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

Authors: B. Aylaj; M. E. Achhab; M. Laabissi
Journal: Abstract and Applied Analysis
 (2008)

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## 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 relatedC0-semigroup.


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## Body

## 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: (1.1)nA+mB→P, whose kinetic is given by r=(−k1CmLn,−k2CmLn)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: (1.2)∂C∂t=−ν∂C∂ξ+D1∂2C∂ξ2−k1CmLn,(1.3)∂L∂t=−ν∂L∂ξ+D2∂2L∂ξ2−k2CmLn,forξ∈]0,l[andt>0, with the following boundary and initial conditions: (1.4)D1∂C∂ξ(0,t)−νC(0,t)+νCin(t)=0=D1∂C∂ξ(l)∀t>0(1.5)D2∂L∂ξ(0,t)−νL(0,t)+νLin(t)=0=D2∂L∂ξ(l,t)∀t>0,(1.6)C(ξ,0)=C0(ξ),L(ξ,0)=L0(ξ)forξ∈]0,l[.Additionally, the existence and uniqueness of the
corresponding equilibrium profile will be proven.In the above equations,D1,D2 are the
dispersion coefficients, ν is the
superficial fluid velocity, t,ξ denote the time
and space independent variables, respectively, l is the length
of the reactor, m and n are two
positive integers, Cin and Lin 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,ν=0, [8]
established the asymptotic behavior of solutions for the second-order reaction
(i.e., n=m=1). N. Alikakos [9]
established global existence  and L∞ bounds of
positive solutions, when m=1 and 1<n<3/2. This latter result has been generalized by [10] for
the case m=1 and n>1.
In practice, the special casesm=n=1,2,3 have been used
as an industrial pulp bleaching model, where the two reactants are chlorine
dioxide (C) and lignin (L). In particular, [3] studied approximate solutions by
using several methods (orthogonal collocation, finite elements, and finite
difference methods), when n=m and D1=D2. The reader can find another model with D1≠D2 in [11], where
the numerical analysis has been done for m=n=1 and D2=4D1, D2=16D1 (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 all0≤ξ≤l and for all t≥0: (1.7)0≤C≤C¯,0≤L≤L¯,(1.8)Cin(t)≤C¯,Lin(t)≤L¯, where C¯ and L¯ are positive
constants.This paper is organized as follows. In Section2, 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

