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

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 𝐶₀-semigroup.

1. Introduction

Several chemical and biochemical processes are typically described by nonlinear coupled partial differential equations “PDE” 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 𝑡. The considered class of models correspond to the following chemical reaction: 𝑛𝐴+𝑚𝐵→𝑃,(1.1) whose kinetic is given by 𝑟=(−𝑘₁𝐶^𝑚𝐿^𝑛,−𝑘₂𝐶^𝑚𝐿^𝑛)^𝑇, where 𝐶 and 𝐿 are the concentrations of the reactants 𝐴 and 𝐵, respectively, 𝑘₁ and 𝑘₂ are the kinetic constants and 𝑚,𝑛 are the order of the reaction to 𝐴 and 𝐵, respectively. More precisely, we study the global existence and uniqueness of the trajectories of the models which describe the evolution of two reactant concentrations 𝐶 and 𝐿: 𝜕𝐶𝜕𝑡=−𝜈𝜕𝜕𝐶𝜉+𝐷¹𝜕²𝐶𝜕𝜉²𝜕𝐿−𝑘₁𝐶^𝑚𝐿^𝑛,(1.2)𝜕𝑡=−𝜈𝜕𝜕𝐿𝜉+𝐷²𝜕²𝐿𝜕𝜉²−𝑘₂𝐶^𝑚𝐿^𝑛,(1.3)for0x000a0𝜉∈]0,𝑙[and𝑡>0𝐷₁𝜕𝐶𝜕𝜉(0,𝑡)−𝜈𝐶(0,𝑡)+𝜈𝐶ⁱⁿ(𝑡)=0=𝐷₁𝜕𝐶𝐷𝜕𝜉(𝑙)∀𝑡>0(1.4)₂𝜕𝐿𝜕𝜉(0,𝑡)−𝜈𝐿(0,𝑡)+𝜈𝐿ⁱⁿ(𝑡)=0=𝐷₂𝜕𝐿𝐶(𝜉,0)=𝐶₀𝜕𝜉(𝑙,𝑡)∀𝑡>0,(1.5)(𝜉),𝐿(𝜉,0)=𝐿₀(𝜉)for𝜉∈]0,𝑙[.(1.6), with the following boundary and initial conditions: 𝐷₁,𝐷₂𝜈𝑡,𝜉

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

In the above equations, 𝑙 are the dispersion coefficients, 𝑚 is the superficial fluid velocity, 𝑛 denote the time and space independent variables, respectively, 𝐶_in is the length of the reactor, 𝐿_in and 𝜈=0 are two positive integers, 𝑛=𝑚=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, 𝑚=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 𝑚=1 bounds of positive solutions, when 𝑛>1 and 𝑚=𝑛=1,2,3. This latter result has been generalized by [10] for the case (𝐶) and (𝐿).
In practice, the special cases 𝑛=𝑚 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 𝑚=𝑛=1 and 𝐷₂=4𝐷₁. The reader can find another model with 𝐷₂=16𝐷₁ in [11], where the numerical analysis has been done for 0≤𝜉≤𝑙 and 𝑡≥0, 0𝐶≤𝐶≤𝐶,0≤𝐿≤𝐿,(1.7)_in(𝑡)≤𝐶,𝐿_in(𝑡)≤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 𝐶 and for all 𝐿: 𝑋‖⋅‖,𝐽=[𝑎,𝑏[(𝑎<𝑏≤+∞), where {𝒯(𝑡);𝑡≥0} and 𝐶₀ 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 𝒜 and let ℬ be a linear contraction Ω-semigroup on 𝑋 generated by Ω. Let 𝐽×𝑋 be a nonlinear continuous operator form 𝐼 into 𝕀, where 𝑋 is a subset of 𝑥̇(𝑡)=𝒜𝑥(𝑥𝑡()𝜏+)=ℬ𝑥(𝑡_𝜏,𝑥(𝑡)),𝜏<𝑡<𝑏,∈Ω(𝜏),(2.1). Ω(𝜏) and Ω 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: Ω(𝑡)={𝑥∈𝑋;(𝑡,𝑥)∈Ω}. where Ω(𝑡)≠∅ denote the section of 𝑡∈𝐽 at 𝑑(𝑥;𝒟)=inf{‖𝑥−𝑦‖,𝑦∈𝒟},, given by 𝑥∈𝑋 Assume that 𝒟 for all 𝑋. Moreover, recall that Ω for Ω=𝐽×𝒟 and Ω is a subset of 𝑥(𝑡)∈Ω(𝑡).

