Mathematical analysis of a chemotaxis-type model of soil carbon dynamic.

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Mathematical analysis of a chemotaxis-type model of soil carbon dynamic.

Alaaeddine Hammoudi, Oana Iosifescu

To cite this version:

Alaaeddine Hammoudi, Oana Iosifescu. Mathematical analysis of a chemotaxis-type model of soil

carbon dynamic.. Chinese Annals of Mathematics - Series B, Springer Verlag, 2018, 39 (2), pp.253-

280. �10.1007/s11401-018-1063-7�. �hal-01818727�

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Mathematical analysis of a chemotaxis-type model of soil carbon dynamic.

Alaaeddine Hammoudi

^∗

, Oana Iosifescu

^∗

Abstract

The goal of this paper is to study the mathematical properties of a new model of soil carbon dynamics which is a reaction-diffusion system with a chemotactic term, with the aim to account for the formation of soil aggregations in the bacterial and microorganism spatial orga- nization (hot spot in soil). This is a spatial and chemotactic version of MOMOS (Modelling Organic changes by Micro-Organisms of Soil), a model recently introduced by M. Pansu and his group. We present here two forms of chemotactic terms, first a “classical” one and second a function which prevents the overcrowding of microorganisms. We prove in each case the existence of a nonnegative global solution, we investigate its uniqueness and the existence of a global attractor for all the solutions.

Keywords: Soil organic carbon dynamics, reaction-diffusion-advection system, positive weak solutions, periodic weak solutions.

Mathematics Subject Classification 2000: 35B09, 35B10, 35D30, 35K51, 35Q92.

1 Introduction.

Chemotaxis is the ability of some bacteria to direct their movement ac- cording to the gradient of chemicals contained in their environment. In soil, some bacteria microorganisms that degrade organic carbon (SOC) are motile and chemotactic. This phenomenon is observed in experiments [1] and on field. Nevertheless to our best knowledge no model of terrestrial carbon cy- cle adresses this issue. Indeed, these models are essentially compartimental

∗Institut de mathématiques - IMAG, Université Montpellier, Case courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France, alaaeddine.hammoudi@univ- montp2.fr, iosifescu@math.univ-montp2.fr

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corresponding naturally to systems of ordinary differential equations (e.g Century, RothC, MOMOS) [2]. They are used globally to estimate soil CO

₂

emissions in local land management and crop optimization, among other things.

Very few prototypes of spatial soil organic model have been proposed. Some of them use systems of partial differential equations: Balesdent et al. [3]

combined vertical directed transport of organic carbon with a degradation phenomenon and diffusion. More recently, Goudjo et al. [4] proposed a three dimensional model for dissolved organic matter using also a system of PDEs. Other authors opted for a finite sequence of interconnected systems of ordinary differential equations each localized in a soil layer (see [5]).

We previously studied the model MOMOS proposed by Pansu (see [6],[7]), which is a nonlinear system of ordinary differential equations [8] written as:

y ˙ = g(t, y) where

y =



 u v w





and

G(t, y) =





−k

₁

(t)u − q(t)u

²

+ k

2

(t)v + k

3

(t)w + f (t) k

₁

(t)u − (k

₂

(t) + k

₄

(t))v

k

₄

(t)v − k

₃

(t)w.





In these equations the unknown u models the alive microbial biomass, whereas the unknowns v and w are soil organic matters with distinct decomposition rates.

In reality, the nonnegative functions k

_i

i ∈ {1, 2, 3, 4}, q and f depend not only on time but also on space because of the variability in soil clay con- tent. The phenomena described by MOMOS can also be subjected to the influence of transport and sedimentation through transport and diffusion.

In order to test the effect of soil heterogeneity we studied in [9] the following reaction-diffusion-advection initial problem:



 

 

∂ui

∂t

− div(A

_i

(t, x)∇u

_i

) + B

_i

(t, x)∇u

_i

= g

_i⁺

(t, x, u), (t, x) ∈ Q

_T

:= (0, T ) × Ω γ (A

i

(t, x)∇u

_i

) · ν + β

i

(t, x)u

i

= 0, (t, x) ∈ Σ

T

:= (0, T ) × ∂Ω

u

_i

(0) = u

_i,0

in Ω

where Ω is a domain in R

ⁿ

representing the soil, A

_i

is a diffusion matrix

and B

_i

a transport vector, for each i = 1, 2, 3.

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In [9] the boundary conditions were either of Dirichlet type (γ = 0, β

i

≡ 1) or of Neumann-Robin type (γ = 1, β

_i

(t, x) ≥ 0). The right hand side term of (1.1) was :

g

⁺

(t, x, u) :=





−k

₁

(t, x)u

₁

− q(t, x)|u

₁

|u

₁

+ k

₂

(t, x)u

₂

+ k

₃

(t, x)u

₃

+ f(t, x) k

₁

(t, x)u

₁

− (k

₂

(t, x) + k

₄

(t, x))u

₂

k

4

(t, x)u

2

− k

3

(t, x)u

3





where we replaced the term q(t, x)u

²₁

with q(t, x)|u

₁

|u

₁

for more accuracy, since q(t, x)u

1

corresponds to a kinetic coefficient that cannot be negative.

We assumed there that the diffusion matrices A

_i

were bounded, symmetric and coercive:

A

i

∈ L

^∞

(Q

T

)

^n×n

,

A

i

(t, x)ζ · ζ ≥ c|ζ|

²

, ∀ζ ∈ R

ⁿ

, a.e in Q

T

, with c > 0 and the transport vectors B

_i

were bounded on Q

_T

:

B

i

∈ L

^∞

(Q

T

)

^N

, |(B

_i

(t, x))

j

| ≤ c

max

a.e in Q

T

, for all 1 ≤ j ≤ n.

Also we assumed that the functions k

j

, β

i

and q were nonnegative and bounded, i.e. for all j = 1, 2, 3, 4 and i = 1, 2, 3

k

_j

, q ∈ L

^∞

(Q

_T

), 0 ≤ k

_j

(t, x), q(t, x) ≤ C

_max

a.e on Q

_T

, β

i

∈ L

^∞

(Σ

T

), 0 ≤ β

i

(t, x) ≤ C

max

a.e on Σ

T

,

where constant C

_max

> 0. Finally we assumed that the initial data and input were nonnegative and bounded :

u

i,0

∈ L

²₊

(Ω), f ∈ L

²₊

(Q

_T

), f (t, x) ≤ C

max

a.e on Q

_T

.

In [9] we proved first that this model had a unique weak solution. We were looking for weak solutions, because initial inputs were not regular enough to give rise to more “regular” solutions. Second, for periodic data, we proved the existence of a maximal and a minimal periodic solution of this system.

In some particular cases, the minimal and the maximal periodic solutions coincide and this function becomes a global attractor for any bounded solution of the periodic system.

