Asymptotic behaviour of state trajectories for a class of tubular reactor non-linear models

doi:10.1093/imamci/dnl013

Advance Access publication on June 30, 2006

Asymptotic behaviour of state trajectories for a class of tubular reactor non-linear models

B. A

YLAJ

†, M. E. A

CHHAB

‡

AND

M. L

AABISSI

§

Laboratoire d’Ing´enierie Math´ematique (LINMA), D´epartement de Math´ematique et Informatique, Facult´e des Sciences, BP 20, El Jadida, Morocco

[Received on 9 September 2005; revised on 21 February 2006; accepted on 13 March 2006]

We prove the global existence of the state trajectories for a class of non-linear systems arising from convection-dispersion-reaction processes. It is also shown that there is at least one steady state in the set of physically feasible states for such systems. The uniqueness and the stability analysis of this steady- state solution are discussed. Our approach is based on the analysis of a non-linear set of partial differential equations, using the upper and lower solutions, dissipativity properties, a subtangential condition and the positivity of the related C

₀

-semigroup.

Keywords: tubular reactor; non-linear distributed parameter systems; equilibrium profile; positive C₀

-semigroup; compact semigroup; dissipativity.

1. Introduction

During the last decades, increasing attention from system theory as well as from mathematical control theory has been dedicated to study chemical and biochemical processes. This interest can be explained by the potential of these processes to give a higher productivity. At the same time, more and more constraints, arising from environmental and safety considerations, need to be satisfied. The mathematical control theory dedicates its major efforts to the design and the implementation of ‘good’ controls and to stabilize the process under consideration (Antoniades & Christofides, 2001; Dochain, 1994; Ray, 1981;

Renou, 2000; Renou et al., 2003). For this purpose, the investigation of the questions of state trajectories analysis and of the existence and stability of steady states is of great importance (Achhab et al., 2004;

Cazenave & Haraux, 1999; Danckwerts, 1970; Laabissi et al., 2001, 2004; Varma & Morbidelli, 1997;

Winkin et al., 2000).

Chemical and biochemical processes are typically described by non-linear coupled partial differen- tial equations (PDE) and hence by distributed parameter models (see Dochain, 1994, and the references within). The source of non-linearities is essentially the kinetics of the reactions involved in the process.

In this paper, we consider a class of gas–liquid reactions whose kinetic is given by r = k Π C

_i^αi

where C

_i

are the concentrations of the reactants involved in the reaction, k is the kinetic constant and α

i

is defined as the order of the reaction with respect to the i th reactant. More precisely, first, we study the global existence of the trajectories of the models which describe the evolution of two reactants’ concentrations

†Email: [email protected]

‡Email: [email protected]

§Corresponding author. Email: [email protected]

© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

C and L:

∂ C

∂ t = −ν ∂ C

∂ξ + D

₁

∂

²

C

∂ξ

²

− k

_C

C

^m

L

ⁿ

, (1.1)

∂ L

∂ t = −ν ∂ L

∂ξ + D

₂

∂

²

L

∂ξ

²

− k

_L

C

^m

L

ⁿ

, (1.2)

for ξ ∈ ] ^{0, l} [ ^{and t} > 0, with the following boundary and initial conditions:

D

₁

∂ C

∂ξ ( 0, t ) − ν C ( 0, t ) + ν C

_in

( t ) = 0 = D

₁

∂ C

∂ξ ( l, t ) , for all t > 0, (1.3) D

₂

∂ L

∂ξ ( 0, t ) − ν L ( 0, t ) + ν L

_in

( t ) = 0 = D

₂

∂ L

∂ξ ( l, t ) , for all t > 0, (1.4) and

C (ξ , 0 ) = C

0

(ξ) , L (ξ , 0 ) = L

0

(ξ) , for ξ ∈ ] ^{0, l} [ ^{. (1.5)} Second, we study the existence and uniqueness of the steady states of those models. The equilibrium profiles must then be solutions of the following system: for ξ ∈ ] ^{0, l} [ ^,

−ν dC

d ξ + D

₁

d

²

C

d ξ

²

− k

_C

C

^m

L

ⁿ

= 0, (1.6)

−ν dL

d ξ + D

₂

d

²

L

d ξ

²

− k

_L

C

^m

L

ⁿ

= 0, (1.7)

D

₁

dC

d ξ ( 0 ) − ν C ( 0 ) + ν C

_in

= 0 = D

₁

dC

d ξ ( l ) , (1.8)

D

₂

dL

d ξ ( 0 ) − ν L ( 0 ) + ν L

_in

= 0 = D

₂

dL

d ξ ( l ) . (1.9)

Finally, we discuss the stability of the steady-state solutions of (1.1)–(1.5).

