existence of solutions for reaction-diffusion systems …saad/cv/art1.pdf · existence of solutions...

Advances in Differential Equations Volume 7, Number 6, June 2002, Pages 743–768

EXISTENCE OF SOLUTIONS FOR REACTION-DIFFUSIONSYSTEMS WITH L1 DATA

M. BendahmaneUMR CNRS 5466, Mathematiques Appliquees de Bordeaux, Universite Bordeaux I

351, cours de la Liberation, F-33405 Talence Cedex

M. LanglaisUMR CNRS 5466, Mathematiques Appliquees de Bordeaux, Case 26

Universite Victor Segalen Bordeaux 2146, rue Leo-saignat, F-33076 Bordeaux Cedex

M. SaadUMR CNRS 5466, Mathematiques Appliquees de Bordeaux, Universite Bordeaux I

351, cours de la Liberation, F-33405 Talence Cedex

(Submitted by: Roger Temam)

Abstract. We are concerned with a system of nonlinear partial differ-ential equations modeling the spread of an epidemic disease through aheterogeneous habitat. Assuming no-flux boundary conditions and L1

data, we prove the existence of at least one weak solution.

1. Introduction

Our motivation is a mathematical model describing the spatial propaga-tion in a domain Ω in RN (N ≥ 1) of Feline Leukemia Virus (FeLV), a felineretro-virus; see [13] and [20] for a model derivation and an analysis of theunderlying system of ordinary differential equations. A prototype of a non-linear system that governs the spreading of FeLV through a cat population ina heterogeneous spatial domain with seasonal variations and external supplyis the following reaction-diffusion-advection system

(S1)

ut(t, x)− div(A1(t, x)∇u(t, x) + u(t, x)K1(t, x)) + r1(t, x, u, v, w)= b(t, x)(u(t, x) + w(t, x))−m(t, x)u(t, x)− σ(t, x, u, v, w) + f(t, x),

vt(t, x)− div(A2(t, x)∇v(t, x) + v(t, x)K2(t, x)) + r2(t, x, u, v, w)= πσ(t, x, u, v, w)− (m(t, x) + α(t, x))v(t, x) + g(t, x),

wt(t, x)− div(A3(t, x)∇w(t, x) + w(t, x)K3(t, x)) + r3(t, x, u, v, w)= (1− π)σ(t, x, u, v, w)−m(t, x)w(t, x) + h(t, x);

Accepted for publication: November 2001.AMS Subject Classifications: 35K57, 35K55, 92D30.

743

744 M. Bendahmane, M. Langlais, and M. Saad

in (0, T )× Ω, together with no-flux boundary conditions on (0, T )× ∂Ω

(S2)

(A1(t, x)∇u(t, x) + u(t, x)K1(t, x)) · η(x) = 0,(A2(t, x)∇v(t, x) + v(t, x)K2(t, x)) · η(x) = 0,(A3(t, x)∇w(t, x) + w(t, x)K3(t, x)) · η(x) = 0;

corresponding to an isolated population, and initial distributions in Ω

(S3) u(0, x) = u0(x), v(0, x) = v0(x) and w(0, x) = w0(x).

Here, u(t, x), v(t, x) and w(t, x) represent the spatial densities at time t andlocation x ∈ Ω of susceptible, infectious and immune individuals. Thus,we recover the time dependent state variables used in [13] and [20], wherespatial considerations are ignored, upon integrating over space∫

Ωu(t, x) dx,

∫Ωv(t, x) dx and

∫Ωw(t, x) dx.

This also shows L1(Ω) is a natural functional space for developing a mathe-matical analysis of the spatial propagation of an epidemic disease.

The diffusitivity matrix Ai is a bounded symmetric and coercive matrix,the transport vector Ki is bounded on (0, T )×Ω, i = 1, 2, 3. The birth rate b,the death ratem and the additional disease induced death rate in the infectedclass α, are assumed to be bounded on (0, T )×Ω. The incidence term σ andthe density dependent mortality terms ri, i = 1, 2, 3, are nonlinear terms.As a typical example, σ can take one of the following forms,

σ(t, x, u, v, w) =

σ1(t, x)uv,σ2(t, x) u v

u+v+w ,

σ3(t, x) (u+v+w)ν

1+(u+v+w)νu v

u+v+w , ν > 0.

In this work, for some p > 1 we taker1(t, x, u, v, w) = k1(t, x) u|u+ v + w|p−1,

r2(t, x, u, v, w) = k2(t, x) v|u+ v + w|p−1,

r3(t, x, u, v, w) = k3(t, x) w|u+ v + w|p−1,

(1.1)

Earlier forms of this problem are considered in [11] and [12] with L∞ data.In the logistic case (p = 2) for constant coefficients and no advection termsexistence results are established in [9] with L∞-data. See also [10].

Assuming σ = 0, ri = 0 for i = 1, 2, 3 and under Dirichlet boundary con-ditions, existence results for the corresponding linear parabolic system withnon regular data are established in [1], [4], [5], [8], [14], [21] and [23] whileuniqueness questions, in the sense of entropic or renormalized formulations,are considered in [3], [22].

reaction-diffusion systems with L1-data 745

2. Notations, Assumptions

2.1. Notations. Here, Ω is a bounded open domain of RN (N ≥ 1) withsmooth boundary ∂Ω, so that locally Ω lies on one side of ∂Ω; η is theouter unit normal to Ω on ∂Ω. We denote D+(Ω) the space of nonnegativefunctions in C∞0 (Ω). The norm in Lp(Ω) is denoted by ‖ ·‖Lp(Ω), 1 ≤ p <∞;then

Lp+(Ω) = u : Ω −→ R+ measurable and∫

Ω|u|p(x)dx <∞,

L∞+ (Ω) = u : Ω −→ R+ measurable and supx∈Ω|u(x)| <∞.

If 1 ≤ p ≤ ∞, W 1,p(Ω) the Sobolev space of functions u on the open setΩ for which u and ∇u belong to Lp(Ω). If X is a Banach space, a < band 1 ≤ p ≤ ∞, Lp(a, b;X) denotes the space of all measurable functionu : (a, b) −→ X such that ‖u(·)‖X belongs to Lp(a, b). Next T is a positivenumber and QT = (0, T )× Ω, ΣT = (0, T )× ∂Ω. We denote C1

c ([0, T )× Ω)the set of all C1-functions with compact support in [0, T )×Ω. Let us recallthe definition of truncated function Tγ . Let γ ∈ R+. We set

Tγ(z) =

−γ if z ≤ −γz if |z| ≤ γγ if z ≥ γ

and we denote Sγ(z) =∫ z

0 Tγ(τ) dτ . We introduce the function φγ = Tγ+1−Tγ , that is

φγ(z) =

1 if z ≥ γ + 1z − γ if γ ≤ z < γ + 10 if − γ ≤ x ≤ γz + γ if − γ − 1 < z ≤ −γ−1 if z ≥ −γ − 1

and we set Ψγ(z) =∫ z

0 φγ(τ) dτ . We note that Tγ and φγ are Lipschitzcontinuous functions.

