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Abstract and Applied Analysis
VolumeΒ 2008, Article IDΒ 127394, 13 pages
http://dx.doi.org/10.1155/2008/127394
Research Article

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

1Center de Recherche INRIA Futurs-Site de Bordeaux-IMB, 351 Cours de la LibΓ©ration, Talent Cedex, Bordeaux 33405, France
2DΓ©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 𝐢0-semigroup.

1. Introduction

Several chemical and biochemical processes are typically described by nonlinear coupled partial differential equations β€œPDE” 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 [1–5]. 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,+∞[. 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 𝑑. The considered class of models correspond to the following chemical reaction: 𝑛𝐴+π‘šπ΅β†’π‘ƒ,(1.1) whose kinetic is given by π‘Ÿ=(βˆ’π‘˜1πΆπ‘šπΏπ‘›,βˆ’π‘˜2πΆπ‘šπΏπ‘›)𝑇, where 𝐢 and 𝐿 are the concentrations of the reactants 𝐴 and 𝐡, respectively, π‘˜1 and π‘˜2 are the kinetic constants and π‘š,𝑛 are the order of the reaction to 𝐴 and 𝐡, respectively. More precisely, we study the global existence and uniqueness of the trajectories of the models which describe the evolution of two reactant concentrations 𝐢 and 𝐿: πœ•πΆπœ•π‘‘=βˆ’πœˆπœ•πΆπœ•πœ‰+𝐷1πœ•2πΆπœ•πœ‰2βˆ’π‘˜1πΆπ‘šπΏπ‘›,(1.2)πœ•πΏπœ•π‘‘=βˆ’πœˆπœ•πΏπœ•πœ‰+𝐷2πœ•2πΏπœ•πœ‰2βˆ’π‘˜2πΆπ‘šπΏπ‘›,(1.3)for0x000a0πœ‰βˆˆ]0,𝑙[and𝑑>0𝐷1πœ•πΆπœ•πœ‰(0,𝑑)βˆ’πœˆπΆ(0,𝑑)+𝜈𝐢in(𝑑)=0=𝐷1πœ•πΆπ·πœ•πœ‰(𝑙)βˆ€π‘‘>0(1.4)2πœ•πΏπœ•πœ‰(0,𝑑)βˆ’πœˆπΏ(0,𝑑)+𝜈𝐿in(𝑑)=0=𝐷2πœ•πΏπœ•πœ‰(𝑙,𝑑)βˆ€π‘‘>0,(1.5)𝐢(πœ‰,0)=𝐢0(πœ‰),𝐿(πœ‰,0)=𝐿0(πœ‰)forπœ‰βˆˆ]0,𝑙[.(1.6), with the following boundary and initial conditions: 𝐷1,𝐷2πœˆπ‘‘,πœ‰

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

In the above equations, 𝑙 are the dispersion coefficients, π‘š is the superficial fluid velocity, 𝑛 denote the time and space independent variables, respectively, 𝐢in is the length of the reactor, 𝐿in and 𝜈=0 are two positive integers, 𝑛=π‘š=1 and 𝐿∞ 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, π‘š=1, [8] established the asymptotic behavior of solutions for the second-order reaction (i.e., 1<𝑛<3/2). N. Alikakos [9] established global existence and π‘š=1 bounds of positive solutions, when 𝑛>1 and π‘š=𝑛=1,2,3. This latter result has been generalized by [10] for the case (𝐢) and (𝐿).
In practice, the special cases 𝑛=π‘š have been used as an industrial pulp bleaching model, where the two reactants are chlorine dioxide 𝐷1=𝐷2 and lignin 𝐷1≠𝐷2. In particular, [3] studied approximate solutions by using several methods (orthogonal collocation, finite elements, and finite difference methods), when π‘š=𝑛=1 and 𝐷2=4𝐷1. The reader can find another model with 𝐷2=16𝐷1 in [11], where the numerical analysis has been done for 0β‰€πœ‰β‰€π‘™ and 𝑑β‰₯0, 0≀𝐢≀𝐢,0≀𝐿≀𝐢𝐿,(1.7)in(𝑑)≀𝐢,𝐿in(𝑑)≀𝐿,(1.8) (see also [12]).
Recently, the existence of global solutions for problems such as (1.2)–(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 𝐢 and for all 𝐿: 𝑋‖⋅‖,𝐽=[π‘Ž,𝑏[(π‘Ž<𝑏≀+∞), where {𝒯(𝑑);𝑑β‰₯0} and 𝐢0 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)–(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)–(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 [13–16].