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, Ω [14, 22]. More generally, the cylindrical case of ℬ(𝑡,⋅) will not be convenient for the study of evolution system satisfying time-dependent constraints, that is, 𝑙_ℬ on Ω (see our problem in Section 3). A noncylindrical (𝑡_𝑛,𝑥_𝑛)∈Ω,𝑡_𝑛↑𝑡 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 𝑥_𝑛→𝑥-dissipative [16].

Theorem 2.1 (see [16]). Suppose that the following conditions are fulfilled: (i)𝑋 is closed from the left, that is, if 𝑛→∞ in (𝑡,𝑥)∈Ω, and for0x000a0all0x000a0(𝑡,𝑥)∈Ω,liminf_ℎ↓0(1/ℎ)𝑑(𝒯(ℎ)𝑥+ℎℬ(𝑡,𝑥),Ω(𝑡+ℎ))=0 in ℬ as Ω, then 𝑙_ℬ∈ℝ⁺;(ii)(ℬ(𝑡,⋅)−𝑙_ℬ𝐼);(iii)Ω(𝑡) is continuous on 𝑡∈𝐽 and there exists Ω such that the operator 𝐽×𝑋 is dissipative on 𝑡∈𝐽,Ω(𝑡)≠∅ for all (𝜏,𝑥_𝜏)∈Ω,. If 𝐽 is a connected subset of Ω(𝑡) such that for all 𝒯(𝑠), then, for each 𝒯(𝑠)(Ω(𝑡))⊂Ω(𝑡) (2.1) has a unique mild solution on 𝑡,𝑠≥0,.

Comment 2. It is shown in [16] that the “subtangential 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 ℬ∶Ω→𝑋 is Ω-invariant, that is, Ω(𝑡) for all 𝒯(𝑠) we have the following lemma.

Lemma 2.2. Let 𝑡,𝑠≥0 be continuous and let lim_ℎ↓0inf1ℎ𝑑𝑥+ℎℬ(𝑡,𝑥);Ω(𝑡+ℎ)=0∀(𝑡,𝑥)∈Ω(2.2) be closed from the left. If lim_ℎ↓0inf1ℎ𝑑𝒯(ℎ)𝑥+ℎℬ(𝑡,𝑥),Ω(𝑡+ℎ)=0,∀(𝑡,𝑥)∈Ω.(2.3) is (𝑡,𝑥)∈Ω,-invariant for all 𝜖>0,, then the following subtangential condition ℎ∈(0,𝜖] implies the condition 𝑦∈Ω(𝑡+ℎ)

Proof. Let ‖𝑦−𝑥−ℎℬ(𝑡,𝑥)‖≤ℎ𝜖 given 𝑢=𝑦−𝑥−ℎℬ(𝑡,𝑥) from condition (2.2) it follows, by [23, Lemma 3] (see also [24, Lemma 1]), that there is 𝑣=(1/ℎ)𝑢 and ‖𝑣‖≤𝜖 such that 𝑦=𝑥+ℎ(𝐵(𝑡,𝑥)+𝑣)∈Ω(𝑡+ℎ). Let now Ω(𝑡) and 𝒯(ℎ)𝑦∈Ω(𝑡+ℎ). We get 𝑑𝒯(ℎ)𝑥+ℎℬ(𝑡,𝑥);Ω(𝑡+ℎ)≤≤≤≤ℎℎ‖‖‖‖𝒯ℎ‖‖‖‖ℬ𝒯𝒯(ℎ((()ℎℎ𝑡𝑥,))𝑥ℬℬ+)((−ℎ𝑡𝑡,,ℬℎ𝑥𝑥𝒯))(𝑡−−,(𝑥ℎℬℬ))ℬ−((𝑡𝑡(,,𝒯𝑡𝑥𝑥,))𝑥(‖‖‖‖ℎ))++−𝑦‖‖ℎℎℎ,𝜖‖‖𝒯.𝒯(ℎ(ℎ)𝑣)𝑣‖‖‖‖,,(2.4) such that 𝐶₀. By the invariance properties of (𝒯(𝑡))_𝑡≥0,, we have Ω. Consequently, (𝑡_𝑛,𝑥_𝑛)∈Ω,𝑡_𝑛↑𝑡 By using the continuity of 𝐽-semigroup 𝑥_𝑛→𝑥 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 𝑛→∞, in (𝑡,𝑥)∈Ω, and Ω(𝑡) in 𝒯(𝑠) as 𝑡,𝑠≥0 then for0x000a0all(𝑡,𝑥)∈Ω,liminf_ℎ↓0(1/ℎ)𝑑(𝑥+ℎℬ(𝑡,𝑥),Ω(𝑡+ℎ))=0;(ii)ℬ is Ω-invariant, for all 𝑙_ℬ∈ℝ⁺;(iii)(ℬ(𝑡,⋅)−𝑙_ℬ𝐼);(iv)Ω(𝑡) is continuous on 𝑡∈𝐽 and there exists Ω such that the operator 𝐽×𝑋 is dissipative on 𝑡∈𝐽,Ω(𝑡)≠∅, for all (𝜏,𝑥_𝜏)∈Ω,. If 𝐽 is a connected subset of 𝐻=𝐿²[0,1]⊕𝐿²[0,1] such that for all 𝑥¹,𝑥₂,𝑦¹,𝑦₂=𝑥¹,𝑦₁_𝐿2+𝑥₂,𝑦₂_𝐿2(3.1), then, for each ‖‖𝑥₁,𝑥₂‖‖=(‖‖𝑥¹‖‖^2𝐿₂+‖‖𝑥₂‖‖^2𝐿₂)^1/2(3.2) (2.1) has a unique mild solution on (𝑥₁,𝑥₂).