In the above equations, D

₁

, D

₂

are the dispersion coefficients, ν the superficial fluid velocity, k

_C

, k

_L

the kinetic constants, t , ξ denote the time- and space-independent variables, respectively, l is the length of the reactor, m and n are two positive integers, C

_in

and L

_in

are the inlet concentrations which will be assumed to be two positive constants. For further discussion of parameters, we refer to Renou (2000).

The non-linear models considered in this paper have been studied in a qualitative manner by several authors. In the case, ν = 0, Martin (1980) established the asymptotic behaviour of solutions for the second-order reaction (i.e. n = m = 1). Alikakos (1979) established global existence and L

^∞

bounds of positive solutions when m = 1 and 1 < n <

³₂

. This latter result has been generalized by Masuda (1983) for the case m = 1 and n > 1.

In practice, the special cases m = 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, Renou (2000)

studied approximate solutions by using several methods (orthogonal collocation, finite elements and

finite-difference methods), when n = m and D

₁

= D

₂

. The reader can find another model with D

₁

=

D

₂

in Van Elk (2001), where the numerical analysis has been done for m = n = 1 and D

₂

= 4D

₁

,

D

₂

= 16D

₁

(see also Danckwerts, 1970).

For technological limitations and economical considerations, it is usually assumed, for all 0 ξ ^l and for all t ^{0, that}

0 ^{C C, 0 L L, (1.10)}

where C and L are positive constants chosen such that

C

_in

^{C, L}

in

^{L. (1.11)}

This assumption will be confirmed by the further analysis.

Let us consider the following state transformation:

z = ξ

l , x

1

= C − C

_in

C

_in

, x

2

= L − L

_in

L

_in

, x

01

= C

₀

− C

_in

C

_in

, x

02

= L

₀

− L

_in

L

_in

. Then, we obtain the new equivalent dimensionless system for all z ∈ ] ^{0, 1} [ ^{and t} > 0

∂ x

1

∂ t = −v ∂ x

1

∂ z + d

₁

∂

²

x

1

∂ z

²

− k

₁

C

_in^m

L

ⁿ_in

( x

₁

+ 1 )

^m

( x

₂

+ 1 )

ⁿ

, (1.12)

∂ x

₂

∂ t = −v ∂ x

₂

∂ z + d

2

∂

²

x

₂

∂ z

²

− k

2

C

_in^m

L

ⁿ_in

( x

1

+ 1 )

^m

( x

2

+ 1 )

ⁿ

, (1.13) with

d

_i

∂ x

i

∂ z ( 0, t ) − v x

_i

( 0, t ) = 0 = d

_i

∂ x

i

∂ z ( 1, t ) , for all t > 0, i = 1, 2, (1.14) and

x

_i

( z, 0 ) = x

_0i

( z ) , for z ∈ ] ^{0, 1} [ ^{and i} = 1, 2. (1.15) The new parameters are related to the original ones by

d

1

= D

₁

l

²

, d

2

= D

₂

l

²

, v = ν

l , k

1

= k

_C

C

_in

, k

2

= k

_L

L

_in

. (1.16)

The corresponding steady-state system is

−v dx

₁

dz + d

₁

d

²

x

₁

dz

²

− k

₁

C

_in^m

L

ⁿ_in

( x

₁

+ 1 )

^m

( x

₂

+ 1 )

ⁿ

= 0, (1.17)

−v dx

₂

dz + d

₂

d

²

x

₂

dz

²

− k

₂

C

_in^m

L

ⁿ_in

( x

₁

+ 1 )

^m

( x

₂

+ 1 )

ⁿ

= 0, (1.18) with

d

_i

dx

i

dz ( 0 ) − v x

_i

( 0 ) = 0 = d

_i

dx

i

dz ( 1 ) , i = 1, 2. (1.19)

This paper is organized as follows: The notations and preliminaries are given in Section 2. In Section 3, we state the main global existence result for System (1.1)–(1.5). The existence of equilibrium profiles is proved in Section 4, and with additional assumptions, we also state uniqueness and stability results.

Finally, the main conclusions are outlined in Section 5. The background of our approach can be found

in Curtain & Zwart (1995), Martin (1976), Pazy (1981) and Pao (1992).