2. Preliminaries

Let 𝑋 be a real Banach space with norm π’œ and let ℬ be a linear contraction Ξ©-semigroup on 𝑋 generated by Ξ©. Let 𝐽×𝑋 be a nonlinear continuous operator form 𝐼 into 𝕀, where 𝑋 is a subset of Μ‡π‘₯(𝑑)=π’œπ‘₯(𝑑)+ℬ(𝑑,π‘₯(𝑑)),𝜏<𝑑<𝑏,π‘₯(𝜏)=π‘₯𝜏∈Ω(𝜏),(2.1). Ξ©(𝜏) and Ξ© denote, respectively, the identity operator of 𝜏∈𝐽 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: Ξ©(𝑑)={π‘₯βˆˆπ‘‹;(𝑑,π‘₯)∈Ω}. where Ξ©(𝑑)β‰ βˆ… denote the section of π‘‘βˆˆπ½ at 𝑑(π‘₯;π’Ÿ)=inf{β€–π‘₯βˆ’π‘¦β€–,π‘¦βˆˆπ’Ÿ},, given by π‘₯βˆˆπ‘‹ Assume that π’Ÿ for all 𝑋. Moreover, recall that Ξ© for Ξ©=π½Γ—π’Ÿ and Ξ© is a subset of π‘₯(𝑑)∈Ω(𝑑).

The semilinear nonautonomous evolution equations have been treated by a number of authors [14, 15, 17–21]. However, one may find that in most cases 𝐽 is cylindrical, that is, Ξ© [14, 22]. More generally, the cylindrical case of ℬ(𝑑,β‹…) will not be convenient for the study of evolution system satisfying time-dependent constraints, that is, 𝑙ℬ on Ξ© (see our problem in Section 3). A noncylindrical (𝑑𝑛,π‘₯𝑛)∈Ω,𝑑𝑛↑𝑑 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 𝐽 is π‘₯𝑛→π‘₯-dissipative [16].

Theorem 2.1 (see [16]). Suppose that the following conditions are fulfilled: (i)𝑋 is closed from the left, that is, if π‘›β†’βˆž in (𝑑,π‘₯)∈Ω, and for0x000a0all0x000a0(𝑑,π‘₯)∈Ω,liminfβ„Žβ†“0(1/β„Ž)𝑑(𝒯(β„Ž)π‘₯+β„Žβ„¬(𝑑,π‘₯),Ξ©(𝑑+β„Ž))=0 in ℬ as Ξ©, then π‘™β„¬βˆˆβ„+;(ii)(ℬ(𝑑,β‹…)βˆ’π‘™β„¬πΌ);(iii)Ξ©(𝑑) is continuous on π‘‘βˆˆπ½ and there exists Ξ© such that the operator 𝐽×𝑋 is dissipative on π‘‘βˆˆπ½,Ξ©(𝑑)β‰ βˆ… for all (𝜏,π‘₯𝜏)∈Ω,. If 𝐽 is a connected subset of Ξ©(𝑑) such that for all 𝒯(𝑠), then, for each 𝒯(𝑠)(Ξ©(𝑑))βŠ‚Ξ©(𝑑) (2.1) has a unique mild solution on 𝑑,𝑠β‰₯0,.

Comment 2. It is shown in [16] that the β€œsubtangential condition” (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 β„¬βˆΆΞ©β†’π‘‹ is Ξ©-invariant, that is, Ξ©(𝑑) for all 𝒯(𝑠) we have the following lemma.

Lemma 2.2. Let 𝑑,𝑠β‰₯0 be continuous and let limβ„Žβ†“01infβ„Žπ‘‘ξ‚€ξ‚π‘₯+β„Žβ„¬(𝑑,π‘₯);Ξ©(𝑑+β„Ž)=0βˆ€(𝑑,π‘₯)∈Ω(2.2) be closed from the left. If limβ„Žβ†“01infβ„Žπ‘‘ξ‚€ξ‚π’―(β„Ž)π‘₯+β„Žβ„¬(𝑑,π‘₯),Ξ©(𝑑+β„Ž)=0,βˆ€(𝑑,π‘₯)∈Ω.(2.3) is (𝑑,π‘₯)∈Ω,-invariant for all πœ–>0,, then the following subtangential condition β„Žβˆˆ(0,πœ–] implies the condition π‘¦βˆˆΞ©(𝑑+β„Ž)