In the present work a new PDEs model is considered to take account of

chemotaxis. The chemotactic movement of bacteria to root exudates is well

known to play an important role in rhizosphere colonisation. Field studies

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with tracers and laboratory experiments using soil columns were both used to demonstrate the effect of chemotaxis on microbial movements. So, the model proposed here can represent the spatial heterogeneity of soil microbial biomass, highlighted by recent observations at submicron scale [1]

The new model derived from a simplified MOMOS ODEs model, which comprised only two differential equations instead of the three originally, where the microbial biomass was u and the organic matter was v. As additional simplifying hypothesis soil temperature, soil moisture, soil texture and organic input were considered to be isotropic and constant with time. Hence, the simplified ODEs model can be expressed as:

( u ˙ = −k

₁

u − qu

²

+ k

2

v

˙

v = −k

₂

v + k

₁

u + f

with the initial conditions (u

₀

, v

₀

) , where k

₁

is the microbial mortality rate, k

2

is the soil carbon degradation rate, q is the metabolic quotient and f is the soil organic carbon input. It can be proven that the unique positive steady state (u

^∗₀

, v

₀^∗

), is stable [8].

The chemotaxis-type model was finded following the conventional Keller- Segel approach [10], using an advection-diffusion system. This comprised two parabolic equations in a smooth domain with no-flux boundary conditions. The advection term was controlled by the gradient of the chemo- attractant. Applying the same principles to our problem leads to the following reaction-diffusion-chemotaxis system (P

_h

)



 



 



∂

_t

u − a∆u = −βdiv(h(u)∇v) − k

₁

u − q|u|u + k

₂

v, (t, x) ∈ Q

_T

∂

_t

v − d∆v = −k

₂

v + k

₁

u + f, (t, x) ∈ Ω

_T

,

∇u · ν = ∇v · ν = 0, (t, x) ∈ Σ

T

, u(0) = u

₀

, in Ω,

v(0) = v

0

, in Ω.

(P

_h

)

The parameter β is the chemotaxis sensitivity, a and d are the diffusion coefficients of microbes and soil organic carbon respectively, Ω is a smooth and bounded domain, and h(·) is a continuous function, involved in the modelling of chemotaxis. As bacteria can release exoenzymes to avoid overcrowding, the function h can be selected to limit overcrowding, as required.

This new model is, therefore, a new variation of the Keller-Segel approach

[10] with the reaction part modified to fit the MOMOS model. In the first

equation of (P

_h

), we change again the term qu

²

by q|u|u, see [9].

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We prove here (see Appendix 1) the existence of Turing patterns that may provide possible explanations for the formation of soil aggregations, for the bacterial and micro-organisms spatial organizations (hotspots in soil) or jus- tify the formation of the microscopic patterns observed by Vogel et al. [1].

Although spatial heterogeneity can be verified visually in a numerical sim- ulation (see Appendix 2), formal mathematical analysis is required to con- firm its emergence and to provide a mathematical proof of the necessary conditions. The mathematical criteria are based on matrices derived from equations and analysed using conditions on the determinant, trace and eigen- values.

Keller-Segel model was the earliest mathematical system involving chemotaxis [10]. Many others models emerged specially in biology and ecology.

Most authors focused their efforts essentially on existence and on asymp- totic behaviour of solutions in one or two dimensional domains in order to avoid blow-up of solutions (see [11], [12], [13], [14], [15] and references therein).

Unlike the classical Keller-Segel model, where equations are coupled only by the chemotactic term, the system of partial differential equations (P

h

) is also coupled through the reaction term. More specifically, the organic matter will not only attract microorganisms, but part of it will be “trans- formed”, under a degradation process, to microorganisms. This mechanism introduces a supplementary linear coupling term in the first equation of this model. Many authors ([11], [12], [13] and references therein) already considered reaction coupling terms, but under some restrictive conditions, which are not verified here. This feed-back in the chemotactic equation is not compatible with mass conservation of microorganisms, unlike in [12], [15].

Furthermore, neither the boundedness of the microorganisms total mass nor the positivity and the boundedness (existence of threshold) of the solution remain immediate, unlike in [13], [11] and [14].

Our main concern here is to prove the existence of a unique solution to this minimal MOMOS model improved by adding chemotaxis effect. We consider two chemotactic functions h, a “classical” one, h(u) = u, and a second one which prevents overcrowding of microorganisms, h(u) = u(M − u) if 0 ≤ u ≤ M and zero otherwise, proposed by Wrzosek [15].

This paper is organized as follows. Section 2 introduces some notations,

results and tools used throughout the paper. Section 3 presents sufficient

conditions to get global solutions, and to prove the existence of an exponen-

tial attractor, in the case where h(u) = u. Section 4 is concerned with the

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second chemotactic function, where the chemotactic term cancels when u achieves the threshold M, which helps to prove that any local solution is ac- tually global. In sections 3 and 4 the domain is two dimensional. In Section 5, still keeping the second form of h and for domains of dimension less than or equal to 3 (the dimension 3 is particularly interesting in applications), we prove the existence of a unique solution, with less restrictions on the initial conditions and forcing term than in section 4. In Appendix 1we prove that chemotactic term in system (P

_h

) is mandatory to obtain Turing patterns and in Appendix 2 we give some numerical simulations.

2 Mathematical preliminary and notations.

Unless it is explicitly indicated, Ω is a bounded region in R

²

of C

³

class, the constants a, β, q, d, k

1

and k

2

are nonnegative, and f is a nonegative function belonging to an admissible space to be fixed later. In all that follows C denotes a positive constant which may vary from line to line.

We recall here some known results (see [16], [17] and references therein) that will help afterwards.

Interpolation space:

For 0 ≤ s

₀

< s < s

₁

< ∞, H

^s

(Ω) is the interpolation space [H

^s⁰

(Ω), H

^s¹

(Ω)]

_θ

with s = (1 − θ)s

₀

+ θs

₁

between H

^s⁰

(Ω) and H

^s¹

(Ω). Furthermore, we have

k.k

_H^s

≤ k.k

^1−θ_Hs0

k.k

^θ_Hs1

. (2.1) Embedding theorem:

When 0 < s < 1, H

^s

(Ω) ⊂ L

^p

(Ω) for

¹_p

=

^1−s₂

with the estimate k · k

_L^p

≤ C

s

k · k

_H^s

When s = 1, H

¹

(Ω) ⊂ L

^q

(Ω) for any 1 ≤ q < ∞ and k · k

_Lq

≤ C

_q,p

k · k

^1−p/q

H¹

k · k

^p/q_Lp

(2.2) where 1 ≤ p ≤ q < ∞.

When s > 1 H

^s

(Ω) ⊂ C(Ω) with continuous embedding.