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 𝑥=(𝑥₁,𝑥₂)∈𝐻,𝑦=(𝑦₁,𝑦₂)∈𝐻 and 𝑥≤𝑦iff0x000a0𝑥₁(𝑧)≤𝑦₁(𝑧),𝑥₂(𝑧)≤𝑦₂(𝑧)fora.e.𝑧∈[0,1].(3.3) in 𝑥,𝑦∈𝐻.

Clearly, the Hilbert space [𝑥,𝑦]=𝑤∈𝐻∶𝑥₁≤𝑤₁≤𝑦₁,𝑥₂≤𝑤₂≤𝑦₂=𝑥¹,𝑦₁×𝑥²,𝑦₂(3.4) is a real Banach lattice, where for all given 𝑥,𝑦 Recall that for every pair [𝑥,𝑦], the set𝑥≤𝑦 is called the order interval between 𝒯 and 𝐻. Clearly, 0≤𝒯𝑥 is nonempty if 0≤𝑥. (for more details, see, e.g., [25]). A bounded linear operator (𝒯(𝑡))_𝑡≥0 on 𝐻 is said to be positive if 𝐶₀-semigroup for all 𝐻 Similarly, a family of bounded linear operators 𝒯(𝑡) of 𝐶₀-semigroup is said to be a positive 𝐻 on 𝒯(𝑡) if 𝑡≥0 is a 𝐶_in(𝑡) on 𝐿_in(𝑡) and 𝒞¹([0,∞[) is a positive operator for all 𝑧=𝜉𝑙,𝑥¹=𝐶−𝐶_in,𝑥₂=𝐿−𝐿_in,𝑥₀₁=𝐶₀−𝐶_in,𝑥₀₂=𝐿₀−𝐿_in.(3.5).