Proof. Let β€–π‘¦βˆ’π‘₯βˆ’β„Žβ„¬(𝑑,π‘₯)β€–β‰€β„Žπœ– given 𝑒=π‘¦βˆ’π‘₯βˆ’β„Žβ„¬(𝑑,π‘₯) from condition (2.2) it follows, by [23, Lemma 3] (see also [24, Lemma 1]), that there is 𝑣=(1/β„Ž)𝑒 and β€–π‘£β€–β‰€πœ– such that 𝑦=π‘₯+β„Ž(𝐡(𝑑,π‘₯)+𝑣)∈Ω(𝑑+β„Ž). Let now Ξ©(𝑑) and 𝒯(β„Ž)π‘¦βˆˆΞ©(𝑑+β„Ž). We get 𝑑≀‖‖‖‖,≀‖‖‖‖,β€–β€–β€–β€–β€–β€–β€–β€–,‖‖‖‖𝒯(β„Ž)π‘₯+β„Žβ„¬(𝑑,π‘₯);Ξ©(𝑑+β„Ž)𝒯(β„Ž)π‘₯+β„Žβ„¬(𝑑,π‘₯)βˆ’π’―(β„Ž)π‘¦β„Žβ„¬(𝑑,π‘₯)βˆ’β„Žπ’―(β„Ž)ℬ(𝑑,π‘₯)βˆ’β„Žπ’―(β„Ž)π‘£β‰€β„Žπ’―(β„Ž)ℬ(𝑑,π‘₯)βˆ’β„¬(𝑑,π‘₯)+β„Žπ’―(β„Ž)π‘£β‰€β„Žπ’―(β„Ž)ℬ(𝑑,π‘₯)βˆ’β„¬(𝑑,π‘₯)+β„Žπœ–.(2.4) such that 𝐢0. By the invariance properties of (𝒯(𝑑))𝑑β‰₯0,, we have Ξ©. Consequently, (𝑑𝑛,π‘₯𝑛)∈Ω,𝑑𝑛↑𝑑 By using the continuity of 𝐽-semigroup π‘₯𝑛→π‘₯ 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)𝑋 is closed from the left, that is, if π‘›β†’βˆž, in (𝑑,π‘₯)∈Ω, and Ξ©(𝑑) in 𝒯(𝑠) as 𝑑,𝑠β‰₯0 then for0x000a0all(𝑑,π‘₯)∈Ω,liminfβ„Žβ†“0(1/β„Ž)𝑑(π‘₯+β„Žβ„¬(𝑑,π‘₯),Ξ©(𝑑+β„Ž))=0;(ii)ℬ is Ξ©-invariant, for all π‘™β„¬βˆˆβ„+;(iii)(ℬ(𝑑,β‹…)βˆ’π‘™β„¬πΌ);(iv)Ξ©(𝑑) is continuous on π‘‘βˆˆπ½ and there exists Ξ© such that the operator 𝐽×𝑋 is dissipative on π‘‘βˆˆπ½,Ξ©(𝑑)β‰ βˆ…, for all (𝜏,π‘₯𝜏)∈Ω,. If 𝐽 is a connected subset of 𝐻=𝐿2[0,1]βŠ•πΏ2[0,1] such that for all π‘₯1,π‘₯2,𝑦1,𝑦2=π‘₯1,𝑦1𝐿2+π‘₯2,𝑦2𝐿2(3.1), then, for each β€–β€–ξ‚€π‘₯1,π‘₯2‖‖‖‖π‘₯=(1β€–β€–2𝐿2+β€–β€–π‘₯2β€–β€–2𝐿2)1/2(3.2) (2.1) has a unique mild solution on (π‘₯1,π‘₯2).

3. Abstract Semigroup Formulation

Throughout the sequel, we assume (𝑦1,𝑦2), the Hilbert space with the usual inner product𝐻 and the induced norm 𝐻 for all π‘₯=(π‘₯1,π‘₯2)∈𝐻,𝑦=(𝑦1,𝑦2)∈𝐻 and π‘₯≀𝑦iff0x000a0π‘₯1(𝑧)≀𝑦1(𝑧),π‘₯2(𝑧)≀𝑦2(𝑧)fora.e.π‘§βˆˆ[0,1].(3.3) in π‘₯,π‘¦βˆˆπ».

Clearly, the Hilbert space [π‘₯,𝑦]=π‘€βˆˆπ»βˆΆπ‘₯1≀𝑀1≀𝑦1,π‘₯2≀𝑀2≀𝑦2=π‘₯1,𝑦1×π‘₯2,𝑦2ξ‚„(3.4) is a real Banach lattice, where for all given π‘₯,𝑦 Recall that for every pair [π‘₯,𝑦], the setπ‘₯≀𝑦 is called the order interval between 𝒯 and 𝐻. Clearly, 0≀𝒯π‘₯ is nonempty if 0≀π‘₯. (for more details, see, e.g., [25]). A bounded linear operator (𝒯(𝑑))𝑑β‰₯0 on 𝐻 is said to be positive if 𝐢0-semigroup for all 𝐻 Similarly, a family of bounded linear operators 𝒯(𝑑) of 𝐢0-semigroup is said to be a positive 𝐻 on 𝒯(𝑑) if 𝑑β‰₯0 is a 𝐢in(𝑑) on 𝐿in(𝑑) and π’ž1([0,∞[) is a positive operator for all πœ‰π‘§=𝑙,π‘₯1=πΆβˆ’πΆin,π‘₯2=πΏβˆ’πΏin,π‘₯01=𝐢0βˆ’πΆin,π‘₯02=𝐿0βˆ’πΏin.(3.5).