Fractional Power of the Laplace operator: (see ([17], Chap 2.7), [12])

Let a

₀

, a

₁

> 0 be constants and L = −a

₁

∆ + a

₀

be the Laplace operator

equipped with the Newman boundary conditions, with the domain D(L) =

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{u ∈ H

²

(Ω);

^∂u_∂ν

= 0 on ∂Ω} = H

_N²

(Ω). Thus L is a positive definite self- adjoint operator of L

²

(Ω). For θ > 0, the fractional power on L is defined and noted L

^θ

and L

^θ

is also a positive definite self-adjoint operator on L

²

(Ω).

More

D(L

^θ

) =

( H

^2θ

(Ω), 0 ≤ θ <

³₄

, H

_N^2θ

(Ω),

³₄

< θ ≤

³₂

, with the norm equivalence.

Some useful inequalities Biler’s lemma (see [18])

Let 0 ≤ u ∈ H

¹

(Ω) and N

_log¹

(u) := k(u + 1)log(u + 1)k

_L1

. For any η > 0, kuk

³_L3

≤ ηkuk

_H1

N

_log¹

(u) + p(η

⁻¹

)kuk

_L1

, (2.3) where p(·) denotes here some increasing function.

Let ε ∈ (0, 1]. It is proved in ([13], (2.10) ÷ (2.12)) that:

k∇(u∇v)k

_L2

≤ C

_ε

kuk

_H1

kvk

_H2+ε

, for all u ∈ H

¹

(Ω), v ∈ H

^2+ε

(Ω) (2.4) k∇(u∇v)k

_L2

≤ C

ε

kuk

_H1+ε

kvk

_H2

, for all u ∈ H

^1+ε

(Ω), v ∈ H

²

(Ω) (2.5) k∇(u∇v)k

_H1

≤ Ckuk

_H2

kvk

_H3

, for all u ∈ H

²

(Ω), v ∈ H

³

(Ω). (2.6) Local Existence

We need first to prove the existence of local solution of (P

_h

). For this purpose, we use the result obtained by Yagi and based on the Galerkin method (see [12], [17]).

Let V and H be seperable Hilbert spaces with dense and compact embedding V ⊂ H. Let V

⁰

be the dual space of V and identify H and H

⁰

to get:

V ⊂ H ⊂ V

⁰

The duality product between V and V

⁰

is denoted by < ·, · >. It coincides with the scalar product on H denoted by (·, ·).

Consider the following Cauchy problem of a semilinear abstract differential equation:

dU

dt + AU = G(U ) + F (t), 0 < t ≤ T

U (0) = U

0

, (2.7)

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in the space V

⁰

.

Here, A is the positive definite self-adjoint operator of H defined by a symmetric sesquilinear form a(U, U ˜ ) on V , with < AU, U > ˜

_V,V⁰

= a(U, U ˜ ).

Assumptions on a(·, ·):

(a.i) ka(U, U ˜ )k

_H

≤ M kU k

_V

k U ˜ k

_V

, U, U ˜ ∈ V, (a.ii) a(U, U) ≥ δkU k

²_V

, U ∈ V,

with constants δ, M > 0. The operator A is also bounded from V to V

⁰

. Assumptions on G(·):

G(.) is a continuous function from V to V

⁰

, which satisfy:

(g.i) For each ζ > 0, there exists an increasing continuous function φ

ζ

: [0, ∞) → [0, ∞) such that:

kG(U )k

_V⁰

≤ ζkU k

_V

+ φ

ζ

(kU k

_H

), U ∈ V

(g.ii) For each ζ > 0, there exists an increasing continuous function ψ

_ζ

: [0, ∞) → [0, ∞) such that:

kG(U ) − G( ˜ U )k

_V⁰

≤ ζkU − U ˜ k

_V

+ ψ

_ζ

(kUk

_H

+ k U ˜ k

_H

) × kU − U ˜ k

_H

(kU k

_V

+ k U ˜ k

_V

+ 1), U, U ˜ ∈ V Finally F (·) ∈ L

²

(0, T ; V

⁰

) is a given function and U

0

∈ H is an initial value.

Then, we have [12]:

Theorem 2.1. Under assumptions (a.i), (a.ii), (g.i) and (g.ii) and for ev- ery F(·) ∈ L

²

(0, T ; V

⁰

) and U

₀

∈ H, there exists a unique local solution U of (2.7) such that:

U ∈ H

¹

(0, T (U

₀

, F ); V

⁰

) ∩ C([0, T (U

₀

, F )]; H) ∩ L

²

(0, T (U

₀

, F ), V ) Here T (U

₀

, F ) is determined by the norm kU

₀

k

_H

and kF k

_L2(0,T;V⁰)

3 First case: h(u) = u.

3.1 Local existence and positivity.

Let ε

0

arbitrarily fixed, ε

0

∈ (0, 1). Then:

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Theorem 3.1. Let u

0

∈ L

²

(Ω), v

0

∈ H

^1+ε⁰

(Ω) and f ∈ L

²

(0, T ; H

^ε⁰

(Ω)) be nonnegative functions. Then (P

_h

) has a unique nonnegative local solution on an interval [0, T

₀

] such that

u ∈ H

¹

(0, T

0

; H

¹

(Ω)

⁰

) ∩ C([0, T

0

]; L

²

(Ω)) ∩ L

²

(0, T

0

, H

¹

(Ω)), v ∈ H

¹

(0, T

0

; H

^ε⁰

(Ω)) ∩ C([0, T

0

]; H

^1+ε⁰

(Ω)) ∩ L

²

(0, T

0

, H

_N^2+ε⁰

(Ω)), where T

0

depends only on kfk

_L2(0,T;H^ε⁰(Ω))

, ku

₀

k

_L2(Ω)

and kv

₀

k

_H1+ε0(Ω)

. Proof. First Step: Construction of a unique local solution

Let A

1

= −a∆ + k

1

and A

2

= −d∆ + k

2

be two operators with the same domain H

_N²

(Ω). A

₁

and A

₂

are positive self-adjoint operators on L

²

(Ω). We can then define their corresponding fractional power operators (see [17], as described in the previous section.

Let V = H

¹

(Ω) × H

_N^2+ε⁰

(Ω) and H = L

²

(Ω) × H

^1+ε⁰

(Ω). Identifying H with its dual space gives: V ⊂ H = H

⁰

⊂ V

⁰

and V

⁰

= (H

¹

(Ω))

⁰

× H

^ε⁰

(Ω) with the duality product:

< U, U > ˜

V,V⁰

= < u, u > ˜

_H¹_,(H¹₎⁰

+ < A

¹⁺

ε0 2

2

v, A

ε0 2

2

v) ˜ >

_L²_,L²

, where U = (u, v) ∈ V and ˜ U = (˜ u, v) ˜ ∈ V

⁰

.

We also set a symmetric sesquilinear form on V × V , a(U, U ˜ ) =

Z

Ω

{a∇u · ∇˜ u + k

₁

u˜ u}dx + (A

¹⁺

ε0 2

2

v, A

¹⁺

ε0 2

2

v) ˜

_L²

for U = (u, v) and ˜ U = (˜ u, v) ˜ ∈ V .

