EXISTENCE OF SOLUTIONS FOR REACTION-DIFFUSION SYSTEMS WITH L1 DATA
M. Bendahmane
UMR CNRS 5466, Math´ematiques Appliqu´ees de Bordeaux, Universit´e Bordeaux I 351, cours de la Lib´eration, F-33405 Talence Cedex
M. Langlais
UMR CNRS 5466, Math´ematiques Appliqu´ees de Bordeaux, Case 26 Universit´e Victor Segalen Bordeaux 2
146, rue L´eo-saignat, F-33076 Bordeaux Cedex M. Saad
UMR CNRS 5466, Math´ematiques Appliqu´ees de Bordeaux, Universit´e Bordeaux I 351, cours de la Lib´eration, 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 a heterogeneous 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 inRN (N ≥1) of Feline Leukemia Virus (FeLV), a feline retro-virus; see [13] and [20] for a model derivation and an analysis of the underlying system of ordinary differential equations. A prototype of a non- linear system that governs the spreading of FeLV through a cat population in a heterogeneous spatial domain with seasonal variations and external supply is 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
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) andw(t, x) represent the spatial densities at time tand location x ∈ Ω of susceptible, infectious and immune individuals. Thus, we recover the time dependent state variables used in [13] and [20], where spatial considerations are ignored, upon integrating over space
Z
Ω
u(t, x) dx, Z
Ω
v(t, x) dx and Z
Ω
w(t, x) dx.
This also showsL1(Ω) is a natural functional space for developing a mathe- matical analysis of the spatial propagation of an epidemic disease.
The diffusitivity matrixAi is a bounded symmetric and coercive matrix, the transport vectorKiis bounded on (0, T)×Ω,i= 1,2,3. The birth rateb, the death ratemand the additional disease induced death rate in the infected classα, are assumed to be bounded on (0, T)×Ω. The incidence termσ and the 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+wu v ,
σ3(t, x)1+(u+v+w)(u+v+w)νν u v
u+v+w, ν >0.
In this work, for somep >1 we take
r1(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] withL∞ data.
In the logistic case (p= 2) for constant coefficients and no advection terms existence results are established in [9] with L∞-data. See also [10].
Assumingσ = 0, ri = 0 fori= 1,2,3 and under Dirichlet boundary con- ditions, existence results for the corresponding linear parabolic system with non regular data are established in [1], [4], [5], [8], [14], [21] and [23] while uniqueness questions, in the sense of entropic or renormalized formulations, are considered in [3], [22].
2. Notations, Assumptions
2.1. Notations. Here, Ω is a bounded open domain of RN (N ≥ 1) with smooth boundary ∂Ω, so that locally Ω lies on one side of ∂Ω; η is the outer unit normal to Ω on ∂Ω. We denoteD+(Ω) the space of nonnegative functions inC0∞(Ω). The norm inLp(Ω) is denoted byk·kLp(Ω), 1≤p <∞;
then
Lp+(Ω) ={u: Ω−→R+ measurable and Z
Ω
|u|p(x)dx <∞}, L∞+(Ω) ={u: Ω−→R+ measurable and sup
x∈Ω|u(x)|<∞}.
If 1 ≤ p ≤ ∞, W1,p(Ω) the Sobolev space of functions u on the open set Ω for which u and ∇u belong to Lp(Ω). If X is a Banach space, a < b and 1 ≤ p ≤ ∞, Lp(a, b;X) denotes the space of all measurable function u: (a, b) −→X such that ku(·)kX belongs to Lp(a, b). Next T is a positive number and QT = (0, T)×Ω, ΣT = (0, T)×∂Ω. We denoteCc1([0, T)×Ω) the set of allC1-functions with compact support in [0, T)×Ω. Let us recall the definition of truncated functionTγ. Letγ ∈R+. We set
Tγ(z) =
−γ ifz≤ −γ z if|z| ≤γ γ ifz≥γ and we denoteSγ(z) =Rz
0 Tγ(τ)dτ. We introduce the functionφγ =Tγ+1− Tγ, that is
φγ(z) =
1 ifz≥γ+ 1 z−γ ifγ ≤z < γ+ 1 0 if −γ ≤x≤γ z+γ if −γ−1< z≤ −γ
−1 ifz≥ −γ−1 and we set Ψγ(z) = Rz
0 φγ(τ) dτ. We note that Tγ and φγ are Lipschitz continuous functions.
2.2. Assumptions. Throughout this paper, the following assumptions are assumed to hold true.
First Ai, Ki and ki (see (1.1)) for i= 1,2,3,α,b,m are given functions defined onQT with values inRN×RN,RN andR+,R+,R+,R+, respectively.
Our basic requirement onAi,Ki,α,b,mand ki is
Ai ∈(L∞(QT))N×N, fori= 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 andkmax, k0>0 such that fori= 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 with respect tou,v etw, and satisfies a growth condition
there exist p >1 and two bounded functions
L, M :RN ×[0,∞)→[0,∞), andl, s, l0 ∈R+ such that l, l0 >0, 1≤s <max(NN+2+1, p) and
|σ(t, x, u, v, w)| ≤L(t, x)(|u|l|v|s+|u|l0|w|s) +M(t, x) a.e. x, t∈QT;
(2.6)
and a nonnegativity condition
σ(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, fori= 1,2,3,one assumesri is given by (1.1) withpas in (2.6), andki
as in (2.3) and (2.5) fori= 1,2,3. Lastπ is a real number with 0< π <1.