In the following, we will assume that π‘§βˆˆ]0,1[ and 𝑑>0 are positive πœ•π‘₯1πœ•π‘‘=βˆ’π‘£πœ•π‘₯1πœ•π‘§+𝑑1πœ•2π‘₯1πœ•π‘§2βˆ’π‘˜1ξ‚€π‘₯1+𝐢in(𝑑)π‘šξ‚€π‘₯2+𝐿in(𝑑)π‘›βˆ’Μ‡πΆin(𝑑),(3.6)πœ•π‘₯2πœ•π‘‘=βˆ’π‘£πœ•π‘₯2πœ•π‘§+𝑑2πœ•2π‘₯2πœ•π‘§2βˆ’π‘˜2ξ‚€π‘₯1+𝐢in(𝑑)π‘šξ‚€π‘₯2+𝐿in(𝑑)π‘›βˆ’Μ‡πΏin(𝑑),(3.7)-functions. Let us consider the following state transformation: π‘‘π‘–πœ•π‘₯π‘–πœ•π‘§(0,𝑑)βˆ’π‘£π‘₯𝑖(0,𝑑)=0=π‘‘π‘–πœ•π‘₯𝑖π‘₯πœ•π‘§(1,𝑑)βˆ€π‘‘>0𝑖=1;2,(3.8)𝑖(𝑧,0)=π‘₯0𝑖(𝑧)forπ‘§βˆˆ]0,1[,𝑖=1;2,(3.9) Then, we obtain the new equivalent system for all 𝑑1=𝐷1𝑙2,𝑑2=𝐷2𝑙2𝜈,𝑣=𝑙.(3.10) and ,Μ‡π‘₯(𝑑)=𝐴π‘₯(𝑑)+𝐡𝑑,π‘₯(𝑑)π‘₯(0)=π‘₯0∈Ω(0),(3.11): Ξ©(𝑑)Ξ© with π‘‘βˆˆβ„+ξ‚€ξ‚€π‘₯Ξ©={𝑑,1,π‘₯2ξ‚ξ‚π‘‡βˆˆβ„+Γ—π»βˆΆβˆ’πΆin(𝑑)≀π‘₯1(𝑧)β‰€πΆβˆ’πΆin(𝑑),βˆ’πΏin(𝑑)≀π‘₯2(𝑧)β‰€πΏβˆ’πΏin(𝑑)a.e.0x000a0π‘§βˆˆ[0,1]}.(3.12) where 𝐴

This PDEs describing the reactor dynamics may be formally written in the abstract form as ξ‚€π‘₯𝐷(𝐴)={π‘₯=1,π‘₯2ξ‚π‘‡βˆˆπ»βˆΆπ‘₯,𝑑π‘₯π‘‘π‘‘π‘§βˆˆπ»are0x000a0absolutely0x000a0continuous,2π‘₯𝑑𝑧2∈𝐻,𝑑𝑖𝑑π‘₯𝑖𝑑𝑧(0)βˆ’πœπ‘₯𝑖(0)=0=𝑑𝑖𝑑π‘₯π‘–βŽ›βŽœβŽœβŽœβŽπ‘‘π‘‘π‘§(1);𝑖=1;2},(3.13)𝐴π‘₯=1𝑑2π‘₯1𝑑𝑧2βˆ’πœπ‘‘π‘₯10𝑑𝑧0𝑑2𝑑2π‘₯2𝑑𝑧2βˆ’πœπ‘‘π‘₯2⎞⎟⎟⎟⎠=𝐴𝑑𝑧1π‘₯100𝐴2π‘₯2ξƒͺ.(3.14) where 𝐡 denote the section of Ξ© at 𝐡(𝑑,π‘₯)=βˆ’π‘˜1ξ‚€π‘₯1+𝐢in(𝑑)π•€π‘šξ‚€π‘₯2+𝐿in(𝑑)π•€π‘›βˆ’Μ‡πΆin(𝑑)𝕀,βˆ’π‘˜2ξ‚€π‘₯1+𝐢in(𝑑)π•€π‘šξ‚€π‘₯2+𝐿in(𝑑)π•€π‘›βˆ’Μ‡πΏin(𝑑)𝕀𝑇.(3.15), which is given in view of (1.7) by 𝐴

The linear operator 𝐻 is defined by 𝑇𝑇(𝑑)=1(𝑑)00𝑇2ξƒͺ(𝑑),(3.16)𝑇1(𝑑) The nonlinear operator 𝑇2(𝑑) is defined on 𝐢0-semigroups by 𝐴1 It is shown in [7] that the linear operator 𝐴2 given by (3.14) is the infinitesimal generator of contraction semigroup on (𝑑,π‘₯)∈Ω,limβ„Žβ†“01β„Žπ‘‘ξ‚€ξ‚π‘₯+β„Žπ΅(𝑑,π‘₯);Ξ©(𝑑+β„Ž)=0.(4.1) where (𝑑,π‘₯)∈Ω. and Ξ©(𝑑) are the Ξ©(𝑑)=Ξ©1(𝑑)Γ—Ξ©2(𝑑) generated, respectively, by Ξ©1(𝑑)=βˆ’πΆinξ‚€(𝑑)𝕀,πΆβˆ’πΆin𝕀,Ξ©(𝑑)2(𝑑)=βˆ’πΏinξ‚€(𝑑)𝕀,πΏβˆ’πΏin𝕀.(𝑑)(4.2) and 𝑋1(𝑑)=π‘₯1+𝐢in(𝑑)𝕀,𝑋2(𝑑)=π‘₯2+𝐿in(𝑑)𝕀,(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)–(3.9) In order to be able to apply Corollary 2.3, we need the following lemmas.