2. Notations and preliminaries

Let ( X , ·) be a real Banach space, (T ( t ))

t0

a C

₀

-semigroup of linear operators such that T ( t ) e

^wt

, for all t 0, for some w ∈ R . A is the infinitesimal generator of (T ( t ))

t0

, N be a continuous function from a closed subset D of X into X and I be the identity operator of X . Recall that

d ( x ; D) = inf { x − y , y ∈ D} . For the following abstract Cauchy problem:

˙

x ( t ) = A x ( t ) + N ( x ( t )) , x ( 0 ) = x

0

∈ D ,

(2.1) we have

T

HEOREM

2.1 (Martin, 1976, p. 355) If the following conditions are satisfied:

(i) D is T ( t ) -invariant, i.e. T ( t )D ⊂ D , for all t ⁰ ; (ii) for all x ∈ D , lim

_h→0+ 1

h

d ( x + h N ( x ); D) = 0 ;

(iii) N is continuous on D and there exists l

N

∈ R

⁺

such that the operator (N − l

N

I ) is dissipative on D , i.e. lim

h→0(x−y)+h(Nx−Ny)−x−y

h

0 for all x, y ∈ D .

Then, (2.1) has a unique mild solution x ( t, x

₀

) on [ ^0, +∞ [ , for all x

₀

∈ D . Furthermore, if ( S ( t ))

t0

is defined on D by S ( t ) x

₀

= x ( t, x

₀

) , for all t ^{0 and x}

0

∈ D , it is a non-linear semigroup on D , with (A + N ) as its generator.

Now, the following result gives sufficient conditions for the existence of equilibrium profiles for (2.1).

T

HEOREM

2.2 (Pruss, 1981) If D is a convex bounded closed subset of ( X , ·) , ( T ( t ))

t0

is a compact semigroup and N : D → X is locally lipschitz and bounded function such that

for all x ∈ D , lim

h→0⁺

1

h d (T ( h ) x + h N ( x ); D) = 0. (2.2) Then the equation

A x + N ( x ) = 0 admits at least one solution x ∈ D (A) ∩ D .

R

EMARK

2.1

(i) Theorem 2.2 is a particular version of Pruss (1981, Theorem D) when the C

₀

-semigroup ( T ( t ))

t0

is compact.

(ii) If D is T ( t ) -invariant set in X , then, by Pruss (1981), the Property (ii) of Theorem 2.1 implies the Condition (2.2).

Throughout the sequel, we assume H = L

²

[ ^{0, 1} ] ⊕ L

²

[ ^{0, 1} ] the Hilbert space with the usual inner product

( x

₁

, x

₂

) , ( y

₁

, y

₂

) = x

₁

, y

₁

_L²

+ x

₂

, y

₂

_L²

and the induced norm

( x

₁

, x

₂

) = ( x

₁

²_L2

+ x

₂

²_L2

)

¹²

, (2.3) for all ( x

₁

, x

₂

) and ( y

₁

, y

₂

) in H . Clearly, the Hilbert space H is a real Banach lattice (for more details, see, e.g. Nagel, 1986) where, for all given x = ( x

₁

, x

₂

) ∈ H , y = ( y

₁

, y

₂

) ∈ H ,

x ^y if and only if x

₁

( z ) ^y

1

( z ) and x

₂

( z ) ^y

2

( z ) for a.e. z ∈ [ ^{0, 1} ] ^.

One recalls that a bounded linear operator T on H is said to be positive if 0 T x, for all 0 ^x.

Similarly, a family of bounded linear operators (T ( t ))

t0

of H is said to be a positive C

₀

-semigroup on H if T ( t ) is a C

₀

-semigroup on H and T ( t ) is a positive operator for all t ^0.

At the end of this section, assume that Λ is a closed interval of R , X = L

²

[ ^{0, 1} ] ^{and define} K (Λ , X ) = { φ ∈ X , φ( z ) ∈ Λ for almost all z ∈ [ ^{0, 1} ] } .

As a convenient criterion for the subtangential condition given by (ii) of Theorem 2.1, we have the following result.

L

EMMA

2.1 (Laabissi et al., 2001) Assume that X = L

²

[ ^{0, 1} ] ^, Λ = [ ^{a, b} ] ^{, f}

c

: Λ → R is a continuous function and f

p

: [ ^{0, 1} ] → R is a non-negative bounded measurable function. If f

c

( a ) ^{0 and f}

c

( b ) 0, then

lim

h→0⁺

1

h d (φ + h B(φ) , K (Λ , X )) = 0,

where the substitution operator B is defined on K (Λ , X ) by [ B(φ) ] ( z ) = f

_p

( z )· f

_c

(φ( z )) , for all z ∈ [ ^{0, 1} ] and for all φ ∈ K (Λ , X ) .