LetX be a real
Banach   space with norm ∥⋅∥,J=[a,b[(a<b≤+∞), and let {𝒯(t);t≥0} be a linear
contraction C0-semigroup on X generated by 𝒜. Let ℬ be a nonlinear
continuous operator form Ω into X, where Ω is a subset of J×X. I and 𝕀 denote,
respectively, the identity operator of X 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:(2.1)x˙(t)=𝒜x(t)+ℬ(t,x(t)),τ<t<b,x(τ)=xτ∈Ω(τ),  where Ω(τ) denote the
section of Ω at τ∈J,  given by Ω(t)={x∈X;(t,x)∈Ω}. Assume that Ω(t)≠∅ for all t∈J. Moreover, recall that d(x;𝒟)=inf⁡{∥x−y∥,y∈𝒟}, for x∈X and 𝒟 is a subset of X.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 Ω is cylindrical,
that is, Ω=J×𝒟 [14, 22]. More
generally, the cylindrical case of Ω will not be
convenient for the study of evolution system satisfying time-dependent
constraints, that is, x(t)∈Ω(t) on J (see our
problem in Section 3). A noncylindrical Ω 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 ℬ(t,⋅) is lℬ-dissipative
[16].Theorem 2.1 (see [16]).
Suppose that the following conditions are fulfilled:(i)
Ω is closed from
the left, that is, if (tn,xn)∈Ω,tn↑t in J, and xn→x in X as n→∞,  then (t,x)∈Ω;(ii)
for all(t,x)∈Ω,liminf⁡h↓0⁢(1/h)d(𝒯(h)x+hℬ(t,x),Ω(t+h))=0;(iii)
ℬ is  continuous  on Ω and there exists  lℬ∈ℝ+ such that the operator  (ℬ(t,⋅)−lℬI) is dissipative on Ω(t) for all t∈J.
If Ω is a connected
subset of J×X such that for
all t∈J,Ω(t)≠∅, then, for each (τ,xτ)∈Ω, (2.1) has a unique mild solution on J.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 whenΩ(t) is 𝒯(s)-invariant,
that is, 𝒯(s)(Ω(t))⊂Ω(t) for all t,s≥0, we have the following lemma.Lemma 2.2.
Letℬ:Ω→X be continuous and let Ω be closed from the left. If Ω(t) is 𝒯(s)-invariant for all t,s≥0, then the
following subtangential condition (2.2)lim⁡h↓0inf⁡1hd(x+hℬ(t,x);Ω(t+h))=0∀(t,x)∈Ω   implies the
condition (2.3)lim⁡h↓0inf⁡1hd(𝒯(h)x+hℬ(t,x),Ω(t+h))=0,∀(t,x)∈Ω.Proof.
Let(t,x)∈Ω, given ϵ>0, from condition (2.2) it follows, by [23, Lemma 3] (see also [24, Lemma 1]), that there is h∈(0,ϵ] and y∈Ω(t+h) such that ∥y−x−hℬ(t,x)∥≤hϵ. Let now u=y−x−hℬ(t,x) and v=(1/h)u. We get ∥v∥≤ϵ such that y=x+h(B(t,x)+v)∈Ω(t+h). By the
invariance properties of Ω(t),  we have 𝒯(h)y∈Ω(t+h). Consequently, (2.4)d(𝒯(h)x+hℬ(t,x);Ω(t+h))≤∥𝒯(h)x+hℬ(t,x)−𝒯(h)y∥,≤∥hℬ(t,x)−h𝒯(h)ℬ(t,x)−h𝒯(h)v∥,≤h∥𝒯(h)ℬ(t,x)−ℬ(t,x)∥+h∥𝒯(h)v∥,≤h∥𝒯(h)ℬ(t,x)−ℬ(t,x)∥+hϵ. By using the continuity of C0-semigroup (𝒯(t))t≥0, the desired result (2.3) is obtained.
Theorem2.1 with Lemma 2.2 obviously imply the following.Corollary 2.3.
Suppose that the following conditions are
fulfilled:(i)
Ω is closed from
the left, that is, if (tn,xn)∈Ω,tn↑t in J, and xn→x in X as n→∞, then (t,x)∈Ω;(ii)
Ω(t) is 𝒯(s)-invariant, for
all t,s≥0;(iii)
for  all(t,x)∈Ω,liminf⁡h↓0⁢(1/h)d(x+hℬ(t,x),Ω(t+h))=0;(iv)
ℬ is  continuous  on Ω and there exists  lℬ∈ℝ+ such that the operator  (ℬ(t,⋅)−lℬI) is dissipative on Ω(t), for all t∈J.
If Ω is a connected
subset of J×X such that for
all t∈J,Ω(t)≠∅, then, for each (τ,xτ)∈Ω,
(2.1) has a unique mild solution on J.