In the following, we will assume that 𝑧∈]0,1[ and 𝑡>0 are positive 𝜕𝑥₁𝜕𝑡=−𝑣𝜕𝜕𝑥𝑧¹+𝑑¹𝜕²𝑥₁𝜕𝑧²𝜕𝑥₂−𝑘₁𝑥¹+𝐶_in(𝑡)^𝑚𝑥²+𝐿_in(𝑡)^𝑛−𝐶̇_in(𝑡),(3.6)𝜕𝑡=−𝑣𝜕𝜕𝑥𝑧²+𝑑²𝜕²𝑥₂𝜕𝑧²−𝑘₂𝑥¹+𝐶_in(𝑡)^𝑚𝑥²+𝐿_in(𝑡)^𝑛−𝐿̇_in(𝑡),(3.7)-functions. Let us consider the following state transformation: 𝑑_𝑖𝜕𝑥_𝑖𝜕𝑧(0,𝑡)−𝑣𝑥^𝑖(0,𝑡)=0=𝑑_𝑖𝜕𝑥_𝑖𝑥𝜕𝑧(1,𝑡)∀𝑡>0𝑖=1;2,(3.8)_𝑖(𝑧,0)=𝑥_0𝑖(𝑧)for𝑧∈]0,1[,𝑖=1;2,(3.9) Then, we obtain the new equivalent system for all 𝑑₁=𝐷¹𝑙²,𝑑₂=𝐷²𝑙²,𝑣=𝜈𝑙.(3.10) and 𝑥̇(𝑡)𝑥=(0𝐴)𝑥=(𝑡𝑥)₀+𝐵𝑡,𝑥(𝑡),∈Ω(0),(3.11): Ω(𝑡)Ω with 𝑡∈ℝ⁺Ω={−𝑡,𝐿_in𝑥₁,𝑥₂^𝑇∈ℝ⁺×𝐻∶−𝐶_in(𝑡)≤𝑥₁(𝑧)≤𝐶−𝐶_in(𝑡),(𝑡)≤𝑥₂(𝑧)≤𝐿−𝐿_in(𝑡)a.e.0x000a0𝑧∈[0,1]}.(3.12) where 𝐴

This PDEs describing the reactor dynamics may be formally written in the abstract form as 𝐷(𝐴)={𝑥=𝑥₁,𝑥₂^𝑇∈𝐻∶𝑥,𝑑𝑥𝑑𝑑𝑧∈𝐻are0x000a0absolutely0x000a0continuous,₂𝑥𝑑𝑧²∈𝐻,𝑑_𝑖𝑑𝑥_𝑖𝑑𝑧(0)−𝜐𝑥^𝑖(0)=0=𝑑_𝑖𝑑𝑥_𝑖𝐴𝑥=⎛⎜⎜⎜⎝𝑑₁𝑑𝑧(1);𝑖=1;2},(3.13)𝑑²𝑥₁𝑑𝑧²−𝜐𝑑𝑥¹0𝑑𝑧𝑑₂0𝑑²𝑥₂𝑑𝑧²−𝜐𝑑𝑥²𝑑𝑧⎞⎟⎟⎟⎠=𝐴¹0𝑥₁𝐴₂0𝑥₂.(3.14) where 𝐵 denote the section of Ω at 𝐵(𝑡,𝑥)=−−𝑘₂𝑘₁𝑥¹+𝐶_in(𝑡)𝕀^𝑚𝑥²+𝐿_in(𝑡)𝕀^𝑛−𝐶̇_in(𝑡)𝕀,𝑥¹+𝐶_in(𝑡)𝕀^𝑚𝑥²+𝐿_in(𝑡)𝕀^𝑛−𝐿̇_in(𝑡)𝕀^𝑇.(3.15), which is given in view of (1.7) by 𝐴

The linear operator 𝐻 is defined by 𝑇(𝑡)=𝑇¹0(𝑡)𝑇₂0(𝑡),(3.16)𝑇₁(𝑡) The nonlinear operator 𝑇₂(𝑡) is defined on 𝐶₀-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 (𝑡,𝑥)∈Ω,lim_ℎ↓01ℎ𝑑𝑥+ℎ𝐵(𝑡,𝑥);Ω(𝑡+ℎ)=0.(4.1) where (𝑡,𝑥)∈Ω. and Ω(𝑡) are the Ω(𝑡)=Ω₁(𝑡)×Ω₂(𝑡) generated, respectively, by ΩΩ₁(𝑡)=−𝐶_in(𝑡)𝕀,𝐶−𝐶_in(𝑡)₂𝕀,(𝑡)=−𝐿_in(𝑡)𝕀,𝐿−𝐿_in(𝑡)𝕀.(4.2) and 𝑋₁(𝑡)=𝑥₁+𝐶_in(𝑡)𝕀,𝑋₂(𝑡)=𝑥₂+𝐿_in(𝑡)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 𝑥∈Ω(𝑡),𝑋(𝑡)=𝑋₁(𝑡),𝑋₂(𝑡)^𝑇∈[0,𝐶𝕀]×[0,𝐿𝕀].(4.4)