2.2. Assumptions. Throughout this paper, the following assumptions areassumed to hold true.

First Ai, Ki and ki (see (1.1)) for i = 1, 2, 3, α, b, m are given functionsdefined onQT with values in RN×RN , RN and R+, R+, R+, R+, respectively.Our basic requirement on Ai, Ki, α, b, m and ki is

Ai ∈ (L∞(QT ))N×N , for i = 1, 2, 3, (2.1)


Ki ∈ (L∞(QT ))N and div(Ki) ∈ L∞(QT ), for i = 1, 2, 3, (2.2)b, α, m and ki ∈ L∞+ (QT ) for i = 1, 2, 3. (2.3)

We assume that there exists a > 0 and kmax, k0 > 0 such that for i = 1, 2, 3

Ai(t, x)ξ · ξ ≥ a|ξ|2 a.e. (t, x) ∈ QT and ∀ξ ∈ RN , (2.4)0 < k0 ≤ ki(t, x) ≤ kmax < +∞ a.e. (t, x) ∈ QT . (2.5)

Second σ : QT × R × R × R → R+ is measurable on QT , continuous withrespect to u, v et w, and satisfies a growth condition

there exist p > 1 and two bounded functionsL,M : RN × [0,∞)→ [0,∞), and l, s, l′ ∈ R+ such thatl, l′ > 0, 1 ≤ s < max(N+2

N+1 , p) and|σ(t, x, u, v, w)| ≤ L(t, x)(|u|l|v|s + |u|l′ |w|s) +M(t, x)

a.e. x, t ∈ QT ;

(2.6)

and a nonnegativity conditionσ(t, x, 0, v, w) = 0 if v ≥ 0 and w ≥ 0,σ(t, x, u, 0, w) ≥ 0 if u ≥ 0 and w ≥ 0,σ(t, x, u, v, 0) ≥ 0 if u ≥ 0 and v ≥ 0.

(2.7)

Next, for i = 1, 2, 3, one assumes ri is given by (1.1) with p as in (2.6), and kias in (2.3) and (2.5) for i = 1, 2, 3. Last π is a real number with 0 < π < 1.

The main difficulty in this work arises from the fact that we only considerL1-data, namely

u0, v0, w0 ∈ L1+(Ω) and f, g, h ∈ L1

+(QT ). (2.8)

3. Main result

In this section we give the definition of a weak solution for nonlinearparabolic systems of type (S1)-(S2)-(S3) with L1 right-hand sides and initialdata. Then, we supply our existence result.

Definition 1. Let 1 ≤ q < N+2N+1 if N ≥ 2 and 1 ≤ q < 4

3 if N =1. A weak solution of (S1)-(S2)-(S3), is a triple (u, v, w) of nonnegativefunctions belonging to Lq(0, T ;W 1,q(Ω))∩Lp(0, T ;Lp(Ω))∩C([0, T ];L1(Ω))such that σ(·, ·, u, v, w) and ri(·, ·, u, v, w), for i = 1, 2, 3 belong to L1(QT )and satisfying

−∫ T

0

∫Ωuϕt dxdt−

∫Ωϕ(0, x)u0(x)dx


+∫ T

0

∫Ω

(A1(t, x)∇u+ uK1(t, x)) · ∇ϕ dxdt

+∫ T

0

∫Ωm(t, x)uϕdxdt−

∫ T

0

∫Ωb(t, x)(u+ w)ϕdxdt

+∫ T

0

∫Ωσ(t, x, u, v, w)ϕ dxdt+

∫ T

0

∫Ωr1(t, x, u, v, w) ϕ dxdt

=∫ T

0

∫Ωf ϕ dxdt,

−∫ T

0

∫Ωvψt dxdt−

∫Ωψ(0, x)v0(x) dx

+∫ T

0

∫Ω

(A2(t, x)∇v + vK2(t, x)) · ∇ψ dxdt

+∫ T

0

∫Ω

(m(t, x) + α(t, x))vψ dxdt− π∫ T

0

∫Ωσ(t, x, u, v, w)ψ dxdt

+∫ T

0

∫Ωr2(t, x, u, v, w)ψ dxdt =

∫ T

0

∫Ωg ψ dxdt,

−∫ T

0

∫Ωwχt dxdt−

∫Ωχ(0, x)w0(x) dx

+∫ T

0

∫Ω

(A3(t, x)∇w + wK3(t, x)) · ∇χ dxdt

+∫ T

0

∫Ωm w χ dxdt− (1− π)

∫ T

0

∫Ωσ(t, x, u, v, w) χ dxdt

+∫ T

0

∫Ωr3(t, x, u, v, w) χ dxdt =

∫ T

0

∫Ωh χ dxdt,

for all ϕ,ψ, χ ∈ C1c ([0, T )× Ω).

Theorem 1. Assume that (1.1)–(2.7) hold. Let u0, v0, w0 ∈ L1+(Ω) and

f, g, h ∈ L1+(QT ). If either one of the following conditions holds

(CN)

(CN1) Ki(t, x) · η(x) ≥ 0 a.e. (t, x) ∈ ΣT , i = 1, 2, 3;(CN2) p ≥ 2;(CN3) N = 1;

then the system (S1)-(S2)-(S3) has a weak solution.Condition (CN1) is a standard assumption (see [21]), in the context of


reaction-diffusion systems. Next, p = 2 is the condition found in our mo-tivating problem ([13], [20]), the logistic case; hence (CN2) is a naturalcondition in this work.

In the one dimensional case, various simplifications occur, as in [16]; thisallows us to get rid of (CN1) and (CN2). Actually, our solution has moreregularity properties when N = 1 than when N ≥ 2; more optimal resultscould be derived, see [2].

The proof is organized as it follows. Section 4 is devoted to a relatedscalar equation. Some a priori estimates for smooth solutions of the systemsare obtained in section 5 and used for completing the proof. In section 6 weprove technical results used in the previous sections.

4. A related scalar equation

We consider a related scalar problem

(E)

ut(t, x)− div(A(t, x)∇u(t, x) + u(t, x)K(t, x))+k(t, x)|u(t, x)|p−1u(t, x)= b(t, x)u(t, x)−m(t, x)u(t, x) + f(t, x) in QT ,

(A(t, x)∇u(t, x) + u(t, x)K(t, x)) · η(x) = 0 on ΣT ,

u(0, x) = u0(x) in Ω.

Here A, K and k have similar properties to Ai, Ki and ki for i = 1, 2, 3respectively.

Our system reduces to this scalar problem when A1 = A, K1 = K, k1 = k,Ai = 0, Ki = 0, ki = 0, for i = 2, 3, σ = 0, g = h = 0 and v0 = w0 = 0.

Definition 2. Let 1 ≤ q < N+2N+1 if N ≥ 2 and 1 ≤ q < 4

3 if N = 1.A weak solution of (E), is a nonnegative function u in Lq(0, T ;W 1,q(Ω) ∩Lp(0, T ;Lp(Ω)) ∩ C([0, T ];L1(Ω)) satisfying

−∫ T

0

∫Ω

uϕt dt+∫ T

0

ϕ(0, x)u0(x) dx+∫ T

0

∫Ω

(A(t, x)∇u+ uK(t, x)) · ∇ϕ dxdt

+∫ T

0

∫Ω

(m(t, x)− b(t, x))uϕ dxdt+∫ T

0

∫Ω

k(t, x)upϕ dxdt =∫ T

0

∫Ω

fϕ dxdt,

for all ϕ ∈ C1c ([0, T )× Ω).

Theorem 2. Assume that (2.1)–(2.5) hold and p > 1. Let u0 ∈ L1+(Ω) and

f ∈ L1+(QT ). If either one of the following condition holds

(CL)

(CL1) K(t, x) · η(x) ≥ 0 a.e. (t, x) ∈ ΣT ;(CL2) p ≥ 2;(CL3) N = 1;


then problem (E) has a weak solution.

Our proof relies on a few approximation techniques, a priori estimates anda limiting process as ε→ 0.Proof. We introduce the following smooth approximations of the data

(u0,ε)0<ε≤1 ⊂ D+(Ω) and (fε)0<ε≤1 ⊂ D+(QT ) such that,u0,ε → u0 in L1(Ω), fε → f in L1(QT ), as ε→ 0;‖u0,ε‖L1(Ω) ≤ ‖u0‖L1(Ω), ‖fε‖L1(QT ) ≤ ‖f‖L1(QT ), 0 < ε ≤ 1.

