is a real Banach lattice, where for all given š‘„,š‘¦ Recall that for every pair [š‘„,š‘¦], the setš‘„ā‰¤š‘¦ is called the order interval between š’Æ and š». Clearly, 0ā‰¤š’Æš‘„ is nonempty if 0ā‰¤š‘„. (for more details, see, e.g., [25]). A bounded linear operator (š’Æ(š‘”))š‘”ā‰„0 on š» is said to be positive if š¶0-semigroup for all š» Similarly, a family of bounded linear operators š’Æ(š‘”) of š¶0-semigroup is said to be a positive š» on š’Æ(š‘”) if š‘”ā‰„0 is a š¶in(š‘”) on šæin(š‘”) and š’ž1([0,āˆž[) is a positive operator for all šœ‰š‘§=š‘™,š‘„1=š¶āˆ’š¶in,š‘„2=šæāˆ’šæin,š‘„01=š¶0āˆ’š¶in,š‘„02=šæ0āˆ’šæin.(3.5).

In the following, we will assume that š‘§āˆˆ]0,1[ and š‘”>0 are positive šœ•š‘„1šœ•š‘”=āˆ’š‘£šœ•š‘„1šœ•š‘§+š‘‘1šœ•2š‘„1šœ•š‘§2āˆ’š‘˜1ī‚€š‘„1+š¶inī‚(š‘”)š‘šī‚€š‘„2+šæinī‚(š‘”)š‘›āˆ’Ģ‡š¶in(š‘”),(3.6)šœ•š‘„2šœ•š‘”=āˆ’š‘£šœ•š‘„2šœ•š‘§+š‘‘2šœ•2š‘„2šœ•š‘§2āˆ’š‘˜2ī‚€š‘„1+š¶inī‚(š‘”)š‘šī‚€š‘„2+šæinī‚(š‘”)š‘›āˆ’Ģ‡šæin(š‘”),(3.7)-functions. Let us consider the following state transformation: š‘‘š‘–šœ•š‘„š‘–šœ•š‘§(0,š‘”)āˆ’š‘£š‘„š‘–(0,š‘”)=0=š‘‘š‘–šœ•š‘„š‘–š‘„šœ•š‘§(1,š‘”)āˆ€š‘”>0š‘–=1;2,(3.8)š‘–(š‘§,0)=š‘„0š‘–(š‘§)forš‘§āˆˆ]0,1[,š‘–=1;2,(3.9) Then, we obtain the new equivalent system for all š‘‘1=š·1š‘™2,š‘‘2=š·2š‘™2šœˆ,š‘£=š‘™.(3.10) and ī‚€ī‚,Ģ‡š‘„(š‘”)=š“š‘„(š‘”)+šµš‘”,š‘„(š‘”)š‘„(0)=š‘„0āˆˆĪ©(0),(3.11): Ī©(š‘”)Ī© with š‘”āˆˆā„+ī‚€ī‚€š‘„Ī©={š‘”,1,š‘„2ī‚ī‚š‘‡āˆˆā„+Ɨš»āˆ¶āˆ’š¶in(š‘”)ā‰¤š‘„1(š‘§)ā‰¤š¶āˆ’š¶in(š‘”),āˆ’šæin(š‘”)ā‰¤š‘„2(š‘§)ā‰¤šæāˆ’šæin(š‘”)a.e.0x000a0š‘§āˆˆ[0,1]}.(3.12) where š“