Proof. Let ℎ₀>0 Observe that ℎ₀𝑘₁𝐶^𝑚−1𝐿^𝑛≤1. is given by ℎ∈(0,ℎ₀),, where 𝑋₁(𝑡)𝕀−ℎ𝑘₁𝑋^1𝑚−1(𝑡)𝑋^𝑛2(𝑡)∈[0,𝐶𝕀].(4.5) Denote 𝑓₁𝑡,𝑋(𝑡)=𝑋¹(𝑡)𝕀−ℎ𝑘₁𝑋^1𝑚−1(𝑡)𝑋^𝑛2(𝑡)−𝐶_in(𝑡+ℎ)𝕀∈Ω₁(𝑡+ℎ).(4.6) we have, for 𝐶_in𝑑𝑥₁+ℎ𝐵₁(𝑡,𝑥),Ω₁(𝑡+ℎ)≤≤𝑑𝑑(𝑋𝑓₁₁(𝑡)−ℎ𝑘₁𝑋^𝑚1(𝑡)𝑋^𝑛2(𝑡)−𝐶_in(𝑡+ℎ)𝕀,Ω₁(𝑡+ℎ))+ℎ𝜖(ℎ)𝑡,𝑋(𝑡),Ω¹(𝑡+ℎ)+ℎ𝜖(ℎ)=ℎ𝜖(ℎ),(4.7) Let 𝜖(ℎ)→0 be sufficiently small such that ℎ→0.
Let, now, lim_ℎ↓01ℎ𝑑𝑥¹+ℎ𝐵₁(𝑡,𝑥);Ω₁(𝑡+ℎ)=0.(4.8) then 𝐿_in, Hence lim_ℎ↓01ℎ𝑑𝑥²+ℎ𝐵₂(𝑡,𝑥);Ω₂(𝑡+ℎ)=0.(4.9) By using the regularity of the inlet function 𝑑𝑥+𝐵(𝑡,𝑥),Ω(𝑡+ℎ)≤𝑑𝑥₁+𝐵₁(𝑡,𝑥),Ω₁(𝑡+ℎ)+𝑑𝑥₂+𝐵₂(𝑡,𝑥),Ω₂(𝑡+ℎ),(4.10), we get 𝑙_𝐵∈ℝ⁺ where (𝐵(𝑡,⋅)−𝑙_𝐵𝐼) as Ω(𝑡) Whence 𝑡≥0 By similar considerations as above, taking into account the regularity of the function 𝑡≥0 we also get 𝑥,𝑦 Observe, now, that Ω(𝑡). 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 𝑔_𝑖(𝑡,𝑥)=−𝑘_𝑖𝑥¹+𝐶_in(𝑡)𝕀^𝑚𝑥²+𝐿_in(𝑡)𝕀^𝑛for𝑖=1,2,(4.11) such that the operator 𝑋₁(𝑡)=𝑥₁+𝐶_in(𝑡)𝕀;𝑋₂(𝑡)=𝑥₂+𝐿_in(𝑡)𝕀;𝑌₁(𝑡)=𝑦₁+𝐶_in(𝑡)𝕀,𝑌₂(𝑡)=𝑦₂+𝐿_in(𝑡)𝕀.(4.12) is dissipative on 𝑥,𝑦∈Ω(𝑡), for each (𝑋_𝑖(𝑡),𝑌_𝑖(𝑡))^𝑇∈[0,𝐶𝕀]×[0,𝐿𝕀].