## 3. Abstract Semigroup Formulation

Throughout the
sequel, we assumeH=L2[0,1]⊕L2[0,1], the Hilbert space with the usual inner product(3.1)〈(x1,x2),(y1,y2)〉=〈x1,y1〉L2+〈x2,y2〉L2 and the induced
norm (3.2)∥(x1,x2)∥=(∥x1∥L22+∥x2∥L22)1/2     for all (x1,x2) and (y1,y2) in H.Clearly, the Hilbert spaceH is a real
Banach lattice, where for all given x=(x1,x2)∈H,y=(y1,y2)∈H,(3.3)x≤y⁢iffx1(z)≤y1(z),x2(z)≤y2(z)fora.e.⁢z∈[0,1]. Recall that for every pair x,y∈H, the set(3.4)[x,y]={w∈H:x1≤w1≤y1,x2≤w2≤y2}=[x1,y1]×[x2,y2] is called the order interval between x and y. Clearly, [x,y] is nonempty if x≤y (for more
details, see, e.g., [25]). A bounded linear operator 𝒯 on H is said to be
positive if 0≤𝒯x for all 0≤x. Similarly, a
family of bounded linear operators (𝒯(t))t≥0 of H is said to be a
positive C0-semigroup on H if 𝒯(t) is a C0-semigroup on H and 𝒯(t) is a positive
operator for all t≥0.In the following, we will assume thatCin(t) and Lin(t) are positive   𝒞1([0,∞[)-functions. Let us consider the following state transformation: (3.5)z=ξl,x1=C−Cin,x2=L−Lin,x01=C0−Cin,x02=L0−Lin. Then, we obtain the new equivalent system for all z∈]0,1[ and t>0: (3.6)∂x1∂t=−v∂x1∂z+d1∂2x1∂z2−k1(x1+Cin(t))m(x2+Lin(t))n−Cin.(t),(3.7)∂x2∂t=−v∂x2∂z+d2∂2x2∂z2−k2(x1+Cin(t))m(x2+Lin(t))n−Lin.(t), with (3.8)di∂xi∂z(0,t)−vxi(0,t)=0=di∂xi∂z(1,t)∀t>0i=1;2,(3.9)xi(z,0)=x0i(z)forz∈]0,1[,i=1;2,  where (3.10)d1=D1l2,d2=D2l2,v=νl.This PDEs describing the reactor dynamics may be
formally written in the abstract form as(3.11)x˙(t)=Ax(t)+B(t,x(t)),x(0)=x0∈Ω(0), where Ω(t) denote the
section of Ω at t∈ℝ+,  which is given
in view of (1.7) by (3.12)Ω={(t,(x1,x2))T∈ℝ+×H:−Cin(t)≤x1(z)≤C¯−Cin(t),−Lin(t)≤x2(z)≤L¯−Lin(t)a.e.z∈[0,1]}.The linear operatorA is defined by (3.13)D(A)={x=(x1,x2)T∈H⁢:x,dxdz∈Hare  absolutely  continuous,d2xdz2∈H⁢,didxidz(0)−υxi(0)=0=didxidz(1);i=1;2},(3.14)Ax=(d1d2x1dz2−υdx1dz00d2d2x2dz2−υdx2dz)=(A1x100A2x2).  The nonlinear
operator B is defined on Ω by (3.15)B(t,x)=(−k1(x1+Cin(t)𝕀)m(x2+Lin(t)𝕀)n−Cin.(t)𝕀,−k2(x1+Cin(t)𝕀)m(x2+Lin(t)𝕀)n−Lin.(t)𝕀)T.   It is shown in
[7] that the linear operator A given by (3.14)
is the infinitesimal generator of contraction semigroup on H(3.16)T(t)=(T1(t)00T2(t)), where T1(t) and T2(t) are the C0-semigroups generated,
respectively, by A1 and A2.