3. Existence of the global solution

The PDEs (1.12)–(1.15) describing the reactor dynamics can be written in the compact form as

˙

x ( t ) = Ax ( t ) + N ( x ( t )) , x ( 0 ) = x

₀

∈ ,

where A is the linear operator defined by D ( A ) =

x = ( x

₁

, x

₂

)

∈ H : x and dx

dz ∈ H are absolutely continuous, d

²

x

dz

²

∈ H and d

i

dx

_i

dz ( 0 ) − υ x

i

( 0 ) = 0 = d

i

dx

_i

dz ( 1 ) , i = 1, 2

, (3.1)

Ax =

⎛

⎝ d

₁^d2x1

dz²

− υ

^dx_dz¹

0 0 d

₂^d2x2

dz²

− υ

^dx_dz²

⎞

⎠

=

A

1

x

1

0 0 A

₂

x

₂

(3.2)

and the non-linear operator N is defined on which is given in view of ( 1.10 ) by =

( x

₁

, x

₂

)

∈ H : − 1 ^x

1

( z ) ^{C − C}

ⁱⁿ

C

_in

and − 1 ^x

2

( z ) ^{L − L}

ⁱⁿ

L

_in

for almost all z ∈ [ ^{0, 1} ]

, (3.3)

and

N ( x ) = (− k

₁

C

_in^m

L

ⁿ_in

( x

₁

+ 1 )

^m

( x

₂

+ 1 )

ⁿ

, − k

₂

C

_in^m

L

ⁿ_in

( x

₁

+ 1 )

^m

( x

₂

+ 1 )

ⁿ

)

. (3.4) It is shown in Winkin et al. (2000) that the linear operator A given by (3.2) is the infinitesimal generator of a C

0

-semigroup of bounded linear operators on H

T ( t ) =

T

₁

( t ) 0 0 T

₂

( t )

, (3.5)

where T

1

( t ) and T

2

( t ) are the C

0

-semigroups generated, respectively, by A

1

and A

2

. For x

_i

∈ D ( A

_i

) , the domain of the generator A

_i

( i = 1, 2 ) , one has

A

i

x

i

, x

i

L²

=

¹

0

d

i

d

²

x

_i

dz

²

( z ) − υ dx

_i

dz ( z )

x

i

( z ) dz

= −

¹

0

d

_i

dx

_i

dz ( z )

2

dz + d

_i

dx

₁

dz ( 1 ) x

_i

( 1 ) − dx

₀

dz ( 0 ) x

_i

( 0 )

− 1

2 υ [ ^x

i²

( 1 ) − x

_i²

( 0 ) ] −υ x

_i²

( 0 ) − 1

2 υ x

_i²

( 1 ) + 1

2 υ x

_i²

( 0 ) = − 1

2 υ x

_i²

( 1 ) − 1 2 υ x

_i²

( 0 )

^0.

Hence,

Ax, x = A

₁

x

₁

, x

₁

L²

+ A

₂

x

₂

, x

₂

L²

^0, ∀ x = ( x

₁

, x

₂

) ∈ D ( A ) = D ( A

₁

) ⊕ D ( A

₂

) . Actually, by Lumer–Philips theorem (Curtain & Zwart, 1995; Pazy, 1981), A is the generator of con- tractions semigroup on H , i.e. T ( t ) ¹ = e

^ωt

for all t ^{0 (i.e.} w = 0).

In order to apply the result given in Theorem 2.1, we need the following lemmas concerning the non-linear operator N given by (3.4), involved in the dynamics (1.12)–(1.15).

L

EMMA

3.1 For each x ∈ , the following subtangential condition holds:

lim

h→0⁺

1

h d ( x + h N ( x ); ) = 0. (3.6)

Proof. First of all, observe that is given by =

1

×

2

, where

1

=

x

₁

∈ L

²

[ ^{0, 1} ] : x

₁

( z ) ∈ Λ

1

=

− 1, C ¯ − C

_in

C

_in

a.e. on [ ^{0, 1} ]

,

2

=

x

₂

∈ L

²

[ ^{0, 1} ] : x

₂

( z ) ∈ Λ

2

=

− 1, L ¯ − L

in

L

in

a.e. on [ ^{0, 1} ]

.

Let x = ( x

₁

, x

₂

) be arbitrarily fixed on .

Now, we apply Lemma 2.1 where

K (Λ

1

, L

²

[ ^{0, 1} ] ) =

1

, f

_p

( z ) = k

₁

L

ⁿ_in

C

_in^m

( x

₂

( z ) + 1 )

ⁿ

, for all z ∈ [ ^{0, 1} ] ^{, and} f

_c

(λ) = −(λ + 1 )

^m

, for all λ ∈ Λ

1

.