Proof. Let 𝑖=1,2. and let 𝑖=1,2 be in ‖‖𝑔_𝑖(𝑡,𝑥)−𝑔_𝑖(𝑡,𝑦)‖‖_𝐿2≤𝑘_𝑖𝐶^2𝑚‖‖𝑋^𝑛2(𝑡)−𝑌^𝑛2(𝑡)‖‖^2𝐿₂+𝐿^2𝑛‖‖𝑋^𝑚1(𝑡)−𝑌^𝑚1(𝑡)‖‖^2𝐿₂^1/2≤𝑘_𝑖𝑛²𝐶^2𝑚𝐿^2𝑛−2‖‖𝑥₂−𝑦₂‖‖^2𝐿₂+𝑚²𝐿^2𝑛𝐶^2𝑚−2‖‖𝑥₁−𝑦₁‖‖^2𝐿₂^1/2≤𝑘_𝑖𝐶^𝑚−1𝐿^𝑛−1max𝑛𝐶;𝑚𝐿‖𝑥−𝑦‖.(4.13) Denote ‖‖𝐵(𝑡,𝑥)−𝐵(𝑡,𝑦)‖‖=(‖‖𝑔₁(𝑡,𝑥)−𝑔₁(𝑡,𝑦)‖‖^2𝐿₂+‖‖𝑔₂(𝑡,𝑥)−𝑔₂(𝑡,𝑦)‖‖^2𝐿₂)^1/2≤max𝑘₁,𝑘₂𝐶^𝑚−1𝐿^𝑛−1max𝑛𝐶;𝑚𝐿‖𝑥−𝑦‖.(4.14) and let also 𝐵(𝑡,⋅) Observe that, for each 𝑙_𝐵Ω(𝑡) for 𝑙_𝐵=max𝑘₁,𝑘₂𝐶^𝑚−1𝐿^𝑛−1max𝑛𝐶;𝑚𝐿.(4.15) Hence, by applying the mean value theorem, for Ω(𝑡)is𝒯(𝑠)-invariant∀𝑡,𝑠≥0.(4.16), we get𝑡,𝑠≥0 Finally, (𝑥,𝑦)^𝑇∈Ω(𝑡) Consequently, −𝐶ⁱⁿ(𝑡)𝕀,−𝐿_in(𝑡)𝕀^𝑇≤(𝑥,𝑦)^𝑇≤𝐶−𝐶_in(𝑡)𝕀,𝐿−𝐿_in(𝑡)𝕀^𝑇.(4.17) is an (𝑇(𝑡))_𝑡≥0-dissipative operator on −𝐶ⁱⁿ(𝑡)𝑇₁(𝑠)𝕀,−𝐿_in(𝑡)𝑇₂(𝑠)𝕀^𝑇≤𝑇(𝑠)(𝑥,𝑦)^𝑇≤𝐶−𝐶_in(𝑡)𝑇₁(𝑠)𝕀,𝐶−𝐶_in(𝑡)𝑇₂(𝑠)𝕀^𝑇.(4.18) [14, page 245], where 𝑇_𝑖(𝑡)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 𝐶≥𝐶_in

Proof. Let 𝐿≥𝐿_in and Ω(𝑡). We have 𝑡≥0. Hence, by using the positivity of (𝑇₁(𝑠)𝑥,𝑇₂(𝑠)𝑦)^𝑇∈Ω(𝑡) [26], we have 𝑡,𝑠≥0. Since, 𝐶_in(𝑡) (see [26]) and by using the inequalities (1.8) (i.e., 𝐿_in(𝑡) and 𝐶¹([0,+∞[)), the invariance of 𝑥₀∈Ω(0), holds for all 𝐵 Thus, Ω, for all Ω

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

Theorem 4.4. Let Ω and 𝑡_𝑛↗𝑡 be positive 𝑥_𝑛∈Ω(𝑡_𝑛)-functions. Then, for every 𝑥_𝑛→𝑥∈𝐻, the problem (3.6)–(3.9) has a unique global mild solution.

Proof. Since (𝑥_𝑛) is continuous function in (𝑥_𝑛) by Corollary 2.3, it is sufficient to prove the condition (i) in Corollary 2.3 and to check that the subset 𝑥_𝑛(𝑧)→𝑥(𝑧) is connected
(a)Let us first show that [0,1] is closed from the left.Let 𝐶_in and 𝐿_in,that𝑥(𝑧)∈[−𝐶_in(𝑡),𝐶−𝐶_in(𝑡)]×[−𝐿_in(𝑡),𝐿−𝐿_in(𝑡)] with [0,1] then there exists a subsequence of 𝑥∈Ω(𝑡) which is also denoted by 𝑡≥0 such that Ω, that is, on [0,+∞[×𝐻 which implies, by continuity of 𝐾=[0,𝐶𝕀]×[0,𝐿𝕀] and 𝐺∶[0,+∞[×𝐾→Ω, that is, on (𝑡,𝑥)∈[0,+∞[×𝐾,, hence 𝐺(𝑡,𝑥)=(𝑡,𝑥₁−𝐶_in(𝑡)𝕀,𝑥₂−𝐿_in(𝑡)𝕀)^𝑇. for each 𝐶_in.(b)Let us, now, check that 𝐿_in is connected in [0,+∞[,:Let 𝐺 and define [0,+1[×𝐾 such that for all 𝐺[0,𝐶𝕀]×[0,𝐿𝕀] Since 𝐻 and Ω=𝐺([0,+∞[×𝐾) are continuous functions in [0,+∞[×𝐻. it follows that 𝐶_in in 𝐿_in is also a continuous function. Observe that 𝑡, is surjective; since Ω(𝑡) is connected in 𝑡, we get that 𝐶_in is also connected in 𝐿_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)–(3.9).