This PDEs describing the reactor dynamics may be formally written in the abstract form as ī‚€š‘„š·(š“)={š‘„=1,š‘„2ī‚š‘‡āˆˆš»āˆ¶š‘„,š‘‘š‘„š‘‘š‘‘š‘§āˆˆš»are0x000a0absolutely0x000a0continuous,2š‘„š‘‘š‘§2āˆˆš»,š‘‘š‘–š‘‘š‘„š‘–š‘‘š‘§(0)āˆ’šœš‘„š‘–(0)=0=š‘‘š‘–š‘‘š‘„š‘–āŽ›āŽœāŽœāŽœāŽš‘‘š‘‘š‘§(1);š‘–=1;2},(3.13)š“š‘„=1š‘‘2š‘„1š‘‘š‘§2āˆ’šœš‘‘š‘„10š‘‘š‘§0š‘‘2š‘‘2š‘„2š‘‘š‘§2āˆ’šœš‘‘š‘„2āŽžāŽŸāŽŸāŽŸāŽ =īƒ©š“š‘‘š‘§1š‘„100š“2š‘„2īƒŖ.(3.14) where šµ denote the section of Ī© at ī‚€šµ(š‘”,š‘„)=āˆ’š‘˜1ī‚€š‘„1+š¶inī‚(š‘”)š•€š‘šī‚€š‘„2+šæinī‚(š‘”)š•€š‘›āˆ’Ģ‡š¶in(š‘”)š•€,āˆ’š‘˜2ī‚€š‘„1+š¶inī‚(š‘”)š•€š‘šī‚€š‘„2+šæinī‚(š‘”)š•€š‘›āˆ’Ģ‡šæinī‚(š‘”)š•€š‘‡.(3.15), which is given in view of (1.7) by š“

The linear operator š» is defined by īƒ©š‘‡š‘‡(š‘”)=1(š‘”)00š‘‡2īƒŖ(š‘”),(3.16)š‘‡1(š‘”) The nonlinear operator š‘‡2(š‘”) is defined on š¶0-semigroups by š“1 It is shown in [7] that the linear operator š“2 given by (3.14) is the infinitesimal generator of contraction semigroup on (š‘”,š‘„)āˆˆĪ©,limā„Žā†“01ā„Žš‘‘ī‚€ī‚š‘„+ā„Žšµ(š‘”,š‘„);Ī©(š‘”+ā„Ž)=0.(4.1) where (š‘”,š‘„)āˆˆĪ©. and Ī©(š‘”) are the Ī©(š‘”)=Ī©1(š‘”)ƗĪ©2(š‘”) generated, respectively, by Ī©1ī‚ƒ(š‘”)=āˆ’š¶inī‚€(š‘”)š•€,š¶āˆ’š¶inī‚š•€ī‚„,Ī©(š‘”)2ī‚ƒ(š‘”)=āˆ’šæinī‚€(š‘”)š•€,šæāˆ’šæinī‚š•€ī‚„.(š‘”)(4.2) and š‘‹1(š‘”)=š‘„1+š¶in(š‘”)š•€,š‘‹2(š‘”)=š‘„2+šæin(š‘”)š•€,(4.3).

4. Global Existence

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

Lemma 4.1. For each š‘„āˆˆĪ©(š‘”),ī‚€š‘‹š‘‹(š‘”)=1(š‘”),š‘‹2ī‚(š‘”)š‘‡āˆˆ[0,š¶š•€]Ɨ[0,šæš•€].(4.4)