Lemma 4.1. For each π‘₯∈Ω(𝑑),𝑋𝑋(𝑑)=1(𝑑),𝑋2(𝑑)π‘‡βˆˆ[0,𝐢𝕀]Γ—[0,𝐿𝕀].(4.4)

Proof. Let β„Ž0>0 Observe that β„Ž0π‘˜1πΆπ‘šβˆ’1𝐿𝑛≀1. is given by β„Žβˆˆ(0,β„Ž0),, where 𝑋1ξ‚€(𝑑)π•€βˆ’β„Žπ‘˜1𝑋1π‘šβˆ’1(𝑑)𝑋𝑛2(𝑑)∈[0,𝐢𝕀].(4.5) Denote 𝑓1𝑑,𝑋(𝑑)=𝑋1ξ‚€(𝑑)π•€βˆ’β„Žπ‘˜1𝑋1π‘šβˆ’1(𝑑)𝑋𝑛2(𝑑)βˆ’πΆin(𝑑+β„Ž)π•€βˆˆΞ©1(𝑑+β„Ž).(4.6) we have, for 𝐢in𝑑π‘₯1+β„Žπ΅1(𝑑,π‘₯),Ξ©1(𝑑+β„Ž)≀𝑑(𝑋1(𝑑)βˆ’β„Žπ‘˜1π‘‹π‘š1(𝑑)𝑋𝑛2(𝑑)βˆ’πΆin(𝑑+β„Ž)𝕀,Ξ©1𝑓(𝑑+β„Ž))+β„Žπœ–(β„Ž)≀𝑑1𝑑,𝑋(𝑑),Ξ©1(𝑑+β„Ž)+β„Žπœ–(β„Ž)=β„Žπœ–(β„Ž),(4.7) Let πœ–(β„Ž)β†’0 be sufficiently small such that β„Žβ†’0.
Let, now, limβ„Žβ†“01β„Žπ‘‘ξ‚€π‘₯1+β„Žπ΅1(𝑑,π‘₯);Ξ©1(𝑑+β„Ž)=0.(4.8) then 𝐿in, Hence limβ„Žβ†“01β„Žπ‘‘ξ‚€π‘₯2+β„Žπ΅2(𝑑,π‘₯);Ξ©2(𝑑+β„Ž)=0.(4.9) By using the regularity of the inlet function 𝑑π‘₯π‘₯+𝐡(𝑑,π‘₯),Ξ©(𝑑+β„Ž)≀𝑑1+𝐡1(𝑑,π‘₯),Ξ©1π‘₯(𝑑+β„Ž)+𝑑2+𝐡2(𝑑,π‘₯),Ξ©2(𝑑+β„Ž),(4.10), we get π‘™π΅βˆˆβ„+ where (𝐡(𝑑,β‹…)βˆ’π‘™π΅πΌ) as Ξ©(𝑑) Whence 𝑑β‰₯0 By similar considerations as above, taking into account the regularity of the function 𝑑β‰₯0 we also get π‘₯,𝑦 Observe, now, that Ξ©(𝑑). 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 𝑔𝑖(𝑑,π‘₯)=βˆ’π‘˜π‘–ξ‚€π‘₯1+𝐢in(𝑑)π•€π‘šξ‚€π‘₯2+𝐿in(𝑑)𝕀𝑛for𝑖=1,2,(4.11) such that the operator 𝑋1(𝑑)=π‘₯1+𝐢in(𝑑)𝕀;𝑋2(𝑑)=π‘₯2+𝐿in(𝑑)𝕀;π‘Œ1(𝑑)=𝑦1+𝐢in(𝑑)𝕀,π‘Œ2(𝑑)=𝑦2+𝐿in(𝑑)𝕀.(4.12) is dissipative on π‘₯,π‘¦βˆˆΞ©(𝑑), for each (𝑋𝑖(𝑑),π‘Œπ‘–(𝑑))π‘‡βˆˆ[0,𝐢𝕀]Γ—[0,𝐿𝕀].