Whence, since B( x

₁

) = N

₁

( x ) , lim

h→0⁺

1

h d ( x

1

+ h N

1

( x );

1

) = 0. (3.7)

By similar considerations as above, we also get lim

h→0⁺

1

h d ( x

₂

+ h N

₂

( x );

2

) = 0, (3.8)

where Lemma 2.1 has been applied with

K (Λ

2

, L

²

[ ^{0, 1} ] ) =

2

, f

_p

( z ) = k

₂

C

_in^m

L

ⁿ_in

( x

₁

( z ) + 1 )

^m

, for all z ∈ [ ^{0, 1} ] ^{, and} f

c

(λ) = −(λ + 1 )

ⁿ

, for all λ ∈ Λ

2

,

and so, B( x

₂

) = N

₂

( x ) . Finally, observe that

d ( x + h N ( x ); ) = d (( x

₁

+ h N

₁

( x ) , x

₂

+ h N

₂

( x ));

1

×

2

) . By using (2.3), we have

d ( x + h N ( x ); ) ^d ( x

1

+ h N

1

( x );

1

) + d ( x

2

+ h N

2

( x );

2

) ,

which combined with (3.7)–(3.8) proves the desired result (3.6).

L

EMMA

3.2 There exists l

_N

∈ R such that the operator ( N − l

_N

I ) is dissipative on .

Proof. See Appendix.

In order to state the invariance condition of the set given by (3.3), we prove the following.

P

ROPOSITION

3.1

T ( t ) ⊂ , for all t ^0.

Proof. Let 1 be the function identically equal to 1 and ( x, y ) ∈ . Equivalently, we can write in a matrix form that

(−1 , −1)

( x, y )

( C ¯ − C

_in

)

C

_in

1 , ( L ¯ − L

_in

) L

_in

1

.

Hence, by using the positivity of ( T ( t ))

t0

, (see Laabissi et al., 2001, for details), we have (− T

₁

( t )1 , − T

₂

( t )1)

^T ( t )( x, y )

( C ¯ − C

_in

) C

in

T

₁

( t )1 , ( L ¯ − L

_in

) L

in

T

₂

( t )1

.

So, since C ¯ ^C

in

and L ¯ ^L

in

as assumed in (1.11), the invariance of holds if T

i

( t )1 1 , for i = 1, 2.

To prove this condition, we apply Lemma 5.2 of Laabissi et al. (2001) to get λ R (λ , A

_i

)1 1 , for all λ > 0 and i = 1, 2. Then, we use the exponential formula (see Pazy, 1981) to conclude.

Now, we are ready to state the main result of this section.

T

HEOREM

3.1 For every x

₀

∈ , System (1.12)–(1.15) has a unique mild solution x ( t, x

₀

) on the interval [ ^0, +∞ [ . Moreover, if we set S ( t ) x

₀

= x ( t, x

₀

) , then ( S ( t ))

t0

is a strongly continuous non- linear semigroup on , generated by the operator A + N .

Proof. First, the invariance of is proved in Proposition 3.1. Second, the Condition (ii) of Theorem 2.1 holds by Lemma 3.1. Concerning the dissipativity, Lemma 3.2 proves that the operator ( N − l

_N

I ) is dissipative on . Finally, the application of Theorem 2.1 ends the proof.

4. Steady-state solutions

This section deals with the existence results and the stability analysis of steady-state solutions for a non-linear tubular reactor models given by (1.12)–(1.15). First of all, we have to solve the following non-linear coupled system of ordinary differential equations:

Ax + N ( x ) = 0, (4.1)

x = ( x

₁

, x

₂

) ∈ ∩ D ( A ) , (4.2)

where D ( A ) , A, and N are given by (3.1)–(3.4), respectively.

For this purpose, we need the following lemma. The proof of this result may be done as the one of Proposition 3.1 of Laabissi et al. (2004) and is therefore omitted.

L

EMMA

4.1 The C

0

-semigroup ( T ( t ))

t0

, given by (3.5), is compact.

An immediate and important consequence of Lemma 4.1 and Theorem 2.2 is the following result.

T

HEOREM

4.1 The tubular reactor modelled by the non-linear coupled PDEs given by (1.12)–(1.15) has at least one equilibrium profile in given by (3.3).

Proof. The semigroup ( T ( t ))

t0

is compact. Moreover, the operator N is Lipschitz and bounded in the closed convex bounded subset of H . By Theorem 2.2, we have only to verify, for all x ∈ ,

lim

h→0⁺

1

h d ( T ( h ) x + h N ( x ); ) = 0.

For this end, we use Lemma 3.1 and Proposition 3.1 combined with (ii) of Remark 2.1.

R

EMARK

4.1 Recall that the growth bound of the semigroup ( T ( t ))

t0

, given by (3.5), is ω( A ) = lim

t→+∞

log T ( t ) t = −v

²

4 min 1

d

₁

, 1 d

₂

.

For more details, we can see Winkin et al. (2000) and the references within. In Lemma 3.2, it is proved that there exists l

N

such that N − l

N

I is dissipative on , where

l

_N

= max ( k

₁

, k

₂

) C ¯

^m−1

L ¯

ⁿ⁻¹

max { n C L ¯

_in

, m LC ¯

_in

} .