Proof. Let ā„Ž0>0 Observe that ā„Ž0š‘˜1š¶š‘šāˆ’1šæš‘›ā‰¤1. is given by ā„Žāˆˆ(0,ā„Ž0),, where š‘‹1ī‚€(š‘”)š•€āˆ’ā„Žš‘˜1š‘‹1š‘šāˆ’1(š‘”)š‘‹š‘›2ī‚(š‘”)āˆˆ[0,š¶š•€].(4.5) Denote š‘“1ī‚€ī‚š‘”,š‘‹(š‘”)=š‘‹1ī‚€(š‘”)š•€āˆ’ā„Žš‘˜1š‘‹1š‘šāˆ’1(š‘”)š‘‹š‘›2ī‚(š‘”)āˆ’š¶in(š‘”+ā„Ž)š•€āˆˆĪ©1(š‘”+ā„Ž).(4.6) we have, for š¶inš‘‘ī‚€š‘„1+ā„Žšµ1(š‘”,š‘„),Ī©1ī‚(š‘”+ā„Ž)ā‰¤š‘‘(š‘‹1(š‘”)āˆ’ā„Žš‘˜1š‘‹š‘š1(š‘”)š‘‹š‘›2(š‘”)āˆ’š¶in(š‘”+ā„Ž)š•€,Ī©1ī‚€š‘“(š‘”+ā„Ž))+ā„Žšœ–(ā„Ž)ā‰¤š‘‘1ī‚€ī‚š‘”,š‘‹(š‘”),Ī©1ī‚(š‘”+ā„Ž)+ā„Žšœ–(ā„Ž)=ā„Žšœ–(ā„Ž),(4.7) Let šœ–(ā„Ž)ā†’0 be sufficiently small such that ā„Žā†’0.
Let, now, limā„Žā†“01ā„Žš‘‘ī‚€š‘„1+ā„Žšµ1(š‘”,š‘„);Ī©1ī‚(š‘”+ā„Ž)=0.(4.8) then šæin, Hence limā„Žā†“01ā„Žš‘‘ī‚€š‘„2+ā„Žšµ2(š‘”,š‘„);Ī©2ī‚(š‘”+ā„Ž)=0.(4.9) By using the regularity of the inlet function š‘‘ī‚€ī‚ī‚€š‘„š‘„+šµ(š‘”,š‘„),Ī©(š‘”+ā„Ž)ā‰¤š‘‘1+šµ1(š‘”,š‘„),Ī©1ī‚ī‚€š‘„(š‘”+ā„Ž)+š‘‘2+šµ2(š‘”,š‘„),Ī©2ī‚(š‘”+ā„Ž),(4.10), we get š‘™šµāˆˆā„+ where (šµ(š‘”,ā‹…)āˆ’š‘™šµš¼) as Ī©(š‘”) Whence š‘”ā‰„0 By similar considerations as above, taking into account the regularity of the function š‘”ā‰„0 we also get š‘„,š‘¦ Observe, now, that Ī©(š‘”). 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 š‘”š‘–(š‘”,š‘„)=āˆ’š‘˜š‘–ī‚€š‘„1+š¶inī‚(š‘”)š•€š‘šī‚€š‘„2+šæinī‚(š‘”)š•€š‘›forš‘–=1,2,(4.11) such that the operator š‘‹1(š‘”)=š‘„1+š¶in(š‘”)š•€;š‘‹2(š‘”)=š‘„2+šæin(š‘”)š•€;š‘Œ1(š‘”)=š‘¦1+š¶in(š‘”)š•€,š‘Œ2(š‘”)=š‘¦2+šæin(š‘”)š•€.(4.12) is dissipative on š‘„,š‘¦āˆˆĪ©(š‘”), for each (š‘‹š‘–(š‘”),š‘Œš‘–(š‘”))š‘‡āˆˆ[0,š¶š•€]Ɨ[0,šæš•€].