This forms is in fact a linear isomorphism A from V to V

⁰

: A =

A

1

0 0 A

₂

and A becomes a positive definite self-adjoint operator in H.

Finally let f (·) ∈ L

²

(0, T, H

^ε⁰

(Ω)) and let G : V → V

⁰

be the mapping:

G(U ) :=

β∇(u∇v) − q|u|u + k

₂

v k

1

u

, with U = (u, v) ∈ V .

Then (P

_h

) is the following semilinear differential equation:

dU

dt + AU = G(U ) + F (t) in V

⁰

, 0 < t ≤ T, (3.1)

U (0) = U

₀

,

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where F (t) = 0

f (t)

.

In order to apply the existence result of Theorem 2.1. to problem (3.1), let us verify the assumptions on a(·, ·) and G(·).

The assumptions on a(·, ·) are classically satisfied (see for example [12]).

For the conditions on G we have that for an arbitrary U = (u, v) ∈ V and δ > 0:

k∇ · (u∇v)k

_(H1)⁰

≤ Ckuk

_L4

k∇vk

_L4

≤ kuk

1 2

L²

kuk

1 2

H¹

kvk

1 2

H¹

kvk

1 2

H²

≤ kuk

1 2

L²

kuk

1 2

H¹

kvk

1+ε0 2

H^1+ε⁰

kvk

1−ε0 2

H^2+ε⁰

≤ CkU k

¹⁺

ε0 2

H

kU k

¹⁻

ε0 2

V

≤ ζ kU k

_V

+ φ

_ζ

(kU k

_H

) and

kvk

_(H1)⁰

≤ CkU k

_H

, ku

²

k

_(H1)⁰

≤ Ckuk

²_L3

≤ ζkuk

_H1

+ φ

_ζ

(kuk

_L2

).

Finally it is clear that:

kuk

_H^ε0

≤ ζkuk

_H1

+ C

_ζ

(kuk

_L2

).

All these inequalities show that the condition (g.i) is fullfiled.

From the embedding theorem we have:

| Z

Ω

(˜ u − u)∇v · ∇ρ dx| ≤ Ck˜ u − uk

_L2

k˜ vk

_H2+ε0

kρk

_H1

| Z

Ω

∇(˜ v − v)u · ∇ρ dx| ≤ Ckuk

_H1

k˜ v − vk

_H1+ε0

kρk

_H1

and using the interpolation theorem and Young inequality we obtain:

ku − uk ˜

_H^ε0

≤ CkU − U ˜ k

^ε_V⁰

kU − U ˜ k

^1−ε_H ⁰

≤ ζkU − U ˜ k

_V

+ C

_ζ

kU − U ˜ k

_H

for an arbitrary ζ > 0. On the other hand we have that:

ku|u| − u|˜ ˜ u|k

_(H1)⁰

≤ C

k(|u| − |˜ u|)uk

_L3/2

+ k(u − u)|˜ ˜ u|k

_L3/2

≤ Cku − uk ˜

_L2

kuk

_L6

+ k˜ uk

_L6

All these inequalities permit to show that condition (g.ii) is fullfiled too.

Second Step: Positivity of the solution

Now let us take the following semilinear system:

dU

dt

+ AU = ˜ G(U ) + F(t), 0 < t ≤ T

U (0) = U

0

, (3.2)

(12)

where A, F and Y

0

are defined as previously, and the mapping ˜ G : V → V

⁰

is defined by:

G(U ˜ ) :=

β∇(u∇v) − q|u|u + k

₂

|v|

k

1

u

.

By Theorem 2.1. there exists a local solution U = (u, v) on [0, T

₀

] × Ω, with T

₀

depending only on U

₀

and F . Let us define u

⁺

= max(u, 0) and u

⁻

= max(−u, 0).

We multiply the first equation by −u

⁻

and we integrate in space. So 1

2 d

dt ku

⁻

k

²_L2

+ ak∇u

⁻

k

²_L2

+ k

1

ku

⁻

k

²_L2

≤ Z

Ω

βu

⁻

∇v∇u

⁻

dx

≤ βk∇vk

_L^∞

Z

Ω

u

⁻

|∇u

⁻

|dx for 0 < t ≤ T

0

.

Using Young inequality we get 1

2 d

dt ku

⁻

k

²_L2

+ ak∇u

⁻

k

²_L2

≤ k∇vk

_L^∞

C

ε

ku

⁻

k

²_L2

+ εk∇u

⁻

k

²_L2

with ε > 0 small enough and C

_ε

> 0.

Taking ε =

_k∇vk^a

L∞

we get d

dt ku

⁻

k

²_L2

≤ Ck∇vk

²_L∞

ku

⁻

k

²_L2

≤ Ckvk

²_H2+ε0

ku

⁻

k

²_L2

Since v ∈ L

²

(0, T

₀

; H

^2+ε⁰

(Ω)) and ku

⁻₀

k

²_L₂

= 0 by Gronwall Lemma we deduce that u is nonnegative on [0, T

₀

]. By classical results on linear parabolic equations v is nonnegative on [0, T

0

] too. So, the nonnegative solution U of (3.2) is also a solution of (3.1).

Remark. If initial conditions U

₀

and data f are not positive, this theorem proves anyway the existence of a local solution. However, as this is an ecology model, only nonnegative solutions make sense.

With minor changes due to our different problem (P

h

),we prove as in [13]

the following theorems.

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Theorem 3.2. Let U

0

= (u

0

, v

0

) ∈ H

¹

(Ω)×H

_N²

(Ω) and f ∈ L

²

(0, T ; H

¹

(Ω)).

Then there exists a unique local solution U = (u, v) of (3.1) on an interval [0, T

_U₀_,f

] such that

u ∈ H

¹

(0, T

_U₀_,f

; L

²

(Ω)) ∩ C([0, T

_U₀_,f

]; H

¹

(Ω)) ∩ L

²

(0, T

_U₀_,f

, H

_N²

(Ω)), v ∈ H

¹

(0, T

U0,f

; H

¹

(Ω)) ∩ C([0, T

U0,f

]; H

_N²

(Ω)) ∩ L

²

(0, T

U0,f

, H

_N³

(Ω)), where T

_U₀_,f

is determined by kf k

_L2(0,T;H¹(Ω))

, ku

₀

k

_H1(Ω)

and kv

₀

k

_H2(Ω)

. Theorem 3.3. Let U

₀

= (u

₀

, v

₀

) ∈ H

_N²

(Ω)×H

_N³

(Ω) and f ∈ L

²

(0, T ; H

_N²

(Ω)).