## 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(t,x)∈Ω,(4.1)lim⁡h↓01hd(x+hB(t,x);Ω(t+h))=0.Proof.
Let(t,x)∈Ω. Observe that Ω(t) is given by Ω(t)=Ω1(t)×Ω2(t), where (4.2)Ω1(t)=[−Cin(t)𝕀,(C¯−Cin(t))𝕀],Ω2(t)=[−Lin(t)𝕀,(L¯−Lin(t))𝕀]. Denote (4.3)X1(t)=x1+Cin(t)𝕀,X2(t)=x2+Lin(t)𝕀, we have, for x∈Ω(t),(4.4)X(t)=(X1(t),X2(t))T∈[0,C¯𝕀]×[0,L¯𝕀]. Let h0>0 be sufficiently
small such that h0k1C¯m−1L¯n≤1.
Let, now,h∈(0,h0), then (4.5)X1(t)(𝕀−hk1X1m−1(t)X2n(t))∈[0,C¯𝕀]. Hence (4.6)f1(t,X(t))=X1(t)(𝕀−hk1X1m−1(t)X2n(t))−Cin(t+h)𝕀∈Ω1(t+h). By using the regularity of the inlet function Cin, we get (4.7)d(x1+hB1(t,x),Ω1(t+h))≤d(X1(t)−hk1X1m(t)X2n(t)−Cin(t+h)𝕀,Ω1(t+h))+hϵ(h)≤d(f1(t,X(t)),Ω1(t+h))+hϵ(h)=hϵ(h), where ϵ(h)→0 as h→0. Whence (4.8)lim⁡h↓01hd(x1+hB1(t,x);Ω1(t+h))=0.  By similar
considerations as above, taking into account the regularity of the function Lin, we also get (4.9)lim⁡h↓01hd(x2+hB2(t,x);Ω2(t+h))=0.  Observe, now, that (4.10)d(x+B(t,x),Ω(t+h))≤d(x1+B1(t,x),Ω1(t+h))+d(x2+B2(t,x),Ω2(t+h)),
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 existslB∈ℝ+ such that the
operator (B(t,⋅)−lBI) is dissipative
on Ω(t) for each t≥0.Proof.
Lett≥0 and let x,y be in Ω(t). Denote (4.11)gi(t,x)=−ki(x1+Cin(t)𝕀)m(x2+Lin(t)𝕀)nfori=1,2, and let also (4.12)X1(t)=x1+Cin(t)𝕀;X2(t)=x2+Lin(t)𝕀;Y1(t)=y1+Cin(t)𝕀,Y2(t)=y2+Lin(t)𝕀. Observe that, for each x,y∈Ω(t),(Xi(t),Yi(t))T∈[0,C¯𝕀]×[0,L¯𝕀] for i=1,2. Hence, by
applying the mean value theorem, for i=1,2, we get(4.13)∥gi(t,x)−gi(t,y)∥L2≤ki(C¯2m∥X2n(t)−Y2n(t)∥L22+L¯2n∥X1m(t)−Y1m(t)∥L22)1/2≤ki(n2C¯2mL¯2n−2∥x2−y2∥L22+m2L¯2nC¯2m−2∥x1−y1∥L22)1/2≤kiC¯m−1L¯n−1max⁡(nC¯;mL¯)∥x−y∥. Finally, (4.14)∥B(t,x)−B(t,y)∥=(∥g1(t,x)−g1(t,y)∥L22+∥g2(t,x)−g2(t,y)∥L22)1/2≤max⁡(k1,k2)C¯m−1L¯n−1max⁡(nC¯;mL¯)∥x−y∥. Consequently, B(t,⋅) is an lB-dissipative
operator on Ω(t) [14, page 245], where
(4.15)lB=max⁡(k1,k2)C¯m−1L¯n−1max⁡(nC¯;mL¯).
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(4.16)Ω(t)is𝒯(s)-invariant∀t,s≥0.Proof.
Lett,s≥0 and (x,y)T∈Ω(t). We have (4.17)(−Cin(t)𝕀,−Lin(t)𝕀)T≤(x,y)T≤((C¯−Cin(t))𝕀,(L¯−Lin(t))𝕀)T. Hence, by using the positivity of (T(t))t≥0 [26], we have (4.18)(−Cin(t)T1(s)𝕀,−Lin(t)T2(s)𝕀)T≤T(s)(x,y)T≤((C¯−Cin(t))T1(s)𝕀,(C¯−Cin(t))T2(s)𝕀)T. Since, Ti(t)𝕀≤𝕀fori=1;2 (see [26]) and
by using the inequalities
(1.8)
(i.e., C¯≥Cin and L¯≥Lin),  the invariance
of Ω(t)    holds for all t≥0. Thus, (T1(s)x,T2(s)y)T∈Ω(t) for all t,s≥0.Now, we are in
a position to state and prove our global existence result for problem (3.6)–(3.9).Theorem 4.4.
LetCin(t) and Lin(t) be positive C1([0,+∞[)-functions.
Then, for every x0∈Ω(0), the problem (3.6)–(3.9) has a unique
global mild solution.Proof.
SinceB is continuous function in Ω, by Corollary 2.3,
it is sufficient to prove the condition (i) in Corollary 2.3
and to check that the subset Ω is connected
(a)
Let us first show thatΩ is closed from
the left.
Lettn↗t and xn∈Ω(tn) with xn→x∈H, then there
exists a subsequence of (xn) which is also
denoted by (xn) such that xn(z)→x(z), that is, on [0,1] which implies,
by continuity of Cin and Lin,thatx(z)∈[−Cin(t),C¯−Cin(t)]×[−Lin(t),L¯−Lin(t)], that is, on [0,1], hence x∈Ω(t) for each t≥0.(b)
Let us, now, check thatΩ is connected in [0,+∞[×H:
LetK=[0,C¯𝕀]×[0,L¯𝕀] and define G:[0,+∞[×K→Ω such that for
all (t,x)∈[0,+∞[×K,G(t,x)=(t,x1−Cin(t)𝕀,x2−Lin(t)𝕀)T. Since Cin and Lin are continuous
functions in [0,+∞[, it follows that G in [0,+1[×K is also a
continuous function. Observe that G is surjective;