Proof. Let š‘–=1,2. and let š‘–=1,2 be in ā€–ā€–š‘”š‘–(š‘”,š‘„)āˆ’š‘”š‘–ā€–ā€–(š‘”,š‘¦)šæ2ā‰¤š‘˜š‘–ī‚€š¶2š‘šā€–ā€–š‘‹š‘›2(š‘”)āˆ’š‘Œš‘›2ā€–ā€–(š‘”)2šæ2+šæ2š‘›ā€–ā€–š‘‹š‘š1(š‘”)āˆ’š‘Œš‘š1ā€–ā€–(š‘”)2šæ2ī‚1/2ā‰¤š‘˜š‘–ī‚€š‘›2š¶2š‘ššæ2š‘›āˆ’2ā€–ā€–š‘„2āˆ’š‘¦2ā€–ā€–2šæ2+š‘š2šæ2š‘›š¶2š‘šāˆ’2ā€–ā€–š‘„1āˆ’š‘¦1ā€–ā€–2šæ2ī‚1/2ā‰¤š‘˜š‘–š¶š‘šāˆ’1šæš‘›āˆ’1ī‚€š‘›maxš¶;š‘ššæī‚ā€–š‘„āˆ’š‘¦ā€–.(4.13) Denote ā€–ā€–ā€–ā€–ā€–ā€–š‘”šµ(š‘”,š‘„)āˆ’šµ(š‘”,š‘¦)=(1(š‘”,š‘„)āˆ’š‘”1ā€–ā€–(š‘”,š‘¦)2šæ2+ā€–ā€–š‘”2(š‘”,š‘„)āˆ’š‘”2ā€–ā€–(š‘”,š‘¦)2šæ2)1/2ī‚€š‘˜ā‰¤max1,š‘˜2ī‚š¶š‘šāˆ’1šæš‘›āˆ’1ī‚€š‘›maxš¶;š‘ššæī‚ā€–š‘„āˆ’š‘¦ā€–.(4.14) and let also šµ(š‘”,ā‹…) Observe that, for each š‘™šµĪ©(š‘”) for š‘™šµī‚€š‘˜=max1,š‘˜2ī‚š¶š‘šāˆ’1šæš‘›āˆ’1ī‚€š‘›maxš¶;š‘ššæī‚.(4.15) Hence, by applying the mean value theorem, for Ī©(š‘”)isš’Æ(š‘ )-invariantāˆ€š‘”,š‘ ā‰„0.(4.16), we getš‘”,š‘ ā‰„0 Finally, (š‘„,š‘¦)š‘‡āˆˆĪ©(š‘”) Consequently, ī‚€āˆ’š¶in(š‘”)š•€,āˆ’šæinī‚(š‘”)š•€š‘‡ā‰¤(š‘„,š‘¦)š‘‡ā‰¤ī‚€ī‚€š¶āˆ’š¶inī‚ī‚€(š‘”)š•€,šæāˆ’šæinī‚š•€ī‚(š‘”)š‘‡.(4.17) is an (š‘‡(š‘”))š‘”ā‰„0-dissipative operator on ī‚€āˆ’š¶in(š‘”)š‘‡1(š‘ )š•€,āˆ’šæin(š‘”)š‘‡2ī‚(š‘ )š•€š‘‡ā‰¤š‘‡(š‘ )(š‘„,š‘¦)š‘‡ā‰¤ī‚€ī‚€š¶āˆ’š¶inī‚š‘‡(š‘”)1ī‚€(š‘ )š•€,š¶āˆ’š¶inī‚š‘‡(š‘”)2ī‚(š‘ )š•€š‘‡.(4.18) [14, page 245], where š‘‡š‘–(š‘”)š•€ā‰¤š•€forš‘–=1;2
Finally, we state the invariance properties of the state trajectories of the model given by (3.6)ā€“(3.9).

Proposition 4.3. One has that š¶ā‰„š¶in

Proof. Let šæā‰„šæin and Ī©(š‘”). We have š‘”ā‰„0. Hence, by using the positivity of (š‘‡1(š‘ )š‘„,š‘‡2(š‘ )š‘¦)š‘‡āˆˆĪ©(š‘”) [26], we have š‘”,š‘ ā‰„0. Since, š¶in(š‘”) (see [26]) and by using the inequalities (1.8) (i.e., šæin(š‘”) and š¶1([0,+āˆž[)), the invariance of š‘„0āˆˆĪ©(0), holds for all šµ Thus, Ī©, for all Ī©

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

Theorem 4.4. Let Ī© and š‘”š‘›ā†—š‘” be positive š‘„š‘›āˆˆĪ©(š‘”š‘›)-functions. Then, for every š‘„š‘›ā†’š‘„āˆˆš», the problem (3.6)ā€“(3.9) has a unique global mild solution.