Then there exists a unique local solution U = (u, v) to (3.1) on an interval [0, T

_U₀_,f

] such that:

u ∈ H

¹

(0, T

U0,f

; H

¹

(Ω)) ∩ C([0, T

U0,f

]; H

_N²

(Ω)) ∩ L

²

(0, T

U0,f

, H

_N³

(Ω)), v ∈ H

¹

(0, T

_U₀_,f

; H

_N²

(Ω)) ∩ C([0, T

_U₀_,f

]; H

_N³

(Ω)) ∩ L

²

(0, T

_U₀_,f

, D(A

²₂

)(Ω)), where T

_U₀_,f

is determined by kf k

_L2(0,T;H²(Ω))

, ku

₀

k

_H2(Ω)

and kv

₀

k

_H3(Ω)

. 3.2 Global existence.

This section is devoted to proving the following result:

Theorem 3.4. Let ε

₀

∈ (0, 1) and let u

₀

∈ L

²

(Ω), v

₀

∈ H

^1+ε⁰

(Ω) and f ∈ L

²

(0, T, H

¹

(Ω)) ∩L

^∞

(0, T ; L

²

(Ω)) be nonnegative functions. Then there exists a unique global and nonnegative solution (u, v) for the system (P

_h

) with h(u) = u such that:

u ∈ H

¹

(0, T ; (H

¹

(Ω))

⁰

) ∩ C([0, T ]; L

²

(Ω)) ∩ L

²

(0, T ; H

¹

(Ω)) v ∈ H

¹

(0, T ; H

^ε⁰

(Ω)) ∩ C([0, T ]; H

^1+ε⁰

(Ω)) ∩ L

²

(0, T ; H

_N^2+ε⁰

(Ω)).

Proof. We proceed in two steps.

First Step

We show that kvk

_H1(Ω)

and N

_log¹

(u) = k(u + 1)log(u + 1)k

_L1(Ω)

are bounded for all t ∈ [0, T

₀

].

We consider the function log(u + 1); since ∇log(u + 1) =

_u+1^∇u

, it follows that log(u + 1) ∈ L

²

(0, T

0

; H

¹

(Ω)). Noting that

d dt

Z

Ω

{(u + 1)log(u + 1) − u} dx = < du

dt , log(u + 1) >

_H1,(H¹)⁰

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we obtain from the first equation of (P

h

) multiplied by log(u + 1) that:

d dt

Z

Ω

{(u + 1)log(u + 1) − u}dx + 4a Z

Ω

|∇ √

u + 1|

²

dx

= β Z

Ω

u

u + 1 ∇u∇v dx + Z

Ω

(−k

₁

u − qu

²

+ k

2

v) log(u + 1)dx.

So using Stokes theorem we deduce:

Z

Ω

u

u + 1 ∇u∇v dx = Z

Ω

(log(u + 1) − u)∆v dx ≤ η

2 k∆vk

²_L2(Ω)

+ 1

2η kuk

²_L2(Ω)

. Since

(k

1

u + qu

²

)log(u + 1) ≥ k

1

((u + 1)log(u + 1) − u), if we denote Ψ(t) = k(u + 1)log(u + 1) − uk

_L1(Ω)

, we get

d

dt Ψ(t) + k

₁

Ψ(t) ≤ η

2 k∆vk

²_L2(Ω)

+ ( k

₂²

2ε + β

²

2η )kuk

²_L2(Ω)

+ ε

2 kvk

²_L2(Ω)

, with arbitary ε, η > 0.

From the second equation of (P

_h

) multiplied respectively by v and ∆v we obtain that

1 2

d dt

Z

Ω

v

²

+ d Z

Ω

|∇v|

²

dx + k

₂

Z

Ω

v

²

dx = k

₁

Z

Ω

u v dx + Z

Ω

vf dx

≤ k

1

A 2 + B

2 kvk

²_L2(Ω)

+ k

1

2A kuk

²_L2(Ω)

+ 1

2B kfk

²_L2(Ω)

, with arbitrary A, B > 0, and

1 2

d dt

Z

Ω

|∇v|

²

+ d Z

Ω

|∆v|

²

dx + k

₂

Z

Ω

|∇v|

²

dx = k

₁

Z

Ω

u∆v dx + Z

Ω

f ∆v dx

≤ k

₁

C 2 + D

2 k∆vk

²_L2(Ω)

+ k

₁

2C kuk

²_L2(Ω)

+ 1

2D kf k

²_L2(Ω)

, with arbitrary C, D > 0.

Choosing ε = d, η = k

2

, A =

_2k^k²

1

, B =

^k₂²

, C =

_2k^d

1

, D =

^d₂

, we deduce d

dt (Ψ(t) + kvk

²_H1(Ω)

) + d

2 k∆vk

²_L2(Ω)

+ k

2

kvk

²_H1(Ω)

+ k

1

Ψ(t)

≤ k

₁²

k

2

+ β

²

2k

2

+ k

₂²

2d + k

₁²

d

kuk

²_L2(Ω)

+ ( 1 k

2

+ 1

d )kf k

²_L2(Ω)

. (3.3)

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By addition of the first two equation of (P

h

) it follows that:

d

dt kuk

_L1(Ω)

+ kvk

_L1(Ω)

+ qkuk

²_L2(Ω)

= kf k

_L1(Ω)

, (3.4) which implies that for all t ∈ [0, T

0

] we have the inequality :

ku(t)k

_L1(Ω)

+ kv(t)k

_L1(Ω)

≤ ku

₀

k

_L1(Ω)

+ kv

₀

k

_L1(Ω)

+ Z

t

0

kf (s)k

_L1(Ω)

ds (3.5) As N

_log¹

(u) = R

Ω

(u + 1)log(u + 1) dx we have Ψ(t) = N

_log¹

(u(t)) − ku(t)k

_L1(Ω)

. Let denote δ := max

1,

¹_q ^k_k²¹

2

+

_2k^β²

2

+

^k_2d²²

+

^k_d²¹

and σ := min(k

₁

, k

₂

) > 0.

Therefore from (3.3), (3.4) and (3.5) we obtain the following inequality:

d dt

N

_log¹

(u(t)) + kv(t)k

²_H1(Ω)

+ (δ − 1)ku(t)k

_L1(Ω)

+ δkv(t)k

_L1(Ω)

+σ N

_log¹

(u(t)) + kv(t)k

²_H1(Ω)

+ (δ − 1)ku(t)k

_L1(Ω)

+ δkv(t)k

_L1(Ω)

≤ σ(δ + 1)(ku

₀

k

_L1(Ω)

+ kv

₀

k

_L1(Ω)

) +δkf (t)k

_L1(Ω)

+ ( 1

k

₂

+ 1

d )kf (t)k

²_L2(Ω)

+ σδ Z

_t

0

kf (s)k

_L1(Ω)

ds

We denote by g(t) = N

_log¹

(u(t))+kv(t)k

²_H1(Ω)

+(δ−1)ku(t)k

_L1(Ω)

+δkv(t)k

_L1(Ω)

. Since g(t) satisfies the following ordinary differential inequality:

g

⁰

(t) + σg(t)

≤ σ(δ + 1)(ku

₀

k

_L1(Ω)

+ kv

₀

k

_L1(Ω)

) +δkf (t)k

_L1(Ω)