(4.1)

Then, classical results, see e.g [17] and [18], provide the existence of a se-quence (uε)0<ε≤1, with uε ∈ L2(0, T ;H1(Ω)) ∩ Lp(QT ) ∩ C([0, T ];L2(Ω)),with ∂tuε ∈ L2(0, T ; (H1(Ω))′), solutions of (E) where u0 and f are replacedby (u0,ε)0<ε≤1 and (fε)0<ε≤1. Let λ > 0 satisfies

λ− b(t, x) ≥ 0 a.e. (t, x) ∈ QT ,λ− b(t, x)− div(K)(t, x) ≥ 0 a.e. (t, x) ∈ QT .

(4.2)

Set uε(t, x) = eλtuε(t, x). Then uε(t, x) satisfies∫ T

0< ∂tuε, ϕ > dt +

∫ T

0

∫Ω

(A(t, x) ∇uε + uε K(t, x)) · ∇ϕ dxdt

+∫ T

0

∫Ω

(λ+m(t, x)− b(t, x)) uε ϕ dxdt+∫ T

0

∫Ωh(t, x, uε) ϕ dxdt

=∫ T

0

∫Ωe−λt fε ϕ dxdt, (4.3)

for all ϕ ∈ L2(0, T ;H1(Ω)) ∩ L∞(QT ). Herein

h(t, x, uε) = k(t, x)e(p−1)λt|uε(t, x)|p−1uε(t, x).

4.1. Nonnegativity.

Lemma 1. The solution uε is nonnegative.

Proof. Let u−ε = sup(−uε, 0). Substituting ϕ = −Tγ(u−ε ) in (4.3), one has∫ t

0< ∂tu

−ε , Tγ(u−ε ) > dτ +

∫ t

0

∫Ω

(A∇u−ε + u−ε K) · ∇Tγ(u−ε )dxdτ

+∫ t

0

∫Ω

(λ+m(t, x)− b(t, x))u−ε Tγ(u−ε )dxdτ

+∫ t

0

∫Ωk(t, x) e(p−1)λt|uε|p−1u−ε Tγ(u−ε ) dxdτ +

∫ t

0

∫Ωe−λτfεTγ(u−ε )dxdτ = 0.


From the choice of λ in (4.2), the nonnegativity of m and fε and usingHolder’s inequality, one gets

d

dt

∫ΩSγ(u−ε (t, x)) dx+

a2

∫Ω|∇Tγ(u−ε (t, x))|2 dx

≤‖K‖2L∞(QT )

2a

∫Ω

1|uε|≤γ|u−ε (t, x))|2 dx.

One observes that

0 ≤ z21|z|≤γ + (2γ|z| − γ2)1|z|>γ = 2Sγ(z), (4.4)

which yields

d

dt

∫ΩSγ(u−ε (t, x)) dx ≤

‖K‖2L∞(QT )

a

∫ΩSγ(u−ε (t, x)) dx.

Since the data u0,ε is nonnegative, we deduce that u−ε = 0.

4.2. A priori estimates. In this section, we are concerned with a prioriestimates satisfied by the sequence (uε)0<ε≤1 which lead to compactnessproperties. We note

Bγ = (t, x), γ ≤ |uε(t, x)| < γ + 1.Lemma 2. There exists α1 > 0 not depending on ε (0 < ε ≤ 1) such thatthe solutions of (4.3) satisfy

‖uε‖Lp(QT ) + ‖uε‖L∞(0,T ;L1(Ω)) ≤ α1. (4.5)

Assuming condition (CL1) or condition (CL2) to hold, there exists α2 > 0not depending on ε (0 < ε ≤ 1), such that

supγ≥0

∫Bγ

|∇uε(t, x)|2dxdt ≤ α2. (4.6)

Assuming condition (CL3) to hold, there exists α3 > 0 et α4 > 0 not de-pending on ε (0 < ε ≤ 1), such that

supγ≥0

∫Bγ

|∂xuε(t, x)|2dxdt ≤ α3 + α4

∫ T

0

∫Ω|uε(t, x)|2 dxdt. (4.7)

A proof is found in Subsection 6.1. A modification of results in [14], [16]yields the following consequence

Corollary 1. Let q satisfy 1 ≤ q < N+2N+1 when N ≥ 2 or 1 ≤ q < 4

3 whenN = 1. Then, there exists C > 0 depending on α1, α2, α3, α4, meas(Ω), Tand q such that

‖uε‖Lq(0,T ;W 1,q(Ω)) ≤ C, 0 < ε ≤ 1. (4.8)


Now, we are interested in the nonlinear term h(t, x, uε). We have

Proposition 1. Let (uε)0<ε≤1 be the sequence of solutions of (4.3). Thenthe sequence (h(t, x, uε))0<ε≤1 satisfies

limγ→∞

sup0<ε≤1

∫|uε|≥γ

h(t, x, uε) dxdt = 0 (4.9)

A proof is found in Subsection 6.2.

4.3. End of the proof of Theorem 2. By Lemma 2 and Corollary 1 thesequence (uε)0<ε≤1 is bounded in Lq(0, T ;W 1,q(Ω)), for 1 ≤ q < N+2

N+1 whenN ≥ 2, and for 1 ≤ q < 4

3 when N = 1. In view of the equation satisfiedby uε this implies that ∂tuε is bounded in L1(0, T ;

(W 1,q(Ω)

)′) + L1(QT ).Therefore, possibly at the cost of extracting subsequences denoted (uε)0<ε≤1,see e.g. [25], we can assume that there exists a u in Lq(0, T ;W 1,q(Ω)), suchthat as ε −→ 0 uε −→ u strongly in Lq(QT ) and a.e. in QT ,

∇uε −→ ∇u weakly in Lq(QT ),h(t, x, uε) −→ h(t, x, u) a.e. in QT ,

(4.10)

The convergence property of h(t, x, uε) in (4.10) is too weak. It can beimproved:

Lemma 3. The sequence (h(t, x, uε))0<ε≤1 converges to h(t, x, u) almosteverywhere in QT and strongly in L1(QT ).

A proof is found in Subsection 6.3. This result is similar to those obtainedin [15] in the context of elliptic problems.

Since h(t, x, uε) = k(t, x)e(p−1)λt|uε(t, x)|p−1uε(t, x) and by Lemma 3, itis clear that the sequence (uε)0<ε≤1 converges strongly in Lp(QT ) for p > 1.We complete the properties of the sequence uε with the following two results.

Lemma 4. The sequence (∇uε)0<ε≤1 converges to ∇u a.e. in QT as ε goesto zero.

Lemma 5. The sequence (uε)0<ε≤1 is a Cauchy sequence in C([0, T ];L1(Ω)).

The proof of Lemma 4 and Lemma 5 are found in Subsection 6.4 andSubsection 6.5 respectively. Now using the following weak formulation

−∫ T

0

∫Ωuεϕt dxdt+

∫ T

0ϕ(0, x)u0,ε(x) dx

+∫ T

0

∫Ω(A(t, x)∇uε+uεK(t, x)) · ∇ϕdxdt+

∫ T

0

∫Ω(m(t, x)− b(t, x))uεϕdxdt


+∫ T

0

∫Ωk(t, x)upεϕdxdt =

∫ T

0

∫Ωfεϕdxdt, (4.11)

with ϕ ∈ C1c ([0, T )× Ω), we can let ε −→ 0 and obtain a weak solution.

5. Sketch of the proof of Theorem 1.

We introduce σ is measurable on QT , continuous with respect to u, v andw.