Proof. Let 𝑖=1,2. and let 𝑖=1,2 be in ‖‖𝑔𝑖(𝑑,π‘₯)βˆ’π‘”π‘–β€–β€–(𝑑,𝑦)𝐿2β‰€π‘˜π‘–ξ‚€πΆ2π‘šβ€–β€–π‘‹π‘›2(𝑑)βˆ’π‘Œπ‘›2β€–β€–(𝑑)2𝐿2+𝐿2π‘›β€–β€–π‘‹π‘š1(𝑑)βˆ’π‘Œπ‘š1β€–β€–(𝑑)2𝐿21/2β‰€π‘˜π‘–ξ‚€π‘›2𝐢2π‘šπΏ2π‘›βˆ’2β€–β€–π‘₯2βˆ’π‘¦2β€–β€–2𝐿2+π‘š2𝐿2𝑛𝐢2π‘šβˆ’2β€–β€–π‘₯1βˆ’π‘¦1β€–β€–2𝐿21/2β‰€π‘˜π‘–πΆπ‘šβˆ’1πΏπ‘›βˆ’1𝑛max𝐢;π‘šπΏξ‚β€–π‘₯βˆ’π‘¦β€–.(4.13) Denote ‖‖‖‖‖‖𝑔𝐡(𝑑,π‘₯)βˆ’π΅(𝑑,𝑦)=(1(𝑑,π‘₯)βˆ’π‘”1β€–β€–(𝑑,𝑦)2𝐿2+‖‖𝑔2(𝑑,π‘₯)βˆ’π‘”2β€–β€–(𝑑,𝑦)2𝐿2)1/2ξ‚€π‘˜β‰€max1,π‘˜2ξ‚πΆπ‘šβˆ’1πΏπ‘›βˆ’1𝑛max𝐢;π‘šπΏξ‚β€–π‘₯βˆ’π‘¦β€–.(4.14) and let also 𝐡(𝑑,β‹…) Observe that, for each 𝑙𝐡Ω(𝑑) for π‘™π΅ξ‚€π‘˜=max1,π‘˜2ξ‚πΆπ‘šβˆ’1πΏπ‘›βˆ’1𝑛max𝐢;π‘šπΏξ‚.(4.15) Hence, by applying the mean value theorem, for Ξ©(𝑑)is𝒯(𝑠)-invariantβˆ€π‘‘,𝑠β‰₯0.(4.16), we get𝑑,𝑠β‰₯0 Finally, (π‘₯,𝑦)π‘‡βˆˆΞ©(𝑑) Consequently, ξ‚€βˆ’πΆin(𝑑)𝕀,βˆ’πΏin(𝑑)𝕀𝑇≀(π‘₯,𝑦)π‘‡β‰€ξ‚€ξ‚€πΆβˆ’πΆin(𝑑)𝕀,πΏβˆ’πΏin𝕀(𝑑)𝑇.(4.17) is an (𝑇(𝑑))𝑑β‰₯0-dissipative operator on ξ‚€βˆ’πΆin(𝑑)𝑇1(𝑠)𝕀,βˆ’πΏin(𝑑)𝑇2(𝑠)𝕀𝑇≀𝑇(𝑠)(π‘₯,𝑦)π‘‡β‰€ξ‚€ξ‚€πΆβˆ’πΆin𝑇(𝑑)1ξ‚€(𝑠)𝕀,πΆβˆ’πΆin𝑇(𝑑)2(𝑠)𝕀𝑇.(4.18) [14, page 245], where 𝑇𝑖(𝑑)𝕀≀𝕀for𝑖=1;2
Finally, we state the invariance properties of the state trajectories of the model given by (3.6)–(3.9).

Proposition 4.3. One has that 𝐢β‰₯𝐢in

Proof. Let 𝐿β‰₯𝐿in and Ξ©(𝑑). We have 𝑑β‰₯0. Hence, by using the positivity of (𝑇1(𝑠)π‘₯,𝑇2(𝑠)𝑦)π‘‡βˆˆΞ©(𝑑) [26], we have 𝑑,𝑠β‰₯0. Since, 𝐢in(𝑑) (see [26]) and by using the inequalities (1.8) (i.e., 𝐿in(𝑑) and 𝐢1([0,+∞[)), the invariance of π‘₯0∈Ω(0), holds for all 𝐡 Thus, Ξ©, for all Ξ©

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

Theorem 4.4. Let Ξ© and 𝑑𝑛↗𝑑 be positive π‘₯π‘›βˆˆΞ©(𝑑𝑛)-functions. Then, for every π‘₯𝑛→π‘₯∈𝐻, the problem (3.6)–(3.9) has a unique global mild solution.

Proof. Since (π‘₯𝑛) is continuous function in (π‘₯𝑛) by Corollary 2.3, it is sufficient to prove the condition (i) in Corollary 2.3 and to check that the subset π‘₯𝑛(𝑧)β†’π‘₯(𝑧) is connected
(a)Let us first show that [0,1] is closed from the left.Let 𝐢in and 𝐿in,thatπ‘₯(𝑧)∈[βˆ’πΆin(𝑑),πΆβˆ’πΆin(𝑑)]Γ—[βˆ’πΏin(𝑑),πΏβˆ’πΏin(𝑑)] with [0,1] then there exists a subsequence of π‘₯∈Ω(𝑑) which is also denoted by 𝑑β‰₯0 such that Ξ©, that is, on [0,+∞[×𝐻 which implies, by continuity of 𝐾=[0,𝐢𝕀]Γ—[0,𝐿𝕀] and 𝐺∢[0,+∞[×𝐾→Ω, that is, on (𝑑,π‘₯)∈[0,+∞[×𝐾,, hence 𝐺(𝑑,π‘₯)=(𝑑,π‘₯1βˆ’πΆin(𝑑)𝕀,π‘₯2βˆ’πΏin(𝑑)𝕀)𝑇. for each 𝐢in.(b)Let us, now, check that 𝐿in is connected in [0,+∞[,:Let 𝐺 and define [0,+1[×𝐾 such that for all 𝐺[0,𝐢𝕀]Γ—[0,𝐿𝕀] Since 𝐻 and Ξ©=𝐺([0,+∞[×𝐾) are continuous functions in [0,+∞[×𝐻. it follows that 𝐢in in 𝐿in is also a continuous function. Observe that 𝑑, is surjective; since Ξ©(𝑑) is connected in 𝑑, we get that 𝐢in is also connected in 𝐿in
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)–(3.9).