5. Equilibrium Profiles

In the steady-state solution analysis, the inlet functions 𝐶_in and 𝐿_in are independent of time −𝑣𝑑𝑥¹𝑑𝑧=𝑑¹𝑑²𝑥₁𝑑𝑧²−𝑣𝑑𝑥₂−𝑘₁𝑥¹+𝐶_in^𝑚𝑥²+𝐿_in^𝑛=0,(5.1)𝑑𝑧=𝑑²𝑑²𝑥₂𝑑𝑧²−𝑘₂𝑥¹+𝐶_in^𝑚𝑥²+𝐿_in^𝑛=0,(5.2) which implies that the domain 𝑑_𝑖𝑑𝑥_𝑖𝑑𝑧(0)−𝑣𝑥^𝑖(0)=0=𝑑_𝑖𝑑𝑥_𝑖Ω(𝑡)=Δ={𝑥₁𝑑𝑧(1),𝑖=1;2,(5.3)−,𝑥₂^𝑇∈𝐻∶−𝐶_in≤𝑥₁(𝑧)≤𝐶−𝐶_in,𝐿_in≤𝑥₂(𝑧)≤𝐿−𝐿_inforalmostall𝑧∈[0,1]}.(5.4) is also independent of 𝐶_in. If we denote by 𝐿_in and Δ 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: 𝒜=𝑑(𝑑²/𝑑𝑧²)−𝑣(𝑑/𝑑𝑧)=𝐴_𝑖𝐷(𝒜)=𝐷(𝐴_𝑖) with 𝑖=1;2𝑏(⋅) The following existence result can be proven as in the case where [0,1] and 𝑢∈𝐿²([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 𝒜𝑢=𝑏𝑢in]0,1[,0x000a0𝑢∈𝐷(𝒜),(5.5).

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

First, since [0,1], we denote 𝑢 with ⟨𝒜𝑢,𝑢⟩_𝐿2=⟨𝑏𝑢,𝑢⟩_𝐿2.(5.6) for ⟨𝒜𝑢,𝑢⟩_𝐿2=¹⁰𝑑𝑑²𝑢𝑑𝑧²(𝑧)−𝑣𝑑𝑢=−₁₀𝑑𝑧(𝑧)𝑢(𝑧)𝑑𝑧,𝑑𝑑𝑢𝑑𝑧(𝑧)²𝑑𝑧+𝑑𝑑𝑢𝑑𝑧(1)𝑢(1)−𝑑𝑑𝑧𝑢(0)𝑢(0)−21𝑣𝑢²=−𝑑‖‖‖𝑑𝑢(1)−𝑢²(0),𝑑𝑧‖‖‖^2𝐿2−12𝑣𝑢²(1)−12𝑣𝑢²≤0.(0),((55..78)).

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 ⟨𝑏𝑢,𝑢⟩_𝐿2=∫¹⁰𝑏(𝑧)𝑢²(𝑧)𝑑𝑧=0.(5.9) satisfies the equations ⟨𝒜𝑢,𝑢⟩_𝐿2=0=𝑑‖‖‖𝑑𝑢𝑑𝑧‖‖‖^2𝐿2+12𝑣𝑢²(1)+12𝑣𝑢²(0).(5.10) then 𝑑𝑢𝑑𝑧(𝑧)𝑢(=0)0=0a.=e.𝑢𝑧(1∈).[0,1],(5.11) in 𝐷(𝒜)⊂𝒞([0,1])..