σ(t, x, u, v, w) =

σ(t, x, u, v, w) if u ≥ 0, v ≥ 0, w ≥ 0,σ(t, x, u, v, 0) if u ≥ 0, v ≥ 0, w < 0,σ(t, x, u, 0, w) if u ≥ 0, v < 0, w ≥ 0,σ(t, x, 0, v, w) if u < 0, v ≥ 0, w ≥ 0,σ(t, x, 0, 0, w) if u < 0, v < 0, w ≥ 0,σ(t, x, u, 0, 0) if u ≥ 0, v < 0, w < 0,σ(t, x, 0, v, 0) if u < 0, v ≥ 0, w < 0,σ(t, x, 0, 0, 0) if u < 0, v < 0, w < 0.

We are concerned with system (S1)-(S2)-(S3), where σ is replaced by σ. Weintroduce the following smooth approximations of the data u0, v0, w0 andf, g, h; let Zε = fε, gε, hε and Z0,ε = u0,ε, v0,ε, w0,ε be such that

Zε ∈ D+(QT ) and Z0,ε ∈ D+(Ω) such that‖Zε‖L1(QT ) ≤ ‖Z‖L1(QT ), Zε → Z in L1(QT ), as ε→ 0;‖Z0,ε‖L1(Ω) ≤ ‖Z0‖L1(Ω), Z0,ε → Z0 in L1(Ω), 0 < ε ≤ 1;

(5.1)

here Z = f, g, h and Z0 = u0, v0, w0. Then classical results, see e.g [17]and [18] provide the existence of a sequence uε, vε, wε ∈ L2(0, T ;H1(Ω)) ∩Lp(QT ) ∩ C([0, T ];L2(Ω)), with ∂tuε, ∂tvε, ∂twε ∈ L2(0, T ; (H1(Ω))′), ofsolutions of (S1)-(S2)-(S3), where u0, v0, w0 and f, g, h are replaced byu0,ε, v0,ε, w0,ε and fε, gε, hε respectively, and σ is replaced by σ. Let λ > 0satisfies λ− b ≥ 0 a.e. (t, x) ∈ QT ,

λ− b− div(K1(t, x)) ≥ 0 a.e. (t, x) ∈ QT ,λ− div(Ki(t, x)) ≥ 0 a.e. (t, x) ∈ QT for i = 2, 3.

(5.2)

We will often write σ(t, x, ·, ·, ·) = σ(·, ·, ·) and ri(t, x, ·, ·, ·) = ri(·, ·, ·) fori = 1, 2, 3 when no confusion can arise. Set uε = eλtuε, vε = eλtvε andwε = eλtwε; then uε, vε and wε satisfies∫ T

0< ∂tuε, ϕ > dt+

∫ T

0

∫Ω

A1∇uε · ∇ϕ dxdt+∫ T

0

∫ΩuεK1 · ∇ϕdxdt

+∫ T

0

∫Ωe−λtσ(eλtuε, eλtvε, eλtwε)ϕdxdt+

∫ T

0

∫Ω

(λ+m− b)uεϕdxdt


−∫ T

0

∫Ωbwεϕdxdt+

∫ T

0

∫Ωr1,λ(uε, vε, wε)ϕdxdt =

∫ T

0

∫Ωe−λtfεϕdxdt,

(5.3)

∫ T

0< ∂tvε, ψ > dt+

∫ T

0

∫Ω

A2∇vε · ∇ψ dxdt+∫ T

0

∫ΩvεK2 · ∇ψ dxdt

+∫ T

0

∫Ω

(m+ λ+ α) vεψ dxdt− π∫ T

0

∫Ωe−λtσ(eλtuε, eλtvε, eλtwε)ψ dxdt

+∫ T

0

∫Ωr2,λ(uε, vε, wε)ψ dxdt =

∫ T

0

∫Ωe−λtgεψ dxdt, (5.4)

∫ T

0< ∂twε, χ > dt+

∫ T

0

∫Ω

A3∇wε · ∇χdxdt+∫ T

0

∫ΩwεK3 · ∇χdxdt

+∫ T

0

∫Ω

(λ+m)wεχdxdt− (1− π)∫ T

0

∫Ωσ(eλtuε, eλtvε, eλtwε)χdxdt

+∫ T

0

∫Ωr3,λ(uε, vε, wε)χdxdt =

∫ T

0

∫Ωe−λthε χ dxdt, (5.5)

for all ϕ, ψ, χ ∈ L2(0, T ;H1(Ω)) ∩ L∞(QT ).Herein ri,λ(t, x, uε, vε, wε) = e(p−1)λt ri(t, x, uε, vε, wε) for i = 1, 2, 3.

5.1. Nonnegativity.

Lemma 6. The solution (uε, vε, wε) is nonnegative.

Proof. Using the definition of σ, we have∫ T

0

∫Ωe−λtσ(eλtuε, eλtvε, eλtwε)Tγ(w−ε )dxdt ≥ 0,∫ T

0

∫Ωe−λtσ(eλtuε, eλtvε, eλtwε)Tγ(v−ε )dxdt ≥ 0,∫ T

0

∫Ωe−λtσ(eλtuε, eλtvε, eλtwε) Tγ(u−ε ) dxdt = 0,

and ∫ T

0

∫Ωr3,λ(uε, vε, wε)Tγ(w−ε )dxdt ≤ 0,∫ T

0

∫Ωr2,λ(uε, vε, wε)Tγ(v−ε )dxdt ≤ 0,

754 M. Bendahmane, M. Langlais, and M. Saad∫ T

0

∫Ωr1,λ(uε, vε, wε) Tγ(u−ε ) dxdt ≤ 0.

Let us substitute in first χ = −Tγ(w−ε ) in (5.5) where w−ε = sup(−wε, 0).Since hε is nonnegative, the choice of λ in (5.2), and using Holder’s inequalityand (4.4), one gets

d

dt

∫ΩSγ(w−ε (t, x)) dx ≤

‖K3‖2L∞(QT )

a

∫ΩSγ(w−ε (t, x)) dx.

Since the data w0,ε is nonnegative, we deduce that w−ε = 0.Next, substitute ϕ = −Tγ(u−ε ) in (5.3). Since fε and wε are nonnegative,

the choice of λ in (5.2) and using Holder’s inequality and (4.4), one gets

d

dt

∫ΩSγ(u−ε (t, x)) dx ≤

‖K1‖2L∞(QT )

a

∫ΩSγ(u−ε (t, x)) dx,

Since the data u0,ε is nonnegative, we deduce that u−ε = 0.Along the same lines one can show that vε(t, x) ≥ 0 a.e. (t, x) ∈ QT .

5.2. A priori estimates.

Proposition 2. Assume that (1.1)–(2.7) hold. Then, there exist c1, c2 notdepending on ε such that the sequences (uε, vε, wε)0<ε≤1 satisfies

‖uε + vε + wε‖L∞(0,T ;L1(Ω)) ≤ c1, (5.6)

‖uε + vε + wε‖Lp(QT ) + ‖σ(eλtuε, eλtvε, eλtwε)‖L1(QT ) ≤ c2. (5.7)

Assuming condition (CN1) or condition (CN2), there exist c3 such that

supγ≥0

(∫γ≤|uε|<γ+1

|∇uε|2 dxdt+∫γ≤|vε|<γ+1

|∇vε|2 dxdt

+∫γ≤|wε|<γ+1

|∇wε|2 dxdt) ≤ c3. 0 < ε ≤ 1. (5.8)

Assuming condition (CN3), there exist c4 and c5 such that

supγ≥0

∫γ≤|zε|<γ+1

|∂xzε|2 dxdt ≤ c4 + c5(‖zε‖2L2(QT )), 0 < ε ≤ 1; (5.9)

herein zε = uε, vε, wε.