According to Martin (1976, Proposition 5.1, p. 357), if ω( A ) + l

_N

< 0, then (1.12)–(1.15) has a unique stable steady state in ∩ D ( A ) .

If we take C = C

_in

and L = L

_in

, the last condition can be written with the original parameters (1.16) as

4 max ( m, n ) max k

_C

C

_in

, k

_L

L

_in

C

_in^m

L

ⁿ_in

< ν

²

min 1

D

₁

, 1 D

₂

.

The sequel of this paper will deal with the important case when d

₁

= d

₂

= d. For this end, we recall some useful results.

Clearly, by using the Sobolev’s imbedding theorem, D ( A ) ⊂ C [ ^{0, 1} ] × C [ ^{0, 1} ] . Hence, every solu- tion of (4.1)–(4.1) is in C

²

[ ^{0, 1} ] × C

²

[ ^{0, 1} ] . Consequently, instead of H , we may take as the state space X = C [ ^{0, 1} ] × C [ ^{0, 1} ] with its norm x = x

1

∞

+ x

2

∞

, for all x = ( x

1

, x

2

)

∈ X .

In this case, the set given by (3.3) will be replaced by

E =

x = ( x

1

, x

2

)

∈ X : − 1 ^x

1

( z ) ^{C − C}

ⁱⁿ

C

_in

and − 1 ^x

2

( z ) ^{L − L}

ⁱⁿ

L

_in

for all z ∈ [ ^{0, 1} ]

. (4.3)

To prove the uniqueness and the asymptotic stability of steady-state solutions to System (1.12)–(1.15), we apply the monotone iterative method as in Pao (1992). More precisely, our approach is based on the analysis of the non-linear operator N given by (3.4), using the monotonicity and the existence of a pair of a lower and an upper solutions of the Problem (1.17)–(1.19).

We consider the coupled system of elliptic boundary-value problems

−A

i

x

_i

= N

i

( x

₁

, x

₂

) in ] ^{0, 1} [ ^{, (4.4)} B

i

x

i

( z ) = 0, for z = 0, 1 ( i = 1, 2 ) , (4.5) where A

i

and B

i

are given, for all i = 1, 2, in the form

A

i

= − a

i

d dz + b

i

d

²

dz

²

B

i

x

_i

( z ) = − a

_i

dx

_i

( z )

dz + b

_i

( 1 − z ) x

_i

( z ) ,

and N = (N

1

, N

2

) is a function from a bounded subset J = J

₁

× J

₂

⊂ X into X and a

_i

, b

_i

are positive constants.

Recall that N

i

is said to be quasi-monotone non-increasing if for fixed x

_i

, N

i

is non-increasing in x

_j

for j = i . Similarly, N = (N

1

, N

2

)

is called quasi-monotone non-increasing in J if both N

1

and N

2

are quasi-monotone non-increasing for all ( x

₁

, x

₂

) ∈ J .

In the sequel, we assume that N is quasi-monotone non-increasing in J .

D

EFINITION

4.1 A pair of functions x ˜ = ( x ˜

₁

, x ˜

₂

) and x ˆ = ( x ˆ

₁

, x ˆ

₂

) in X ∩ C

²

( ] ^{0, 1} [ ) × C

²

( ] ^{0, 1} [ ) are called ordered upper and lower solutions of (4.4)–(4.5) if they satisfy the relation x ˜ x and if ˆ

A

1

x ˜

₁

+ N

1

(˜ x

₁

, x ˆ

₂

) ⁰ A

1

x ˆ

₁

+ N

1

( x ˆ

₁

, x ˜

₂

) A

2

x ˜

2

+ N

2

(ˆ x

1

, x ˜

2

) ⁰ A

2

x ˆ

2

+ N

2

( x ˜

1

, x ˆ

2

)

B

i

x ˜

_i

⁰ B

i

x ˆ

_i

, on z = 0, 1, i = 1, 2.

In order to prove the uniqueness of the solutions to Problem (4.4)–(4.5), we need these results.

T

HEOREM

4.2 (Pao, 1992, p. 534) Let x ˜ = (˜ x

₁

, x ˜

₂

) and x ˆ = (ˆ x

₁

, x ˆ

₂

) be ordered upper and lower solutions of (4.4)–(4.5) and let N be the C

¹

-function in J . Then there exists a minimal ( x

₁

, x ¯

2

) and a maximal ( x ¯

1

, x

₂

) steady-state solutions in J such that x ¯

1

^x

1

and x

₂

x ¯

2

. Moreover, if ( x

₁^∗

, x

₂^∗

) is any other solution in J , then it satisfies ( x

₁

, x ¯

2

) ( x

₁^∗

, x

₂^∗

) ( x ¯

1

, x

₂

) .