Proof. Since (š‘„š‘›) is continuous function in (š‘„š‘›) by Corollary 2.3, it is sufficient to prove the condition (i) in Corollary 2.3 and to check that the subset š‘„š‘›(š‘§)ā†’š‘„(š‘§) is connected
(a)Let us first show that [0,1] is closed from the left.Let š¶in and šæin,thatš‘„(š‘§)āˆˆ[āˆ’š¶in(š‘”),š¶āˆ’š¶in(š‘”)]Ɨ[āˆ’šæin(š‘”),šæāˆ’šæin(š‘”)] with [0,1] then there exists a subsequence of š‘„āˆˆĪ©(š‘”) which is also denoted by š‘”ā‰„0 such that Ī©, that is, on [0,+āˆž[Ɨš» which implies, by continuity of š¾=[0,š¶š•€]Ɨ[0,šæš•€] and šŗāˆ¶[0,+āˆž[Ɨš¾ā†’Ī©, that is, on (š‘”,š‘„)āˆˆ[0,+āˆž[Ɨš¾,, hence šŗ(š‘”,š‘„)=(š‘”,š‘„1āˆ’š¶in(š‘”)š•€,š‘„2āˆ’šæin(š‘”)š•€)š‘‡. for each š¶in.(b)Let us, now, check that šæin is connected in [0,+āˆž[,:Let šŗ and define [0,+1[Ɨš¾ such that for all šŗ[0,š¶š•€]Ɨ[0,šæš•€] Since š» and Ī©=šŗ([0,+āˆž[Ɨš¾) are continuous functions in [0,+āˆž[Ɨš». it follows that š¶in in šæin is also a continuous function. Observe that š‘”, is surjective; since Ī©(š‘”) is connected in š‘”, we get that š¶in is also connected in šæin
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)ā€“(3.9).

5. Equilibrium Profiles

In the steady-state solution analysis, the inlet functions š¶in and šæin are independent of time āˆ’š‘£š‘‘š‘„1š‘‘š‘§=š‘‘1š‘‘2š‘„1š‘‘š‘§2āˆ’š‘˜1ī‚€š‘„1+š¶inī‚š‘šī‚€š‘„2+šæinī‚š‘›=0,(5.1)āˆ’š‘£š‘‘š‘„2š‘‘š‘§=š‘‘2š‘‘2š‘„2š‘‘š‘§2āˆ’š‘˜2ī‚€š‘„1+š¶inī‚š‘šī‚€š‘„2+šæinī‚š‘›=0,(5.2) which implies that the domain š‘‘š‘–š‘‘š‘„š‘–š‘‘š‘§(0)āˆ’š‘£š‘„š‘–(0)=0=š‘‘š‘–š‘‘š‘„š‘–ī‚€š‘„š‘‘š‘§(1),š‘–=1;2,(5.3)Ī©(š‘”)=Ī”={1,š‘„2ī‚š‘‡āˆˆš»āˆ¶āˆ’š¶inā‰¤š‘„1(š‘§)ā‰¤š¶āˆ’š¶in,āˆ’šæinā‰¤š‘„2(š‘§)ā‰¤šæāˆ’šæinforalmostallš‘§āˆˆ[0,1]}.(5.4) is also independent of š¶in. If we denote by šæin and Ī” the values of š‘‘1=š‘‘2=š‘‘. and š‘‘1=š‘‘2=š‘‘ which correspond to the steady-state solutions, the corresponding steady-state system to the models (3.6)ā€“(3.9) is given by the following equations: š’œ=š‘‘(š‘‘2/š‘‘š‘§2)āˆ’š‘£(š‘‘/š‘‘š‘§)=š“š‘–š·(š’œ)=š·(š“š‘–) with š‘–=1;2š‘(ā‹…) The following existence result can be proven as in the case where [0,1] and š‘¢āˆˆšæ2([0,1]) 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)ā€“(3.9) has at least one equilibrium profile in š’œš‘¢=š‘š‘¢in]0,1[,0x000a0š‘¢āˆˆš·(š’œ),(5.5).