since [0,C¯𝕀]×[0,L¯𝕀] is connected in H, we get that Ω=G([0,+∞[×K) is also
connected in [0,+∞[×H.
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 functionsCin and Lin are independent
of time t, which implies
that the domain Ω(t) is also
independent of t. If we denote by Cin¯ and Lin¯ the values of Cin and Lin 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:
(5.1)−vdx1dz=d1d2x1dz2−k1(x1+Cin¯)m(x2+Lin¯)n=0,(5.2)−vdx2dz=d2d2x2dz2−k2(x1+Cin¯)m(x2+Lin¯)n=0,  with (5.3)didxidz(0)−vxi(0)=0=didxidz(1),i=1;2,(5.4)Ω(t)=Δ={(x1,x2)T∈H:−Cin¯≤x1(z)≤C¯−Cin¯,−Lin¯≤x2(z)≤L¯−Lin¯for almost allz∈[0,1]}. The following existence result can be proven as in the case where Cin and Lin 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 Δ.The sequel of this paper will deal with the uniqueness
analysis of steady states in the important case whered1=d2=d.First, sinced1=d2=d, we denote 𝒜=d(d2/dz2)−v(d/dz)=Ai with D(𝒜)=D(Ai) for i=1;2.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.
Letb(⋅) be a bounded nonnegative
function defined in [0,1]. If u∈L2([0,1]) satisfies the equations (5.5)𝒜u=buin]0,1[,u∈D(𝒜),  then u=0 in [0,1].Proof.
Letu be the solution of problem (5.5), then (5.6)〈𝒜u,u〉L2=〈bu,u〉L2.
We have, (5.7)〈𝒜u,u〉L2=∫01[dd2udz2(z)−vdudz(z)]u(z)dz,=−∫01d[dudz(z)]2dz+d[dudz(1)u(1)−dudz(0)u(0)]−12v[u2(1)−u2(0)],=−d∥dudz∥L22−12vu2(1)−12vu2(0),(5.8)≤0. Since b(z) is nonnegative
function in [0,1], then by (5.8) and taking into account (5.6)
(5.9)〈bu,u〉L2=∫01b(z)u2(z)dz=0.
Which implies,
in view of (5.6)-(5.7), that (5.10)〈𝒜u,u〉L2=0=d∥dudz∥L22+12vu2(1)+12vu2(0). Then, we get (5.11)dudz(z)=0a.e.⁢z∈[0,1],u(0)=0=u(1). Clearly, by
using the Sobolev imbedding theorem, D(𝒜)⊂𝒞([0,1]). Therefore, u=0 since u∈D(𝒜).Theorem 5.3.
Ford1=d2=d, the steady-state problem given by (5.1)–(5.3) has a unique
solution in Δ.Proof.
Letx=(x1,x2)Tandy=(y1,y2)T be solutions to (5.1)–(5.3) on [0,1]. To obtain the desired result, we will be showing that x=y. Let (5.12)g(x1,x2)=−(x1+Cin¯𝕀)m(x2+Lin¯𝕀)n,w1=y1−x1∈D(𝒜),w2=x2−y2∈D(𝒜). Then (5.13)−𝒜w1=k1(g(y1,y2)−g(x1,x2))=k1(y1+Cin¯)m[(x2+Lin¯)n−(y2+Lin¯)n]+k1(x2+Lin¯)n[(x1+Cin¯)m−(y1+Cin¯)m]  Hence, by
applying the mean value theorem, we get
(5.14)−𝒜w1=k1n(y1+Cin¯)mξ2n−1w2−mk1(x2+Lin¯)nξ1m−1w1, where (ξ1,ξ2) are some
intermediate values between (0,0) and (C¯,L¯).
By similar considerations as above, we also get(5.15)−𝒜w2=−k2(g(y1,y2)−g(x1,x2))=−k2n(y1+Cin¯)mξ2n−1w2+mk2(x2+Lin¯)nξ1m−1w1,  for the same ξ1 and ξ2.
Now, we have the following system:(5.16)−𝒜w1=−a1w1+b1w2,(5.17)−𝒜w2=a2w1−b2w2, where, for i=1;2,(5.18)ai(z)=mki(x2(z)+Lin¯)nξ1m−1(z),bi(z)=nki(y1(z)+Cin¯)mξ2n−1(z).
Multiplying
(5.16) by k2 and (5.17) by k1, we get by addition of both equations that (5.19)𝒜w=0,w∈D(𝒜), where w=k2w1+k1w2. By Lemma 5.2,
this system has a unique solution w=0 in [0,1]. Now, let
(5.20)−𝒜w2=a2w1−b2w2 and
substituting the expression (5.21)w1=−k2−1k1w2 yields (5.22)𝒜w2=cw2, where c(z)=a1(z)+b2(z). Observe that,
for i=1;2,  (5.23)0≤ai(z)≤mkiL¯nC¯m−1,0≤bi(z)≤nkiC¯mL¯n−1.
Letλ=max⁡(mL¯,nC¯)max⁡(k1,k2)C¯m−1L¯n−1, then we have 0≤c(z)≤2λ. By Lemma  5.2 we get w2=0. Thus it follows, by (5.21), that w1=0, which ensures
the desired result, that is, x=y.


## 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,x(t)∈Ω(t). Moreover, the
set of physically meaningful admissible states Ω(t) 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.



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*Source: 127394-2008-02-25.xml*