+ ( 1

k

₂

+ 1

d )kf(t)k

²_L2(Ω)

+ σδ Z

t

0

kf (s)k

_L1(Ω)

ds = C, with C > 0 depending only on kfk

_L∞(0,T;L²(Ω))

, ku

₀

k

_L2(Ω)

and kv

₀

k

_H1+ε0(Ω)

, we get

g(t) ≤ e

^−σt

g(0) + C, for all t ≥ 0. (3.6) Thus the inequality

N

_log¹

(u(t)) + kv(t)k

²_H1(Ω)

≤ N

_log¹

(u

0

) + kv

₀

k

²_H1(Ω)

+ (δ − 1)ku

₀

k

_L1(Ω)

+ δkv

₀

k

_L1(Ω)

+ C, (3.7)

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holds for all t ∈ [0, T

0

], where the last constant C > 0 is independent of T

0

and depends only on kfk

_L∞(0,T;L²(Ω))

, ku

₀

k

_L2(Ω)

and kv

₀

k

_H1+ε0(Ω)

. Second Step.

We take t

1

∈ (0, T

0

) so that v(t

1

) ∈ H

_N²

(Ω) and u(t

1

) ∈ H

¹

(Ω) and we set u(t

₁

) = u

₁

and v(t

₁

) = v

₁

. From Theorem 3.1 we allready know that such a time t

₁

exists, arbitrary small. In this step t varies in [t

₁

, T

₀

]. From the first equation of (P

h

) we have:

1 2

d

dt kuk

²_L2

+ ak∇uk

²_L2

+ k

1

kuk

²_L2

+ qkuk

³_L3

= Z

Ω

uv dx + β 2

Z

Ω

u

²

∆v dx.

From Young inequality and interpolation inequality (2.1) we get Z

Ω

u

²

∆v dx ≤ ηk∆vk

³_L3

+ η

^−1/2

kuk

³_L3(Ω)

≤ ηCkvk

²_H3

kvk

¹_H1

+ η

^−1/2

kuk

³_L3

, with η > 0 arbitrary Therefore (3.7) together with this yields that

Z

Ω

u

²

∆v dx ≤ ηCkvk

²_H3

+ η

^−1/2

kuk

³_L3

, In addition

Z

Ω

uv dx ≤ χkuk

³_L3(Ω)

+ χ

^−1/2

kvk

^3/2_H₁_(Ω)

, with χ > 0 arbitrary . Using Biler’s Lemma (2.3) we verify from (3.7) that

kuk

³_L3(Ω)

≤ ηC kuk

²_H1

+ p(η

⁻¹

),

with p a positive increasing function, depending on kf k

_L∞(0,T;L²(Ω))

, ku

₀

k

_L2(Ω)

and kv

₀

k

_H1+ε0(Ω)

as well as the constant C > 0.

Thus we deduce the following inequality 1

2 d

dt kuk

²_L2

+ ak∇uk

²_L2

+ k

1

ku(t)k

²_L2

+ qkuk

³_L3

(3.8)

≤ ξ(kvk

²_H3

+ kuk

²_H1

) + p(ξ

⁻¹

),

with p a positive increasing function depending on kf k

_L∞(0,T;L²(Ω))

, ku

₀

k

_L2(Ω)

and kv

₀

k

_H1+ε0(Ω)

, ξ > 0 an arbitrary constant.

On the other hand we consider v as a solution of the Cauchy problem d

dt v + A

2

v = k

1

u + f t

1

≤ t ≤ T

0

v(t

₁

) ∈ H

¹

(Ω)

(17)

in the space H

¹

(Ω). Since k

1

u + f ∈ L

²

(t

1

, T

0

; H

¹

(Ω)) and v

1

∈ D(A

₂

) = H

_N²

(Ω) it follows that v ∈ L

²

(t

1

, T

0

; D(A

^3/2₂

) ∩ H

¹

(t

1

, T

0

; D(A

^1/2₂

) and

d

dt A

^1/2₂

v = −A

^3/2₂

v + k

₁

A

^1/2₂

u + A

^1/2₂

f, t

₁

≤ t ≤ T

₀

. Therefore

d

dt kA

₂

vk

²_L2

+ kA

^3/2₂

vk

²_L2

≤ C{kA

^1/2₂

uk

²_L2

+ kA

^1/2₂

f k

²_L2

}.

As D(A

³²

) ⊂ H

³

(Ω), we obtain d

dt kA

₂

vk

²_L2

+ δkvk

²_H3

≤ C{kuk

²_H1

+ kf k

²_H1

}, (3.9) with some δ > 0. Let a

₁

= min(a, k

₁

) > 0. We now sum up (3.8) multiplied by

^2C_a

1

, where C > 0 is the constant appearing in (3.9), and (3.9). Then it follows that:

d dt

n C a

1

kuk

²_L2

+ kA

₂

vk

²_L2

o

+ C(1 − ξ 2C a

1

)kuk

²_H1

+ (δ − ξ 2C a

1

)kvk

²_H3

≤ C

1

{kf (t)k

²_H1

+ p(ξ

⁻¹

)}, (3.10) with some constant C

1

> 0 independent of T

0

. Choosing ξ small enough we conclude that:

Z

s t1

(kv(t)k

²_H3

+ ku(t)k

²_H1

) dt ≤ C

2

{ku

₁

k

²_L2

+ kv

₁

k

²_H2

+ Z

T

t1

(kf (t)k

²_H1

+ 1) dt}

with some constant C

₂

> 0 dependent on kf k

_L∞(0,T;L²(Ω))

and the initial condition U

0

through ku

₀

k

_L2

and kv

₀

k

_H1+ε0

, but independent of T

0

. The norms kuk

_L2(t1,T0;H¹(Ω))

and kvk

_L2(t1,T0;H³(Ω))

do not depend on T

0

and hence those of kuk

_C([t

1,T0];L²(Ω))

and kvk

_C([t

1,T0];H²(Ω))

do not depend either.

In particular this shows that the solution (u, v) can be extended as a weak solution beyond the T

0

.

3.3 Exponential attractor.

Let suppose that f is a positive constant function.Then we have the follow-

ing:

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Proposition 3.5. Let u

0

∈ H

_N²

(Ω) and v

0

∈ H

_N³

(Ω) be nonnegative functions. Let u, v be the global solution of (P

_h

). Then, with some continuous increasing function p(·) the following estimate holds:

ku(t)k

_H2(Ω)

+ kv(t)k

_H3(Ω)

≤ p(ku

₀

k

_H2(Ω)

+ kv

₀

k

_H3(Ω)

+ f ), for 0 < t < ∞.