A proof is found in Subsection 6.6.As we did in Subsection 4.2, we deduce immediately that the sequences

(uε)0<ε≤1, (vε)0<ε≤1 and (wε)0<ε≤1 are bounded in Lq(0, T ;W 1,q(Ω)). Sim-ilarly to the scalar case, ∂tuε, ∂tvε, ∂twε is bounded in L1(0, T ; (W 1,q(Ω))′) +L1(QT ), therefore, possibly at the cost of extracting subsequences denoted


(uε, vε, wε), see e.g. [25], we can assume that there exist u, v and w inLq(0, T ;W 1,q(Ω)) such that as ε goes to 0

uε −→ u strongly in Lq(QT ) and a.e. in QT ,

vε −→ v strongly in Lq(QT ) and a.e. in QT ,

wε −→ w strongly in Lq(QT ) and a.e. in QT ,

σ(eλtuε, eλtvε, eλtwε) −→ σ(eλtu, eλtv, eλtw) a.e. in QT ,

r1,λ(t, x, uε, vε, wε) −→ r1,λ(t, x, u, v, w) a.e. in QT ,

r2,λ(t, x, uε, vε, wε) −→ r2,λ(t, x, u, v, w) a.e. in QT ,

r3,λ(t, x, uε, vε, wε) −→ r3,λ(t, x, u, v, w) a.e. in QT .

(5.10)

Now, we are interested in the nonlinear terms σ, r1,λ, r2,λ and r3,λ. Havingin mind for using Vitali’s Theorems and pass to the limit. Let us analyzethe behaviour of the nonlinear terms when (2.6) holds.

Proposition 3. The sequences

(σ(eλtuε, eλtvε, eλtwε))0<ε≤1, (r1,λ(uε, vε, wε))0<ε≤1,

(r2,λ(uε, vε, wε))0<ε≤1, (r3,λ(uε, vε, wε))0<ε≤1

satisfy

limγ→∞

sup0<ε≤1

(∫|uε|≥γ

[r1,λ(uε, vε, wε) + σ(eλtuε, eλtvε, eλtwε

]dxdt

)= 0,

(5.11)and

σ(eλtuε, eλtvε, eλtwε) −→ σ(eλtu, eλtv, eλtw), (5.12)

ri,λ(uε, vε, wε) −→ ri,λ(u, v, w), for i = 1, 2, 3 (5.13)

almost everywhere in QT and strongly in L1(QT ).

A proof of Proposition 3 is found in Subsection 6.7.We complete the properties of the sequences (uε)0<ε≤1, (vε)0<ε≤1 and

(wε)0<ε≤1 with the following two results.

Lemma 7. The sequences (∇uε)0<ε≤1, (∇vε)0<ε≤1 and (∇wε)0<ε≤1 con-verge to ∇u, ∇v and ∇w almost everywhere in QT as ε goes to zero.

Lemma 8. The sequences (uε)0<ε≤1, (vε)0<ε≤1 and (wε)0<ε≤1 are Cauchysequences in C([0, T ];L1(Ω)).

The proof of Lemma 7 and Lemma 8 are similar to the proofs of Lemma 4and Lemma 5, when the terms ri,λ for i = 1, 2, 3 and fε− b wε−σ, π σ+ gε,


(1 − π) σ + hε are replaced by h and fε respectively in (4.3). Finally, bypassing to the limit ε −→ 0 in the following weak formulation

−∫ T

0

∫Ωuεϕt dt−

∫Ωϕ(0, x)u0,ε(x) dx+

∫ T

0

∫Ω

(A1∇uε + uεK1) · ∇ϕdxdt

+∫ T

0

∫Ωm(t, x)uεϕdxdt−

∫ T

0

∫Ωb(t, x)(uε + wε)ϕdxdt

+∫ T

0

∫Ωσ(t, x, uε, vε, wε)ϕdxdt+

∫ T

0

∫Ωr1(t, x, uε, vε, wε)ϕdxdt

=∫ T

0

∫Ωfεϕdxdt,

−∫ T

0

∫Ωvεψt dt−

∫Ωψ(0, x)v0,ε(x) dx+

∫ T

0

∫Ω

(A2∇vε + vε K2) · ∇ψ dxdt

+∫ T

0

∫Ω

(m(t, x) + α(t, x))vεψ dxdt− π∫ T

0

∫Ωσ(t, x, uε, vε, wε)ψ dxdt

+∫ T

0

∫Ωr2(t, x, uε, vε, wε)ψ dxdt =

∫ T

0

∫Ωgεψ dxdt,

−∫ T

0

∫Ωwεχt dt−

∫Ωχ(0, x)w0,ε(x) dx+

∫ T

0

∫Ω

(A3∇wε + wεK3) · ∇χdxdt

+∫ T

0

∫Ωm(t, x)wεχdxdt− (1− π)

∫ T

0

∫Ωσ(t, x, uε, vε, wε)χdxdt

+∫ T

0

∫Ωr3(t, x, uε, vε, wε)χdxdt =

∫ T

0

∫Ωhεχdxdt,

with ϕ, ψ, χ ∈ C1c ([0, T )×Ω), obtaining in this way that the limit (u, v, w)

is a solution of system (S1)-(S1)-(S1) in the sense of Definition 1.

6. Technical results

6.1. Proof of Lemma 2. Since Tγ is a Lipschitz continuous function anduε ∈ L2(0, T ;H1(Ω)), one has Tγ(uε) ∈ L2(0, T ;H1(Ω)), see [6], moreover,

∇Tγ(uε) = 1|uε|≤γ∇uε.Thus, we choose ϕ = T1(uε) as a test function in (4.3); after integration byparts and using Holder’s inequality, it follows that∫

ΩS1(uε)(t, x) dx+

∫ t

0

∫Ωh(τ, x, uε)T1(uε)dxdτ


≤ ‖f‖L1(QT ) +∫

ΩS1(u0,ε) dx+

‖K‖2L∞(QT ) meas(QT )

2a

≤ ‖f‖L1(QT ) + ‖u0‖L1(Ω) +‖K‖2L∞(QT ) meas(QT )

2a,

and ∫|uε|>1

|h(τ, x, uε)| dxdt =∫|uε|>1

T1(uε)h(τ, x, uε) dxdt

≤∫ t

0

∫ΩT1(uε) h(τ, x, uε) dxdt. (6.1)

Since |s| ≤ 2|S1(s)|+ 1, ∀s and 0 ≤ S1(s) ≤ |s|, we deduce (4.5).Proof of (4.6) when (CL1) holds. We introduce a function Fγ , such thatFγ(0) = 0 and F ′γ(s) = s φ′γ(s) (see [24], page 17). Note that∫ T

0

∫Ωuε K · ∇φk(uε) dxdt =

∫ T

0

∫Ω

K · ∇Fk(uε) dxdt

=∫ T

0

∫∂Ω

K · η Fk(uε) dxdt−∫ T

0

∫Ω

div(K) Fk(uε) dxdt.

One also has 0 ≤ Fγ(s) ≤ s φγ(s) for s ≥ 0. Hence, by the choice of λ in(4.2) and the nonnegativity of uε, yield∫ T

0

∫Ω

(λ− b(t, x))uεφγ(uε) dxdt−∫ T

0

∫Ω

div(K) Fk(uε) dxdt ≥ 0.

Substituting ϕ = φγ(uε) in (4.3), the choices of λ in (4.2), the nonnegativityof uε and K · η ≥ 0 on ΣT , yield∫ T

0< ∂tuε, φγ(uε) > dt +

∫ T

0

∫Ω

A∇uε · ∇φγ(uε) dxdt

≤∫ T

0

∫Ωe−λt fε φγ(uε) dxdt ≤ ‖f‖L1(QT ), (6.2)

we have∫ T

0

∫Ω

A∇uε · ∇φγ(uε) dxdt =∫Bγ

A∇uε · ∇uε dxdt ≥ a∫Bγ

|∇uε|2 dxdt,

where Bγ is defined prior to Lemma 2. Hence

a∫Bγ

|∇uε|2 dxdt ≤ ‖f‖L1(QT ) +∫

ΩΨγ(u0,ε) dx ≤ ‖f‖L1(Ω) + ‖u0‖L1(Ω),

so that, α2 = 1a(‖f‖L1(Ω) + ‖u0‖L1(Ω)).