5. Equilibrium Profiles

In the steady-state solution analysis, the inlet functions 𝐢in and 𝐿in are independent of time βˆ’π‘£π‘‘π‘₯1𝑑𝑧=𝑑1𝑑2π‘₯1𝑑𝑧2βˆ’π‘˜1ξ‚€π‘₯1+𝐢inξ‚π‘šξ‚€π‘₯2+𝐿in𝑛=0,(5.1)βˆ’π‘£π‘‘π‘₯2𝑑𝑧=𝑑2𝑑2π‘₯2𝑑𝑧2βˆ’π‘˜2ξ‚€π‘₯1+𝐢inξ‚π‘šξ‚€π‘₯2+𝐿in𝑛=0,(5.2) which implies that the domain 𝑑𝑖𝑑π‘₯𝑖𝑑𝑧(0)βˆ’π‘£π‘₯𝑖(0)=0=𝑑𝑖𝑑π‘₯𝑖π‘₯𝑑𝑧(1),𝑖=1;2,(5.3)Ξ©(𝑑)=Ξ”={1,π‘₯2ξ‚π‘‡βˆˆπ»βˆΆβˆ’πΆin≀π‘₯1(𝑧)β‰€πΆβˆ’πΆin,βˆ’πΏin≀π‘₯2(𝑧)β‰€πΏβˆ’πΏinforalmostallπ‘§βˆˆ[0,1]}.(5.4) is also independent of 𝐢in. If we denote by 𝐿in and Ξ” the values of 𝑑1=𝑑2=𝑑. and 𝑑1=𝑑2=𝑑 which correspond to the steady-state solutions, the corresponding steady-state system to the models (3.6)–(3.9) is given by the following equations: π’œ=𝑑(𝑑2/𝑑𝑧2)βˆ’π‘£(𝑑/𝑑𝑧)=𝐴𝑖𝐷(π’œ)=𝐷(𝐴𝑖) with 𝑖=1;2𝑏(β‹…) The following existence result can be proven as in the case where [0,1] and π‘’βˆˆπΏ2([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)–(3.9) has at least one equilibrium profile in π’œπ‘’=𝑏𝑒in]0,1[,0x000a0π‘’βˆˆπ·(π’œ),(5.5).

The sequel of this paper will deal with the uniqueness analysis of steady states in the important case where 𝑒=0

First, since [0,1], we denote 𝑒 with βŸ¨π’œπ‘’,π‘’βŸ©πΏ2=βŸ¨π‘π‘’,π‘’βŸ©πΏ2.(5.6) for βŸ¨π’œπ‘’,π‘’βŸ©πΏ2=ξ€œ10𝑑𝑑2𝑒𝑑𝑧2(𝑧)βˆ’π‘£π‘‘π‘’ξ‚„ξ€œπ‘‘π‘§(𝑧)𝑒(𝑧)𝑑𝑧,=βˆ’10𝑑𝑑𝑒𝑑𝑧(𝑧)2𝑑𝑧+𝑑𝑑𝑒𝑑𝑧(1)𝑒(1)βˆ’π‘‘π‘’ξ‚„βˆ’1𝑑𝑧(0)𝑒(0)2𝑣𝑒2(1)βˆ’π‘’2ξ‚„,β€–β€–β€–(0)=βˆ’π‘‘π‘‘π‘’β€–β€–β€–π‘‘π‘§2𝐿2βˆ’12𝑣𝑒21(1)βˆ’2𝑣𝑒2(0),(5.7)≀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 𝑏(𝑧) be a bounded nonnegative function defined in [0,1],. If βŸ¨π‘π‘’,π‘’βŸ©πΏ2=∫10𝑏(𝑧)𝑒2(𝑧)𝑑𝑧=0.(5.9) satisfies the equations βŸ¨π’œπ‘’,π‘’βŸ©πΏ2β€–β€–β€–=0=𝑑𝑑𝑒‖‖‖𝑑𝑧2𝐿2+12𝑣𝑒21(1)+2𝑣𝑒2(0).(5.10) then 𝑑𝑒𝑑𝑧(𝑧)=0a.e.π‘§βˆˆ[0,1],𝑒(0)=0=𝑒(1).(5.11) in 𝐷(π’œ)βŠ‚π’ž([0,1])..