C

OROLLARY

4.1 (Pao, 1992, p. 535) ( x ¯

1

, x

₂

) = ( x

₁

, x ¯

2

) if and only if Problem (4.4)–(4.5) has a unique solution in J .

Now, we consider the non-linear coupled system (1.17)–(1.19). This system is a particular case of the System (4.4)–(4.5), with N = N , a

_i

= υ and b

_i

= d

_i

= d. The non-linear operator N is quasi- monotone non-increasing in E . Hence, we have

T

HEOREM

4.3 For d

₁

= d

₂

= d, the steady-state problem given by (1.17)–(1.19) has a unique solution x

_ss

in E.

Proof. By using (1.11) and Definition 4.1, it is easy to verify that the pair x ˜ =

C−Cin

Cin

,

^L−_L^Lin

in

and ˆ

x = (− 1, − 1 ) are ordered upper and lower solutions of problem given by (1.17)–(1.19). Moreover, observe that N is a C

¹

-function in E . Since this function is quasi-monotone non-increasing in E , hence by Theorem 4.2, there exists ( x

₁

, x ¯

₂

) and ( x ¯

₁

, x

₂

) in E such that x ¯

₁

^x

1

, x

₂

x ¯

₂

. Hence, by Corollary 4.1, we shall obtain the desired result showing that x ¯

₁

= x

₁

and x

₂

= ¯ x

₂

.

Let w

1

= ¯ x

₁

− x

₁

^{0 and} w

2

= ¯ x

₂

− x

₂

^{0. Then}

− A

₁

w

1

= k

₁

( g (¯ x

₁

, x

₂

) − g ( x

₁

, x ¯

₂

))

− A

2

w

2

= k

2

( g ( x

₁

, x ¯

2

) − g ( x ¯

1

, x

₂

)) , and, for z = 0, 1,

B w

i

( z ) = − d d w

i

( z )

dz + v( 1 − z )w

i

( z ) = 0 ( i = 1, 2 ) ,

where g ( x

₁

, x

₂

) = − C

_in^m

L

ⁿ_in

( x

₁

+ 1 )

^m

( x

₂

+ 1 )

ⁿ

. Multiplying the first equation by k

₂

and the second one by k

₁

, we get by addition of both equations

− d d

²

w dz

²

+ v d w

dz = 0, for z ∈ ] ^{0, 1} [ ^, and

B w( z ) = − d d w( z )

dz + v( 1 − z )w( z ) = 0, for z = 0, 1,

where w = k

₂

w

1

+ k

₁

w

2

. This system has a unique solution w = 0 in [ ^{0, 1} ] . The non-negative property of w

1

and w

2

implies that w

i

= 0, for i = 1, 2, which ensures the desired result.

As a consequence of the above theorem, we have the following stability result for (1.17)–(1.19).

T

HEOREM

4.4 Let d

₁

= d

₂

= d, for any x

₀

∈ E , the solution x of (1.12)–(1.15) converges to steady- state solution x

_ss

of (1.17)–(1.19), i.e. lim x ( t , z ) = x

_ss

( z ) as t → ∞ , for all z ∈ [ ^{0, 1} ] ^.

Proof. By Theorem 4.3, the solution of (1.17)–(1.19) is unique in E given by (4.3); then the convergence of the time-dependent solution of (1.12)–(1.15) to corresponding steady-state solution follows from Pao

(1992, Theorem 5.3, p. 538).

5. Conclusion

This paper has addressed the questions of the existence and the uniqueness of the state trajectories for a class of non-linear convection-diffusion-reaction systems. It has also been proved that the trajectories are positive and the set of physically meaningful admissible states is invariant under the dynamics of the reactions. In addition, it has been proved that there exists at least one steady state among the physi- cally feasible states for such systems. With additional assumptions, the uniqueness and the asymptotic stability analysis of steady states have been established in the set E .

An important question is the stability analysis of equilibrium profiles for System (1.12)–(1.15) in the set . A further open question is the study of the asymptotic behaviour of the state trajectories for System (1.1)–(1.5) when the inlet concentrations C

_in

and L

_in

are time varying. These questions are under investigation.

Acknowledgements

This paper presents research results of the Moroccan ‘PROgramme Th´ematique d’Appui `a la Recherche Scientifique’ PROTARS III, initiated by the Moroccan ‘Centre National de la Recherche Scientifique et Technique’. The work has been partially carried out within the framework of a collaboration agreement between CESAME (Universit´e Catholique de Louvain, Belgium) and LINMA of the Faculty of sciences (Universit´e Chouaib Doukkali, Morocco), funded by the Belgian Secretary of the State for Development Cooperation and by the CIUF (Conseil Interuniversitaire de la Communaut´e Franc¸aise, Belgium). The scientific responsibility rests with its authors. The work of the author is supported by a Research Grant from the Agence Universitaire de la Francophonie.