Proof of (4.6) when (CL2) holds. We still choose ϕ = φγ(uε) as atest function in (4.3); using the first part of (4.2) and the nonnegativity ofh(t, x, uε), we are led to∫ T

0< ∂tuε, φγ(uε) > dt+

∫ T

0

∫Ω

A∇uε · ∇φγ(uε) dxdt (6.3)

+∫ T

0

∫ΩuεK · ∇φγ(uε) dxdt ≤

∫ T

0

∫Ωe−λtfεφγ(uε) dxdt ≤ ‖f‖L1(QT ).

If p = 2, we have

|∫ T

0

∫Ωuε K · ∇φγ(uε) dxdt| = |

∫Bγ

uε K · ∇uε dxdt|

≤ a2

∫Bγ

|∇uε|2 dxdt+‖K‖2L∞(QT )

2a

∫Bγ

|uε|2 dxdt. (6.4)

Moreover, if p > 2, we are

|∫ T

0

∫Ωuε K · ∇φγ(uε) dxdt| = |

∫Bγ

uε K · ∇uε dxdt|

≤ a2

∫Bγ


2a

∫Bγ

|uε|2 dxdt

≤ a2

∫Bγ


2a(∫Bγ

|uε|p dxdt)2p )(mes(Bγ))1− 2

p .

We deduce from (4.5) and p ≥ 2 that

|∫ T

0

∫Ωuε K · ∇φγ(uε) dxdt| ≤

a2

∫ ∫Bγ

|∇uε|2 dxdt+ c, (6.5)

Then, we have α2 = 2a(‖f‖L1(Ω) + ‖u0‖L1(Ω) + c).

Proof of (4.7) when (CL3) hold. Using ϕ = φγ(uε) in (4.3), we havefrom (6.4)

|∫ T

0

∫ΩuεK ∂xφγ(uε) dxdt|

≤ a2

∫ ∫Bγ

|∂xuε|2 dxdt+‖K‖2L∞(QT )

2a

∫ T

0

∫Ω|uε|2 dxdt.

Then from (6.3), we have α3 = 2a(‖f‖L1(QT ) +‖u0‖L1(Ω)) et α4 =

‖K‖2L∞(QT )

a2 .


6.2. Proof of Proposition 1. Note that∫|uε|≥γ

|h(t, x, uε)| dxdt =1γ

∫|uε|≥γ

Tγ(uε)h(t, x, uε) dxdt

≤ 1γ

∫ t

0

∫ΩTγ(uε) h(t, x, uε) dxdt. (6.6)

Using ϕ = Tγ(uε) in (4.3), we have∫ΩSγ(uε)(t, x)dx+

∫ T

0

∫Ω

A∇uε · ∇Tγ(uε) dxdt

+∫ T

0

∫ΩuεK · ∇Tγ(uε) dxdt+

∫ t

0

∫ΩTγ(uε)h(t, x, uε) dxdt

≤∫ t

0

∫Ω|fεTγ(uε)| dxdt+

∫ΩSγ(u0,ε) dx.

Now ∫ T

0

∫Ω

A∇uε · ∇Tγ(uε) dxdt ≥ a∫ ∫

|uε|≤γ|∇uε|2 dxdt ≥ 0.

From nonnegativity of uε, one deduces

0 ≤∫ t

0

∫ΩTγ(uε)h(t, x, uε) dxdt

≤∫ t

0

∫Ω|fεTγ(uε)|dxdt+

∫ΩSγ(v0,ε)dx+ ‖K‖L∞(QT )

∫ t

0

∫Ω|Tγ(uε)∇uε| dxdt.

Let M > 0. According to [19], we use the following, ∀s0 ≤ Sγ(s) ≤M2 + γ|s|.1|s|>M, (6.7)

0 ≤ |Tγ(s)| ≤M + γ.1|s|>M. (6.8)

We conclude that∫|uε|≥γ

|h(t, x, uε)| dxdt ≤M

γ‖fε‖L1(QT ) +

∫|uε|>M

|fε|dxdt

+M2

γ‖u0,ε‖L1(Ω) +

∫|u0,ε|>M

|u0,ε|dx

+ ‖K‖L∞(QT )(M

γ‖∇uε‖L1(QT ) +

∫|uε|>M

|∇uε| dxdt).

We know (uε)0<ε≤1 is bounded in Lq(0, T ;W 1,q(Ω)) for some q ≥ 1 and(fε)0<ε≤1, (u0,ε)0<ε≤1 are converging in L1(QT ), L1(Ω) respectively, and, on


the other hand,

meas|uε| > M ≤ 1M‖uε‖L1(Ω), (6.9)

tends to 0 as M goes to∞, we can choose M large enough so that the terms

sup0<ε≤1

∫|u0,ε|>M

|u0,ε|dx, sup0<ε≤1

∫|uε|>M

|fε|dx, sup0<ε≤1

∫|uε|>M

|∇uε| dxdt

are arbitrarily small. This completes the proof.

6.3. Proof of Lemma 3. By (4.10), it follows that

h(t, x, uε(t, x)) −→ h(t, x, u(t, x))

a.e. in QT as ε tends to 0. This sequence is bounded in L1(QT ) by (4.5).Thus, by a theorem of Vitali, see e.g. [7], the sequence (h(t, x, uε))0<ε≤1 isstrongly convergent in L1(QT ) if one can show that (h(t, x, uε)0<ε≤1 lies inweakly compact subset of L1(QT ).

To this end, let E be a measurable set in QT . We split∫E|h(t, x, uε)| dxdt =

∫E∩|uε|<γ

|h(t, x, uε)| dxdt+∫E∩|uε|>γ

|h(t, x, uε)| dxdt

≤ γp(∫Ek(t, x)e(p−1)λt dxdt) +

∫|uε|>γ

|h(t, x, uε)| dxdt

≤ γpkmax e(p−1)λt(meas(E)) +

∫|uε|>γ

|h(t, x, uε)| dxdt.

By Proposition 1,∫|uε|>γ |h(t, x, uε)| dxdt tends to 0, uniformly in ε as

γ −→∞. We conclude that

lim|E|−→0

∫E|h(t, x, uε)| dxdt = 0.

Hence, (h(t, x, uε))0<ε≤1 is a compact subset of L1(QT ), which completesthe proof of Lemma 3.

6.4. Proof of Lemma 4. It suffices to show that (∇uε)0<ε≤1 is a Cauchysequence in measure, see [7], i.e., ∀µ > 0

meas(t, x); |(∇uε′ −∇uε)(t, x)| ≥ µ −→ 0

as ε, ε′ −→ 0. Let γ > 0 and δ > 0, we have (see [22])

(t, x); |(∇uε′ −∇uε)(t, x)| ≥ µ ⊂∇uε ≥ γ ∪ ∇uε′ ≥ γ ∪ |uε − uε′ | ≥ δ

∪|∇uε −∇uε′ | ≥ µ, |∇uε| ≤ γ, |∇uε′ | ≤ γ, |uε − uε′ | ≤ δ. (6.10)


Let us denote G ⊂ G1 ∪ G2 ∪ G3 ∪ G4 the subsets of QT in (6.10). ByCorollary 4.1. and (4.10), we easily conclude for the first three sets. Indeed,one has

meas(G1) ≤ 1γ

∫ T

0

∫Ω|∇uε| dxdt ≤

C

γ,

and an analogous estimate holds for G2. Hence, by choosing γ large enough,meas(G1) + meas(G2) is arbitrarily small. Similarly we get

meas(G3) ≤ 1δ‖uε − uε′‖L1(QT ),

which, for δ > 0 fixed, tends to 0 when ε, ε′ −→ 0 since by (4.10) uε is aCauchy sequence in L1(QT ). Then, the proof is completed by choosing δso that meas(G4) is arbitrarily small, uniformly with respect to ε, ε′. Weobserve that

meas(G4) ≤ 1µ2

∫G4

|∇uε −∇uε′ |2 dxdt

≤ 1µ2

∫|uε−uε′ |≤δ

|∇uε −∇uε′ |2 dxdt =1µ2

∫ T

0

∫Ω|∇(Tδ(uε − uε′))|2 dxdt.