The sequel of this paper will deal with the uniqueness analysis of steady states in the important case where š‘¢=0

First, since [0,1], we denote š‘¢ with āŸØš’œš‘¢,š‘¢āŸ©šæ2=āŸØš‘š‘¢,š‘¢āŸ©šæ2.(5.6) for āŸØš’œš‘¢,š‘¢āŸ©šæ2=ī€œ10ī‚ƒš‘‘š‘‘2š‘¢š‘‘š‘§2(š‘§)āˆ’š‘£š‘‘š‘¢ī‚„ī€œš‘‘š‘§(š‘§)š‘¢(š‘§)š‘‘š‘§,=āˆ’10š‘‘ī‚ƒš‘‘š‘¢ī‚„š‘‘š‘§(š‘§)2ī‚ƒš‘‘š‘§+š‘‘š‘‘š‘¢š‘‘š‘§(1)š‘¢(1)āˆ’š‘‘š‘¢ī‚„āˆ’1š‘‘š‘§(0)š‘¢(0)2š‘£ī‚ƒš‘¢2(1)āˆ’š‘¢2ī‚„,ā€–ā€–ā€–(0)=āˆ’š‘‘š‘‘š‘¢ā€–ā€–ā€–š‘‘š‘§2šæ2āˆ’12š‘£š‘¢21(1)āˆ’2š‘£š‘¢2(0),(5.7)ā‰¤0.(5.8).

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 š‘(š‘§) be a bounded nonnegative function defined in [0,1],. If āŸØš‘š‘¢,š‘¢āŸ©šæ2=āˆ«10š‘(š‘§)š‘¢2(š‘§)š‘‘š‘§=0.(5.9) satisfies the equations āŸØš’œš‘¢,š‘¢āŸ©šæ2ā€–ā€–ā€–=0=š‘‘š‘‘š‘¢ā€–ā€–ā€–š‘‘š‘§2šæ2+12š‘£š‘¢21(1)+2š‘£š‘¢2(0).(5.10) then š‘‘š‘¢š‘‘š‘§(š‘§)=0a.e.š‘§āˆˆ[0,1],š‘¢(0)=0=š‘¢(1).(5.11) in š·(š’œ)āŠ‚š’ž([0,1])..

Proof. Let š‘¢=0 be the solution of problem (5.5), then š‘¢āˆˆš·(š’œ). We have, š‘‘1=š‘‘2=š‘‘,Ī”. Since š‘„=(š‘„1,š‘„2)š‘‡andš‘¦=(š‘¦1,š‘¦2)š‘‡ is nonnegative function in [0,1]. then by (5.8) and taking into account (5.6) š‘„=š‘¦. Which implies, in view of (5.6)-(5.7), that š‘”ī‚€š‘„1,š‘„2ī‚ī‚€š‘„=āˆ’1+š¶inš•€ī‚š‘šī‚€š‘„2+šæinš•€ī‚š‘›,š‘¤1=š‘¦1āˆ’š‘„1āˆˆš·(š’œ),š‘¤2=š‘„2āˆ’š‘¦2āˆˆš·(š’œ).(5.12) Then, we get āˆ’š’œš‘¤1=š‘˜1ī‚€š‘”ī‚€š‘¦1,š‘¦2ī‚ī‚€š‘„āˆ’š‘”1,š‘„2ī‚ī‚=š‘˜1ī‚€š‘¦1+š¶inī‚š‘šš‘„ī‚ƒī‚€2+šæinī‚š‘›āˆ’ī‚€š‘¦2+šæinī‚š‘›ī‚„+š‘˜1ī‚€š‘„2+šæinī‚š‘›š‘„ī‚ƒī‚€1+š¶inī‚š‘šāˆ’ī‚€š‘¦1+š¶inī‚š‘šī‚„(5.13) Clearly, by using the Sobolev imbedding theorem, āˆ’š’œš‘¤1=š‘˜1š‘›ī‚€š‘¦1+š¶inī‚š‘ššœ‰2š‘›āˆ’1š‘¤2āˆ’š‘šš‘˜1ī‚€š‘„2+šæinī‚š‘›šœ‰1š‘šāˆ’1š‘¤1,(5.14) Therefore, (šœ‰1,šœ‰2) since (0,0)