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Appendix

Appendix A. Proof of Lemma 3.2

Proof. It is sufficient to prove that N is lipschitz operator on (Martin, 1976, p. 245).

Let x = ( x

₁

, x

₂

)

and y = ( y

₁

, y

₂

)

be in , then

N ( x ) − N ( y )

²

= N

₁

( x ) − N

₁

( y )

²_L2

+ N

₂

( x ) − N

₂

( y )

²_L2

. (A.1) Now, let δ be arbitrarily fixed in the open interval ] ^0, +∞ [ ^{and define}

N

_δ

( x ) = −δ C

_in^m

L

ⁿ_in

( x

₁

+ 1 )

^m

( x

₂

+ 1 )

ⁿ

.

To obtain the desired result, it suffices to show that N

_δ

is a lipschitz operator on . Denote X

1

= C

in

( x

1

+ 1 ) , X

2

= L

in

( x

2

+ 1 ) , Y

1

= C

in

( y

1

+ 1 ) and Y

2

= L

in

( y

2

+ 1 ) . Let us assume α β and p 1. By applying the mean value theorem to g (λ) = λ

^p

on [ α , β ] ^{, there} exists a ζ ∈ (α , β) such that α

^p

− β

^p

= p ζ

^p−1

(α − β) . Moreover, if α ^0, β ^{0, we have}

|α

^p

− β

^p

| ^p |α − β |ζ

^p−1

^p |α − β| max { α

^p−1

, β

^p−1

} .

Putting

α = X

₁

( z ) , β = Y

₁

( z ) , for a.e. z ∈ [ ^{0, 1} ] ^{and p} = m, we get

| X

^m₁

( z ) − Y

₁^m

( z )| ^mC

in

| x

₁

( z ) − y

₁

( z )| max { X

₁^m−1

( z ) , Y

₁^m−1

( z )} , since

− 1 ^x

1

( z ) ^{C ¯ − C}

ⁱⁿ

C

in

and − 1 ^y

1

( z ) ^{C ¯ − C}

ⁱⁿ

C

in

, for almost all z ∈ [ ^{0, 1} ] ^. Hence, we get by using (1.11),

| X

^m₁

( z ) − Y

₁^m

( z )| ^mC

in

C ¯

^m−1

| x

₁

( z ) − y

₁

( z )| , for all m 1, for a.e. z ∈ [ ^{0, 1} ] ^. By similar considerations, as above, we also get

| X

ⁿ₂

( z ) − Y

₂ⁿ

( z )| ^{n L}

in

L ¯

ⁿ⁻¹

| x

₂

( z ) − y

₂

( z )| , for all n 1, for a.e. z ∈ [ ^{0, 1} ] ^. Whence,

N

_δ

( x ) − N

_δ

( y )

²_L2

= δ

²

X

^m₁

X

ⁿ₂

− Y

₁^m

Y

₂ⁿ

²_L2

= δ

²

X

^m₁

X

ⁿ₂

− X

₁^m

Y

₂ⁿ

+ X

₁^m

Y

₂ⁿ

− Y

₁^m

Y

₂ⁿ

²_L2

δ

²

C ¯

^2m

X

ⁿ₂

− Y

₂ⁿ

²_L2

+ δ

²

L ¯

²ⁿ

X

₁^m

− Y

₁^m

²_L2

ⁿ

²

δ

²

C ¯

^2m

L ¯

²ⁿ⁻²

L

²_in

x

₂

− y

₂

²_L2

+ m

²

δ

²

L ¯

²ⁿ

C ¯

^2m−2

C

_in²

x

₁

− y

₁

²_L2

. Consequently,

N

_δ

( x ) − N

_δ

( y )

²_L2

δ

²

C ¯

^2m−2

L ¯

²ⁿ⁻²

max { n

²

C ¯

²

L

²_in

, m

²

L ¯

²

C

_in²

} x − y

²

. (A.2) Therefore, N

_δ

is a lipschitz operator on , for all δ > 0. In particular, (A.2) holds for δ ∈ { k

1

, k

2

} (i.e.

N

k₁

( x ) = N

1

( x ) and N

k₂

( x ) = N

2

( x ) ). Finally, by using (A.1), we have

N ( x ) − N ( y ) ^max ( k

₁

, k

₂

) C ¯

^m−1

L ¯

ⁿ⁻¹

max { n C L ¯

_in

, m LC ¯

_in

} x − y . Consequently, N − l

_N

I is a dissipative operator on , where

l

_N

= max ( k

₁

, k

₂

) C ¯

^m−1

L ¯

ⁿ⁻¹

max { n C L ¯

_in

, m LC ¯

_in

} .