Let ϕ = Tδ(uε − uε′) in equation (4.6) satisfied by uε and uε′ . We have∫ t

0< (uε − uε′)t, Tδ(uε − uε′) > dτ

+∫ t

0

∫Ω

[A(∇uε −∇uε′)] · ∇Tδ(uε − uε′)dxdτ

+∫ t

0

∫Ω

(uε − uε′)K · ∇Tδ(uε − uε′)dxdτ

+∫ t

0

∫Ω

(λ− b+m)(uε − uε′)Tδ(uε − uε′)dxdτ

+∫ t

0

∫Ω

(h(t, x, uε)− h(t, x, uε′)Tδ(uε − uε′) dxdt

=∫ t

0

∫Ωe−λt(fε − fε′) Tδ(uε − uε′) dxdτ. (6.11)

Since Tγ(s) ≤ δ and 0 ≤ Sδ(s) ≤ δ|s|, integrating (6.11), and using thecoercivity of A and from the monotonicity of h(t, x, uε) in uε, yield

a∫ t

0

∫Ω|∇Tδ(uε − uε′)|2 dxdt ≤ δ(‖fε − fε‖L1(QT ) + ‖u0,ε − u0,ε′‖L1(QT )

+ ‖K‖L1(QT )‖∇uε −∇uε‖L1(QT )). (6.12)


Therefore, by using (4.1) and the bounds (4.8) uniform in ε, on ‖uε‖L1(QT )

and ‖∇uε‖L1(QT ), we deduce from (6.12) that

a∫ t

0

∫Ω|∇Tδ(uε−uε′)|2dxdt≤2δ(‖f‖L1(QT )+‖u0‖L1(QT ))+2δ(C‖∇uε‖L1(QT )).

goes to 0 as δ goes to 0, uniformly in ε, ε′. This completes the proof.

6.5. Proof of Lemma 5. We set Hε,ε′ = h(t, x, uε)−h(t, x, uε′) and Fε,ε′ =fε − f ′ε. We multiply equation (4.3) by ϕ = T1(uε − uε′). Subtracting theresulting relations, yields∫ t

0< (uε − uε′)t, T1(uε − uε′) > dτ

+∫ t

0

∫Ω

[A(∇uε −∇uε′)] · ∇T1(uε − uε′)dxdτ

+∫ t

0

∫Ω

(uε − uε′)K · ∇T1(uε − uε′)dxdτ

+∫ t

0

∫Ω

(λ− b+m) (uε − uε′)T1(uε − uε′)dxdτ

+∫ t

0

∫ΩHε,ε′T1(uε − uε′) dxdt

=∫ t

0

∫Ωe−λtFε,ε′ T1(uε − uε′) dxdτ ≤

∫ t

0

∫Ω| Fε,ε′ | dxdτ. (6.13)

Using the coercivity of A, the choice of λ in (4.2) and integrating between0 and t, it follows from the monotonicity of h(t, x, uε) in uε that∫

Ω[S1(uε − uε′)](t, x) dx−

∫ΩS1(u0,ε(x)− u0,ε′(x)) dx

≤ ‖Fε,ε′‖L1(QT ) + ‖K‖L∞(QT )‖∇uε −∇u′ε‖L1(QT ). (6.14)

By Lemma .4 and Corollary 1, ∇uε is bounded Lq(QT ) and is convergenta.e. in QT , which implies that ∇uε is strongly convergent in Lq0(QT ) for1 ≤ q0 < q, and in particular in L1(QT ). Furthermore, fε and u0,ε areconvergent sequences in L1(QT ) and L1(Ω) respectively. Hence, it is clearthat

limε,ε′−→0

∫Ω

[S1(uε − uε′)](t, x) dx = 0, ∀t ≤ T.

Last, by (4.4)∫Ω|uε − uε′ |(t, x) dx =

∫|uε−uε′ |≥1

|uε − uε′ |(t, x) dx+∫|uε−uε′ |≤1

|uε − uε′ |(t, x) dx


≤ (∫|uε−uε′ |<1

|uε − uε′ |2(t, x) dx)12 meas(Ω)

12 + 2

∫Ω

[S1(uε − uε′)](t, x)dx

≤√

2(∫

Ω[S1(uε − uε′)](t, x) dx)

12 meas(Ω)

12 + 2

∫Ω

[S1(uε − uε′)](t, x) dx,

tends to 0 as ε, ε′ −→ 0, which proves that uε is a Cauchy sequence inC([0, T ];L1(Ω)).

6.6. Proof of Proposition 2. We choose ϕ = ψ = χ = 1 as a test functionin (5.3), (5.4) and (5.5) respectively; adding up the resulting equations, givesd

dt

∫Ω(uε + vε + wε)dx+

∫Ωe(p−1)λt(k1uε + k2vε + k3wε)|uε + vε + wε|p−1dx

+∫

Ω(λ− b)(uε + wε) dx+

∫Ωm(uε + wε) dx+

∫Ω

(m+ α) vε dx

=∫

Ωe−λt(fε + gε + hε) dx ≤ ‖(f + g + h)(·, t)‖L1(Ω). (6.15)

The solutions are nonnegative; by the choice of λ in (5.2), it follows thatd

dt

∫Ω|(uε + vε + wε)(x, ·)| dx ≤ ‖(f + g + h)(·, t)‖L1(Ω). (6.16)

Hence, (5.6) follows.Next substituting ϕ = 1 in equation (5.3), we have∫ T

0< ∂tuε, 1 > dt+

∫ T

0

∫Ωr1,λ(uε, vε, wε) dxdt−

∫ T

0

∫Ωb wε dxdt

+∫ T

0

∫Ωe−λtσ(eλtuε, eλtvε, eλtwε) dxdt+

∫ T

0

∫Ω

(λ+m− b)uε dxdt

=∫ T

0

∫Ωe−λtfε dxdt ≤ ‖f‖L1(QT ). (6.17)

Since the solution (uε, vε, wε) is nonnegative, wε is bounded in L∞(0, T ;L1(Ω)) and by the choice of λ in (5.2), we deduce

‖σ(eλtuε, eλtvε, eλtwε)‖L1(QT )

≤ eλT (‖u0‖L1(Ω) + ‖f‖L1(QT ) + ‖b‖L∞(QT )c1T ). (6.18)

Now we prove the second estimate in (5.7). After integrating (6.15), wededuce ∫ T

0

∫Ωe(p−1)λt(k1uε + k2vε + k3wε)|uε + vε + wε|p−1 dxdt

≤ ‖u0 + v0 + w0‖L1(Ω) + ‖f + g + h‖L1(QT ), (6.19)


then, using condition (2.5) as ki, i = 1, 2, 3, yields

k0

∫ T

0

∫Ω|uε+vε+wε|pdxdt ≤ ‖u0+v0+w0‖L1(Ω)+‖f+g+h‖L1(QT ), (6.20)

which completes the proof of estimate (5.7).The proof of estimates (5.8)-(5.9) are similar to the proofs of (4.6)-(4.7),

using the already known estimates (5.7) as the nonlinear terms.