Proof. Using (3.10) we deduce the existence of two constants σ > 0 and C > 0 such that

ku(t)k

²_L2

+ kv(t)k

²_H2

≤ Ce

^−σt

(ku

₀

k

²_L2

+ kv

₀

k

²_H2

)

+ p(f + N

_log¹

(u

₀

) + kv

₀

k

_H1

) (3.11) Multiplying the first equation of (P

_h

) by ∆u and integrating over Ω, gives

1 2

d

dt k∇uk

²_L2

+ ak∆uk

²_L2

+ k

1

k∇uk

²_L2

(3.12)

≤β(εk∆uk

²_L2

+ 1 2ε

Z

Ω

|∇u|

²

|∇v|

²

dx + 1 2ε

Z

Ω

|u|

²

|∆v|

²

dx) + ε

⁰

k∇uk

²_L2

+ C

_ε⁰

k∇vk

²_L2

,

where ε, ε

⁰

and C

_ε⁰

are positive constants derived from Young inequality.

Using technical inequalities proved in ([13] proposition 4.1) we obtain 1

2 d

dt k∇uk

²_L2

+ (a − βε)k∆uk

²_L2

+ (k

₁

− ε

⁰

)k∇uk

²_L2

≤ β 2ε (

Z

Ω

|∇u|

²

|∇v|

²

dx + Z

Ω

|u|

²

|∆v|

²

dx) + C

_ε⁰

k∇v(t)k

²_L2

≤ β

2ε (ηk∆uk

²_L2

+ p(kuk

_L2

+ kvk

_H2

+ η

⁻¹

)) + C

_ε⁰

k∇vk

²_L2

. Taking η = ε

²

, ε =

_2β^a

leads to

d

dt k∇uk

²_L2

+ ak∆uk

²_L2

+ 2(k

1

− ε

⁰

)k∇uk

²_L2

≤ β

²

a p(kuk

_L2

+ kvk

_H2(Ω)

) + C

ε⁰

k∇vk

²_L2

. (3.13) Take the second equation of (P

h

) operated by ∆, choose ∆

²

v as a test function and integrate the product in Ω. After some calculations as in [13]

we have d

dt k∇∆vk

²_L2

+ dk∆

²

vk

²_L2)

+ 2k

₂

k∇∆vk

²_L2

≤ k

₁²

d k∆uk

²_L2

. (3.14)

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We sum (3.14) multiplied by γ and (3.13). Thus we obtain:

d

dt (k∇uk

²_L2

+ γk∇∆vk

²_L2

) + γdk∆

²

vk

²_L2

+ (a − γk

₁²

d )k∆uk

²_L2

+ 2(k

₁

− ε

⁰

)(k∇uk

²_L2

+ k

₂

γ

k

₁

− ε

⁰

k∇∆vk

²_L2

) ≤ p(kuk

_L2

+ kvk

_H2

).

Then for γ and ε

⁰

small enough, there exists a positive constant σ

⁰

such that d

dt (k∇uk

²_L2

+ γk∇∆vk

²_L2

) + σ

⁰

(k∇uk

²_L2

+ γk∇∆v(t)k

²_L2

)

≤ p(kuk

_L2

+ kvk

_H2

). (3.15) So, we can find χ > 0 such as (3.11) is valid when σ = χ and

ku(t)k

²_H1(Ω)

+ kv(t)k

²_H3

≤ e

^−χt

(ku

₀

k

²_H1

+ kv

₀

k

²_H3

)

+ p(f + ku

₀

k

_L2

+ kv

₀

k

_H2

). (3.16) We verify also that

Z

t 0

(k∆

²

v(s)k

²_L2

+ ku(s)k

²_H2

)ds ≤ C(kv

₀

k

²_H3

+ ku

₀

k

²_H1

)

+ tp(f + ku

₀

k

_L2

+ kv

₀

k

_H2

).

Finally, taking the first equation of (P

_h

) operated by ∇ and multiplied by

∇∆u, gives as in [13]

1 2

d

dt k∆uk

²_L2

+ ak∇∆uk

²_L2

= β Z

Ω

∇(∇ · u∇v) · ∇∆udx + k

₁

Z

Ω

∇u · ∇∆udx + 2q Z

Ω

u∇u · ∇∆udx − k

₂

Z

Ω

∇v · ∇∆udx, (3.17) that is

1 2

d

dt k∆uk

²_L2(Ω)

+ ak∇∆uk

²_L2(Ω)

≤ a

2 k∇∆uk

²_L2

+ C Z

Ω

|∇(∇ · (u∇v)|

²

dx + C(

Z

Ω

|u∇u|

²

dx + k∇vk

²_L2

). (3.18) The terms R

Ω

|∇(∇ · (u∇v)|

²

dx and R

Ω

|u∇u|

²

dx of (3.18) can be estimated (see [13], proof of proposition 4.1, step 6) by

ηk∇∆uk

²_L2

+ p(kuk

_H1

+ kvk

_H3

, +η

⁻¹

)

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With an arbitrary η > 0. Thus we obtain:

1 2

d

dt k∆uk

²_L2

+ a

2 k∇∆uk

²_L2

+ ζ k∆uk

²_L2

≤ ηk∇∆uk

²_L2

+ p(kuk

_H1

+ kvk

_H3

+ η

⁻¹

). (3.19) Hence we can find a constant χ > 0 such that (3.16) is valid and

ku(t)k

²_H2

≤ e

^−χt

ku

₀

k

²_H2

+ p(f + ku

₀

k

_H1

+ kv

₀

k

_H3

). (3.20)

To prove the existence of an exponential attractor, we will use the following result:

Proposition 3.6. Let u

₀

∈ L

²

(Ω), v

₀

∈ H

^1+ε⁰

(Ω) be nonnegative functions.

Then there exists a continuous increasing function p(·), independent of u

0

and v

0

such that

ku(t)k

²_H2

+ kv(t)k

²_H3

≤ p(f + N

_log¹

(u

₀

) + kv

₀

k

_H1(Ω)

+ t

⁻¹

)

Proof. Since the proof follows exactly the same ideas and technical difficul- ties as in the proof of Theorem 4.6 [13] we skip it here.

We can now prove the existence of an exponential attractor: Let H = L

²

(Ω) × H

¹

(Ω) and consider the initial value problem

dU

dt + AU = G(U ) U (0) = U

0

(E)

in H, with A as in section 3.1 and D(A) = H

_n²

(Ω) × H

_n³

(Ω) and G(U ) :=

β∇(u∇v) − q|u|u + k

₂

v k

₁

u + f

Let K = {(u, v) ∈ L

²₊

(Ω) × H

₊^1+ε⁰

(Ω)} be the space of initial values and U

₀

∈ K.

We proved already the existence of a unique global solution U = (u, v) continuous with respect to the initial condition U

₀

. We define then a continuous semigroup {S(t)

_t≥0

} on K by S(t)U

₀

= U (t). For a fixed t > 0, S(t) maps K into K ∩ D(A).

Let denote B

_r

:= {(u, v) ∈ K ; ku

₀

k

_L2

+ kv

₀

k

_H1+ε0

≤ r} a bounded ball of K

with radius r > 0.