Proof. Let 𝑒=0 be the solution of problem (5.5), then π‘’βˆˆπ·(π’œ). We have, 𝑑1=𝑑2=𝑑,Ξ”. Since π‘₯=(π‘₯1,π‘₯2)𝑇and𝑦=(𝑦1,𝑦2)𝑇 is nonnegative function in [0,1]. then by (5.8) and taking into account (5.6) π‘₯=𝑦. Which implies, in view of (5.6)-(5.7), that 𝑔π‘₯1,π‘₯2π‘₯=βˆ’1+𝐢inπ•€ξ‚π‘šξ‚€π‘₯2+𝐿in𝕀𝑛,𝑀1=𝑦1βˆ’π‘₯1∈𝐷(π’œ),𝑀2=π‘₯2βˆ’π‘¦2∈𝐷(π’œ).(5.12) Then, we get βˆ’π’œπ‘€1=π‘˜1𝑔𝑦1,𝑦2π‘₯βˆ’π‘”1,π‘₯2=π‘˜1𝑦1+𝐢inξ‚π‘šπ‘₯2+𝐿inξ‚π‘›βˆ’ξ‚€π‘¦2+𝐿in𝑛+π‘˜1ξ‚€π‘₯2+𝐿in𝑛π‘₯1+𝐢inξ‚π‘šβˆ’ξ‚€π‘¦1+𝐢inξ‚π‘šξ‚„(5.13) Clearly, by using the Sobolev imbedding theorem, βˆ’π’œπ‘€1=π‘˜1𝑛𝑦1+𝐢inξ‚π‘šπœ‰2π‘›βˆ’1𝑀2βˆ’π‘šπ‘˜1ξ‚€π‘₯2+𝐿inξ‚π‘›πœ‰1π‘šβˆ’1𝑀1,(5.14) Therefore, (πœ‰1,πœ‰2) since (0,0)

Theorem 5.3. For (𝐢,𝐿). the steady-state problem given by (5.1)–(5.3) has a unique solution in βˆ’π’œπ‘€2=βˆ’π‘˜2𝑔𝑦1,𝑦2π‘₯βˆ’π‘”1,π‘₯2=βˆ’π‘˜2𝑛𝑦1+𝐢inξ‚π‘šπœ‰2π‘›βˆ’1𝑀2+π‘šπ‘˜2ξ‚€π‘₯2+𝐿inξ‚π‘›πœ‰1π‘šβˆ’1𝑀1,(5.15)

Proof. Let πœ‰1 be solutions to (5.1)–(5.3) on πœ‰2. To obtain the desired result, we will be showing that βˆ’π’œπ‘€1=βˆ’π‘Ž1𝑀1+𝑏1𝑀2,(5.16)βˆ’π’œπ‘€2=π‘Ž2𝑀1βˆ’π‘2𝑀2,(5.17) Let 𝑖=1;2, Then π‘Žπ‘–(𝑧)=π‘šπ‘˜π‘–ξ‚€π‘₯2(𝑧)+𝐿inξ‚π‘›πœ‰1π‘šβˆ’1𝑏(𝑧),𝑖(𝑧)=π‘›π‘˜π‘–ξ‚€π‘¦1(𝑧)+𝐢inξ‚π‘šπœ‰2π‘›βˆ’1(𝑧).(5.18) Hence, by applying the mean value theorem, we get
where π‘˜1, are some intermediate values between π’œπ‘€=0,π‘€βˆˆπ·(π’œ),(5.19) and 𝑀=π‘˜2𝑀1+π‘˜1𝑀2.
By similar considerations as above, we also get 𝑀=0 for the same [0,1]. and βˆ’π’œπ‘€2=π‘Ž2𝑀1βˆ’π‘2𝑀2(5.20)
Now, we have the following system: 𝑀1=βˆ’π‘˜2βˆ’1π‘˜1𝑀2(5.21)π’œπ‘€2=𝑐𝑀2,(5.22) where, for 𝑐(𝑧)=π‘Ž1(𝑧)+𝑏2(𝑧).𝑖=1;2
Multiplying (5.16) by 0β‰€π‘Žπ‘–(𝑧)β‰€π‘šπ‘˜π‘–πΏπ‘›πΆπ‘šβˆ’1,0≀𝑏𝑖(𝑧)β‰€π‘›π‘˜π‘–πΆπ‘šπΏπ‘›βˆ’1.(5.23) and (5.17) by πœ†=max(π‘šπΏ,𝑛𝐢)max(π‘˜1,π‘˜2)πΆπ‘šβˆ’1πΏπ‘›βˆ’1 we get by addition of both equations that 0≀𝑐(𝑧)≀2πœ†. where 𝑀2=0. By Lemma 5.2, this system has a unique solution 𝑀1=0, in π‘₯=𝑦. Now, let π‘₯(𝑑)∈Ω(𝑑). and substituting the expression Ξ©(𝑑) yields 𝐿𝑝 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)–(1.6). This question is under investigation.

Acknowledgments

This paper presents research results of the Moroccan β€œProgramme ThΓ©matique d'Appui Γ  la Recherche Scientifique” PROTARS III, initiated by the Moroccan β€œCentre National de la Recherche Scientifique et Technique” (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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