6.7. Proof of Proposition 3. Note that∫|uε|≥γ

r1,λ(uε, vε, wε) dxdt+∫|uε|≥γ

e−λtσ(eλtuε, eλtvε, eλtwε) dxdt

=1γ

∫|uε|≥γTγ(uε)r1,λ(uε, vε, wε) dxdt+

1γ

∫|uε|≥γTγ(uε)σ(eλtuε, eλtvε, eλtwε) dxdt

≤ 1γ

∫ T

0

∫ΩTγ(uε) (r1,λ(uε, vε, wε) + σ(eλtuε, eλtvε, eλtwε)) dxdt. (6.21)

Using ϕ= Tγ(uε) in equation (5.3), we have∫ΩSγ(uε)(t, x)dx+

∫ T

0

∫Ω

A1∇uε · ∇Tγ(uε) dxdt

+∫ T

0

∫ΩuεK1 · ∇Tγ(uε) dxdt+

∫ T

0

∫Ωr1,λ(uε, vε, wε)Tγ(uε)dxdt

+∫ t

0

∫Ωe−λtσ(eλtuε, eλtvε, eλtwε)Tγ(uε) dxdt

≤∫

ΩSγ(u0,ε) dx+

∫ t

0


∫ t

0

∫Ω|b wε Tγ(uε)| dxdt.

Since ∀s, Sγ(s) ≥ 0 and using the coercivity of A1, yield

0 ≤∫ T

0

∫Ωr1,λ(uε, vε, wε) Tγ(uε) dxdt

+∫ T

0

∫Ωe−λtσ(eλtuε, eλtvε, eλtwε)Tγ(uε) dxdt

≤∫

ΩSγ(u0,ε) dx+

∫ T

0


∫ T

0

∫Ω|bwεTγ(uε)| dxdt

+ ‖K1‖L∞(QT )

∫ T

0

∫Ω|Tγ(uε)∇uε| dxdt. (6.22)


Using (6.7)-(6.8), we deduce∫|uε|≥γ

r1,λ(uε, vε, wε) dxdt+∫|uε|≥γ

|e−λtσ(eλtuε, eλtvε, eλtwε)| dxdt

≤ M2

γ‖u0,ε‖L1(Ω) +

∫|u0,ε|>M

|u0,ε|dx+M

γ‖fε‖L1(QT ) +

∫|uε|>M

|fε| dxdt

+ ‖b‖L∞(QT )M

γ‖wε‖L1(QT ) + ‖b‖L∞(QT )

∫|uε|>M

|wε| dxdt

+M

γ‖K1‖L∞(QT )‖∇uε‖L1(QT ) + ‖K1‖L∞(QT )

∫|uε|>M

|∇uε| dxdt. (6.23)

We know (uε)0<ε≤1 is bounded in Lq(0, T ;W 1,q(Ω)) for some q ≥ 1 and(u0,ε)0<ε≤1, (fε)0<ε≤1 and (wε)0<ε≤1 are converging in L1(Ω), L1(QT ) andL1(QT ) respectively. On the other hand,

meas|uε| > M ≤ 1M‖uε‖L1(Ω), (6.24)

tends to 0 as M goes to∞, we can choose M large enough so that the terms

sup0<ε≤1

∫|uε|>M

|fε|dx, sup0<ε≤1

∫|u0,ε|>M

|u0,ε|dx,

sup0<ε≤1

∫|uε|>M

|∇uε| dxdt, sup0<ε≤1

∫|uε|>M

|wε|dx(6.25)

are arbitrarily small. This completes the proof of (5.11).By (5.10), it follows that σ(eλtuε, eλtvε, eλtwε) −→ σ(eλtu, eλtv, eλtw) and

r1,λ(t, x, uε, vε, wε) −→ r1,λ(t, x, u, v, w) a.e. in QT as ε tends to 0, thesesequence are bounded in L1(QT ) by (5.7). Thus, by a theorem of Vi-tali, see e.g. [7], the sequence (eλtuε, eλtvε, eλtwε))0<ε≤1 and the sequence(r1,λ(t, x, uε, vε, wε))0<ε≤1 are strongly convergent in L1(QT ) if one can showthat both sequences (eλtuε, eλtvε, eλtwε))0<ε≤1 and (r1,λ(t,x, uε, vε, wε))0<ε≤1

lie in weakly compact subsets of L1(QT ).To this end, let B be a measurable set in QT . We split∫ ∫Bσ(eλtuε, eλtvε, eλtwε) dxdt

=∫ ∫

B∩|uε|<γ|σ(eλtuε, eλtvε, eλtwε)| dxdt+

∫ ∫B∩|uε|>γ

|σ(eλtuε, eλtvε, eλtwε)| dxdt

≤∫ ∫

B∩|uε|<γ|σ(eλtuε, eλtvε, eλtwε)| dxdt+

∫ ∫|uε|>γ|σ(eλtuε, eλtvε, eλtwε)| dxdt.


First∫ ∫|uε|>γ |σ(eλtuε, eλtvε, eλtwε)| dxdt tends to 0, uniformly in ε as γ

tend to ∞ according to (5.11). Next, from the growth condition (2.6), wehave∫ ∫

B∩|uε|<γ|σ(eλtuε, eλtvε, eλtwε)| dxdt

≤ ‖L‖L∞(QT )

∫ ∫B

(γl |vε|s + γt |wε|s) dxdt+∫ ∫

B|M(t, x)| dxdt.

Since 1 ≤ s < Max(N+2N+1 , p) (see (2.6)), using (5.10), we conclude that

limmeas(B)→0

sup0<ε≤1

∫ ∫B∩|uε|<γ

|σ(eλtuε, eλtvε, eλtwε)| dxdt = 0.

Next, one has,∫ ∫Br1,λ(t, x, uε, vε, wε) dxdt

=∫ ∫

B∩|uε|<γ|r1,λ(t, x, uε, vε, wε)| dxdt+

∫ ∫B∩|uε|>γ

|r1,λ(t, x, uε, vε, wε)| dxdt

≤∫ ∫

B∩|uε|<γ|r1,λ(t, x, uε, vε, wε)| dxdt+

∫ ∫B∩|uε|>γ

|r1,λ(t, x, uε, vε, wε)| dxdt.

Where, by (5.11),∫ ∫|uε|>γ |r1,λ(t, x, uε, vε, wε)| dxdt tend to 0, as γ −→∞.

Then∫ ∫B∩|uε|<γ

|r1,λ(t, x, uε, vε, wε)| dxdt

≤ ‖k1‖L∞(QT )

∫ ∫B∩|uε|<γ

(|uε| |uε + vε + wε|p−1) dxdt

≤ ‖k1‖L∞(QT )(∫ ∫

B∩|uε|<γ|uε|p dxdt)

1p (∫ ∫

B∩|uε|<γ|uε + vε + wε|p dxdt)

p−1p

≤ γ ‖k1‖L∞(QT )(mes(B))1p (∫ ∫

B|uε + vε + wε|p dxdt)

p−1p . (6.26)

Using (5.10), we conclude that

limmeas(B)→0

sup0<ε≤1

∫ ∫B∩|uε|<γ

|r1,λ(t, x, uε, vε, wε)| dxdt = 0,

which completes the proof of estimate (5.11), (5.12) and (5.13) for i = 1.The proof of estimate (5.13) for i = 2, 3 is similar to the proof of estimates(5.11) and (5.13) for i = 1.

existence of solutions for reaction-diffusion systems …saad/cv/art1.pdf · existence of solutions...

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