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Proposition 3.7. There exists a universal constant C > 0 such that the following statement holds: for each r > 0 there exists a time t

_r

> 0 such that

sup

t≥tr

sup

U0∈Br

kS(t)U

₀

k

_H2(Ω)×H³(Ω)

≤ C

Proof. Fix 0 < r < ∞. By t

r

and C

r

we shall denote some time and positive constant which depend on r but are uniform in U

0

∈ B

r

, respectively. By the Proposition 3.6, there exist a time t

_r

and a constant C

_r

such that for t ≥ t

r

ku(t)k

²_H2

+ kv(t)k

²_H3

≤ C

r

(3.21) The desired estimate will be established step by step.

Let us add the first equation of (P

h

) and the second one multiplied by 2 and let us integrate in space the result. If Φ(t) := ku(t)k

_L1

+ 2kv(t)k

_L1

we obtain:

d

dt Φ(t) + k

2

2 Φ(t) = Z

Ω

(−qu

²

+k

1

u + k

2

2 u) dx +f |Ω| ≤ { 1

4q (k

1

+ k

2

2 )

²

+f }|Ω|.

Thus Φ(t) ≤

Φ(0) − 2 k

₂

1 4q (k

₁

+ k

₂

2 )

²

+ f

|Ω| e

⁻^k²²^t

+ 2 k

₂

1 4q (k

₁

+ k

₂

2 )

²

+ f

|Ω|,

and we deduce

ku(t)k

_L1

+ 2kv(t)k

_L1

≤ C(C

r

e

^−ct

+ 1),

with C, c > 0 universal constants and C

_r

> 0 a constant depending in r.

This shows that there exists a time denoted by t

r

such that for all t ≥ t

r

ku(t)k

_L1

+ kv(t)k

_L1

≤ C (3.22) with C > 0 a universal constant.

From (3.6) and (3.22) it follows that

g(t) ≤ (g(0) − C)e

^−σ(t−t^r⁾

+ C for t ≥ t

r

.

Then there exists another time t

_r

and another universal constant C > 0 such that

kv(t)k

_H1

≤ C and N

_log¹

(u(t)) ≤ C for t ≥ t

r

.

(22)

From (3.11) and (3.21) we deduce that

kv(t)k

_H2(Ω)

+ ku(t)k

_L2(Ω)

≤ C

r

e

^−σ(t−t^r⁾

+ C for t ≥ t

r

.

and that there exist another time t

r

and another constant C > 0, such that kv(t)k

_H2(Ω)

+ ku(t)k

_L2(Ω)

≤ C for t ≥ t

_r

.

Finally using (3.20), (3.16) and repeat the argument we finish the proof.

Let B = {(u, v) ∈ H

_N²

(Ω) × H

_N³

(Ω) / kuk

_H2(Ω)

+ kvk

_H3(Ω)

≤ C} ∩ K with C the constant appearing in proposition 3.7. We proved that B is a compact absorbing set for ({S(t)}

t≥0

, K ). Hence by Temam([19]), there exist a global attractor A ⊂ K, where A is a compact and connected subset of K.

Let H = ∪

t≥t_B

S(t)B where t

_B

is such that S(t)B ⊂ B. Then H is a compact set of K with A ⊂ H ⊂ K. Since H is absorbing and positively invariant for {S(t)

_t≥0

} we apply to the dynamical system ({S(t)}

_t≥0

, H ) the following

Theorem. (Theorem 3.1 [20])

Let Γ(t, U

0

) = S(t)U

0

be a mapping from [0, T ] × H into H . If G satisfies kG(U ) − G(V )k ≤ kA

¹²

(U − V )k, U, V ∈ H (C

₁

) and Γ is such that

kΓ(t, U

₀

)−Γ(s, V

₀

)k ≤ C

T

(|t−s|+kU

₀

−V

₀

k

_H

, t, s ∈ [0, T ], U

0

, V

0

∈ H (C

2

) for each T > 0, then there is an exponential attractor M for ({S(t)}, H ).

Thus we obtain

Theorem 3.8. There exists an exponential attractor M of the dynamical system ({S(t)}

_t≥0

, H ) in H

Proof. Since the forcing term f is constant and the reaction coupling of the

first equation of (E) is linear in U : k

₂

v, the proof is the same as provided

in ([13], Theorem 5.1).

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4 Second case: h(u) = u(M − u).

Let M be a positive constant and consider a continuous function ˜ h of h such as

˜ h(u) = u(M − u) if 0 ≤ u ≤ M,

˜ h(u) = 0 otherwise. (4.1)

Then we have the following:

Proposition 4.1. Let ε

0

> 0 and f be a nonnegative function in L

²

(0, T ; H

¹

(Ω))∩

L

^∞

(Ω

T

). For each nonnegative initial condition (u

0

, v

0

) in L

²

(Ω)×H

^1+ε⁰

(Ω) there exists a constant T

₀

such that 0 < T

₀

≤ T and a unique nonnegative solution (u, v) of (P

h˜

) such that:

u ∈ H

¹

(0, T

0

; (H

¹

(Ω))

⁰

) ∩ C([0, T

0

]; L

²

(Ω)) ∩ L

²

(0, T

0

; H

¹

(Ω)) v ∈ H

¹

(0, T

₀

; H

^ε⁰

(Ω)) ∩ C([0, T

₀

]; H

^1+ε⁰

(Ω)) ∩ L

²

(0, T

₀

; H

_ν^2+ε⁰

(Ω)) Proof. The proof is essentially the same as in section 3.2.

Moreover we can prove the following:

Lemma 4.2. Let suppose that M ≥ (

^kfk^L^∞_q^(Ω^T⁾

)

¹²

and M

⁰

=

^qM²_k^+k¹^M

2

> 0.

If the initial condition (u

₀

, v

₀

) satisfies almost everywhere in Ω the following inequalities:

0 ≤ u

₀

(x) ≤ M, 0 ≤ v

₀

(x) ≤ M

⁰

, then the solution (u, v) of (P

_˜_h

) satisfies:

0 ≤ u(t, x) ≤ M 0 ≤ v(t, x) ≤ M

⁰

, almost everywhere in Ω

_T

.

Proof. Let define ˜ u = M − u and ˜ v = M

⁰

− v. Thus we get:

˜

u

t

= a∆˜ u − βdiv(˜ h(˜ u)∇˜ v) − (2qM + k

1

)˜ u + q˜ u

²

+ k

2

v ˜ + qM

²

+ k

1

M − k

2

M

⁰

˜

v

t

= d∆˜ v − k

2

v ˜ + k

1

u ˜ + k

2

M

⁰

− k

1

M − f As M ≥ (

^kfk^L^∞_q^(Ω^T⁾

)

¹²

, and M

⁰

=

^qM²_k^+k¹^M

2

we obtain qM

²

+ k

₁

M − k

₂

M

⁰

= 0 and

k

₂

M

⁰

− k

₁

M − f ≥ 0.