Proof. Let 𝑢=0 be the solution of problem (5.5), then 𝑢∈𝐷(𝒜). We have, 𝑑₁=𝑑₂=𝑑,Δ. Since 𝑥=(𝑥₁,𝑥₂)^𝑇and𝑦=(𝑦₁,𝑦₂)^𝑇 is nonnegative function in [0,1]. then by (5.8) and taking into account (5.6) 𝑥=𝑦. Which implies, in view of (5.6)-(5.7), that 𝑤₁𝑔𝑥₁,𝑥₂=−𝑥¹+𝐶_in𝕀^𝑚𝑥²+𝐿_in𝕀^𝑛,=𝑦₁−𝑥₁∈𝐷(𝒜),𝑤₂=𝑥₂−𝑦₂∈𝐷(𝒜).(5.12) Then, we get −𝒜𝑤₁==𝑘𝑘¹₁𝑔𝑦¹,𝑦₂−𝑔𝑥¹,𝑥₂+𝑘₁𝑦¹+𝐶_in^𝑚𝑥²+𝐿_in^𝑛−𝑦₂+𝐿_in^𝑛𝑥²+𝐿_in^𝑛𝑥¹+𝐶_in^𝑚−𝑦₁+𝐶_in^𝑚(5.13) Clearly, by using the Sobolev imbedding theorem, −𝒜𝑤₁=𝑘₁𝑛𝑦₁+𝐶_in^𝑚𝜉^2𝑛−1𝑤₂−𝑚𝑘₁𝑥²+𝐿_in^𝑛𝜉^1𝑚−1𝑤₁,(5.14) Therefore, (𝜉₁,𝜉₂) since (0,0)

Theorem 5.3. For (𝐶,𝐿). the steady-state problem given by (5.1)–(5.3) has a unique solution in −𝒜𝑤₂==−−𝑘𝑘₂₂𝑔𝑦¹,𝑦₂−𝑔𝑥¹,𝑥₂𝑛𝑦₁+𝐶_in^𝑚𝜉^2𝑛−1𝑤₂+𝑚𝑘₂𝑥²+𝐿_in^𝑛𝜉^1𝑚−1𝑤₁,(5.15)

Proof. Let 𝜉₁ be solutions to (5.1)–(5.3) on 𝜉₂. To obtain the desired result, we will be showing that −−𝒜𝒜𝑤𝑤₁₂=−𝑎₁𝑤₁+𝑏₁𝑤₂,(5.16)=𝑎₂𝑤₁−𝑏₂𝑤₂,(5.17) Let 𝑖=1;2, Then 𝑎𝑏_𝑖(𝑧)=𝑚𝑘_𝑖𝑥²(𝑧)+𝐿_in^𝑛𝜉^1𝑚−1(𝑧),_𝑖(𝑧)=𝑛𝑘_𝑖𝑦¹(𝑧)+𝐶_in^𝑚𝜉^2𝑛−1(𝑧).(5.18) Hence, by applying the mean value theorem, we get
where 𝑘₁, are some intermediate values between 𝒜𝑤=0,𝑤∈𝐷(𝒜),(5.19) and 𝑤=𝑘₂𝑤₁+𝑘₁𝑤₂.
By similar considerations as above, we also get 𝑤=0 for the same [0,1]. and −𝒜𝑤₂=𝑎₂𝑤₁−𝑏₂𝑤₂(5.20)
Now, we have the following system: 𝑤₁=−𝑘²⁻¹𝑘₁𝑤₂(5.21)𝒜𝑤₂=𝑐𝑤₂,(5.22) where, for 𝑐(𝑧)=𝑎₁(𝑧)+𝑏₂(𝑧).𝑖=1;2
Multiplying (5.16) by 0≤𝑎_𝑖(𝑧)≤𝑚𝑘_𝑖𝐿^𝑛𝐶^𝑚−1,0≤𝑏_𝑖(𝑧)≤𝑛𝑘_𝑖𝐶^𝑚𝐿^𝑛−1.(5.23) and (5.17) by 𝜆=max(𝑚𝐿,𝑛𝐶)max(𝑘₁,𝑘₂)𝐶^𝑚−1𝐿^𝑛−1 we get by addition of both equations that 0≤𝑐(𝑧)≤2𝜆. where 𝑤₂=0. By Lemma 5.2, this system has a unique solution 𝑤₁=0, in 𝑥=𝑦. Now, let 𝑥(𝑡)∈Ω(𝑡). and substituting the expression Ω(𝑡) yields 𝐿^𝑝 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 “Programme Thématique d'Appui à la Recherche Scientifique” PROTARS III, initiated by the Moroccan “Centre 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 (Université Catholique de Louvain, Belgium) and LINMA of the Faculty of sciences (Univesité Chouaib Doukkali, Morocco), funded by the Belgian Secretary of the State for Development Cooperation and by the CIUF (Conseil Interuniversitaire de la Communauté Française, Belgium). The work of B. Aylaj is supported by a research grant from the Agence Universitaire de la Francophonie.