Theorem 5.3. For (š¶,šæ). the steady-state problem given by (5.1)ā€“(5.3) has a unique solution in āˆ’š’œš‘¤2=āˆ’š‘˜2ī‚€š‘”ī‚€š‘¦1,š‘¦2ī‚ī‚€š‘„āˆ’š‘”1,š‘„2ī‚ī‚=āˆ’š‘˜2š‘›ī‚€š‘¦1+š¶inī‚š‘ššœ‰2š‘›āˆ’1š‘¤2+š‘šš‘˜2ī‚€š‘„2+šæinī‚š‘›šœ‰1š‘šāˆ’1š‘¤1,(5.15)

Proof. Let šœ‰1 be solutions to (5.1)ā€“(5.3) on šœ‰2. To obtain the desired result, we will be showing that āˆ’š’œš‘¤1=āˆ’š‘Ž1š‘¤1+š‘1š‘¤2,(5.16)āˆ’š’œš‘¤2=š‘Ž2š‘¤1āˆ’š‘2š‘¤2,(5.17) Let š‘–=1;2, Then š‘Žš‘–(š‘§)=š‘šš‘˜š‘–ī‚€š‘„2(š‘§)+šæinī‚š‘›šœ‰1š‘šāˆ’1š‘(š‘§),š‘–(š‘§)=š‘›š‘˜š‘–ī‚€š‘¦1(š‘§)+š¶inī‚š‘ššœ‰2š‘›āˆ’1(š‘§).(5.18) Hence, by applying the mean value theorem, we get
where š‘˜1, are some intermediate values between š’œš‘¤=0,š‘¤āˆˆš·(š’œ),(5.19) and š‘¤=š‘˜2š‘¤1+š‘˜1š‘¤2.
By similar considerations as above, we also get š‘¤=0 for the same [0,1]. and āˆ’š’œš‘¤2=š‘Ž2š‘¤1āˆ’š‘2š‘¤2(5.20)
Now, we have the following system: š‘¤1=āˆ’š‘˜2āˆ’1š‘˜1š‘¤2(5.21)š’œš‘¤2=š‘š‘¤2,(5.22) where, for š‘(š‘§)=š‘Ž1(š‘§)+š‘2(š‘§).š‘–=1;2
Multiplying (5.16) by 0ā‰¤š‘Žš‘–(š‘§)ā‰¤š‘šš‘˜š‘–šæš‘›š¶š‘šāˆ’1,0ā‰¤š‘š‘–(š‘§)ā‰¤š‘›š‘˜š‘–š¶š‘ššæš‘›āˆ’1.(5.23) and (5.17) by šœ†=max(š‘ššæ,š‘›š¶)max(š‘˜1,š‘˜2)š¶š‘šāˆ’1šæš‘›āˆ’1 we get by addition of both equations that 0ā‰¤š‘(š‘§)ā‰¤2šœ†. where š‘¤2=0. By Lemma 5.2, this system has a unique solution š‘¤1=0, in š‘„=š‘¦. Now, let š‘„(š‘”)āˆˆĪ©(š‘”). and substituting the expression Ī©(š‘”) yields šæš‘ 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)ā€“(1.6). This question is under investigation.

Acknowledgments

This paper presents research results of the Moroccan ā€œProgramme ThĆ©matique d'Appui Ć  la Recherche Scientifiqueā€ PROTARS III, initiated by the Moroccan ā€œCentre National de la Recherche Scientifique et Techniqueā€ (CNRST). The scientific responsibility rests with its authors. The work has been partially carried out within the framework of a collaboration agreement between CESAME (UniversitĆ© Catholique de Louvain, Belgium) and LINMA of the Faculty of sciences (UnivesitĆ© Chouaib Doukkali, Morocco), funded by the Belgian Secretary of the State for Development Cooperation and by the CIUF (Conseil Interuniversitaire de la CommunautĆ© FranƧaise, Belgium). The work of B. Aylaj is supported by a research grant from the Agence Universitaire de la Francophonie.

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Copyright © 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.


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