The main difficulty in this work arises from the fact that we only consider L1-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 nonlinear parabolic systems of type (S1)-(S2)-(S3) withL1right-hand sides and initial data. Then, we supply our existence result.
Definition 1. Let 1 ≤ q < N+2N+1 if N ≥ 2 and 1 ≤ q < 43 if N = 1. A weak solution of (S1)-(S2)-(S3), is a triple (u, v, w) of nonnegative functions belonging toLq(0, T;W1,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
− Z T
0
Z
Ω
uϕt dxdt− Z
Ω
ϕ(0, x)u0(x)dx
+ Z T
0
Z
Ω
(A1(t, x)∇u+uK1(t, x))· ∇ϕ dxdt +
Z T 0
Z
Ω
m(t, x)uϕ dxdt− Z T
0
Z
Ω
b(t, x)(u+w)ϕ dxdt +
Z T 0
Z
Ω
σ(t, x, u, v, w)ϕ dxdt+ Z T
0
Z
Ω
r1(t, x, u, v, w)ϕ dxdt
= Z T
0
Z
Ω
f ϕ dxdt,
− Z T
0
Z
Ω
vψtdxdt− Z
Ω
ψ(0, x)v0(x)dx +
Z T
0
Z
Ω
(A2(t, x)∇v+vK2(t, x))· ∇ψ dxdt +
Z T
0
Z
Ω
(m(t, x) +α(t, x))vψ dxdt−π Z T
0
Z
Ω
σ(t, x, u, v, w)ψ dxdt
+ Z T
0
Z
Ω
r2(t, x, u, v, w)ψ dxdt= Z T
0
Z
Ω
g ψ dxdt,
− Z T
0
Z
Ω
wχtdxdt− Z
Ω
χ(0, x)w0(x)dx +
Z T
0
Z
Ω
(A3(t, x)∇w+wK3(t, x))· ∇χ dxdt +
Z T
0
Z
Ω
m w χ dxdt−(1−π) Z T
0
Z
Ω
σ(t, x, u, v, w)χ dxdt
+ Z T
0
Z
Ω
r3(t, x, u, v, w)χ dxdt= Z T
0
Z
Ω
h χ dxdt, for all ϕ, ψ, χ∈Cc1([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 natural condition in this work.
In the one dimensional case, various simplifications occur, as in [16]; this allows us to get rid of (CN1) and (CN2). Actually, our solution has more regularity properties when N = 1 than when N ≥ 2; more optimal results could be derived, see [2].
The proof is organized as it follows. Section 4 is devoted to a related scalar equation. Some a priori estimates for smooth solutions of the systems are obtained in section 5 and used for completing the proof. In section 6 we prove 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 = 1,2,3 respectively.
Our system reduces to this scalar problem whenA1 =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 < 43 if N = 1.
A weak solution of (E), is a nonnegative function u in Lq(0, T;W1,q(Ω)∩ Lp(0, T;Lp(Ω))∩C([0, T];L1(Ω)) satisfying
− Z T
0
Z
Ω
uϕtdt+ Z T
0
ϕ(0, x)u0(x)dx+ Z T
0
Z
Ω
(A(t, x)∇u+uK(t, x))· ∇ϕ dxdt +
Z T 0
Z
Ω
(m(t, x)−b(t, x))uϕ dxdt+ Z T
0
Z
Ω
k(t, x)upϕ dxdt= Z T
0
Z
Ω
f ϕ dxdt,
for all ϕ∈Cc1([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 and a 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 inL1(Ω), fε→f inL1(QT), asε→0;
ku0,εkL1(Ω)≤ ku0kL1(Ω), kfεkL1(QT)≤ kfkL1(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))0), solutions of (E) whereu0 andf are replaced by (u0,ε)0<ε≤1 and (fε)0<ε≤1. Letλ >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) Setuε(t, x) =eλtu˜ε(t, x). Then ˜uε(t, x) satisfies
Z T
0
< ∂tu˜ε, ϕ > dt + Z T
0
Z
Ω
(A(t, x)∇u˜ε+ ˜uε K(t, x))· ∇ϕ dxdt +
Z T
0
Z
Ω
(λ+m(t, x)−b(t, x)) ˜uεϕ dxdt+ Z T
0
Z
Ω
h(t, x,u˜ε)ϕ dxdt
= Z T
0
Z
Ω
e−λtfεϕ dxdt, (4.3)
for all ϕ∈L2(0, T;H1(Ω))∩L∞(QT). Herein
h(t, x,u˜ε) =k(t, x)e(p−1)λt|˜uε(t, x)|p−1u˜ε(t, x).
4.1. Nonnegativity.
Lemma 1. The solutionu˜ε is nonnegative.
Proof. Let ˜u−ε = sup(−u˜ε,0). Substitutingϕ=−Tγ(˜u−ε) in (4.3), one has Z t
0
< ∂tu˜−ε, Tγ(˜u−ε)> dτ + Z t
0
Z
Ω
(A∇˜u−ε + ˜u−εK)· ∇Tγ(˜u−ε)dxdτ +
Z t
0
Z
Ω
(λ+m(t, x)−b(t, x))˜u−εTγ(˜u−ε)dxdτ +
Z t
0
Z
Ω
k(t, x) e(p−1)λt|˜uε|p−1u˜−εTγ(˜u−ε)dxdτ+ Z t
0
Z
Ω
e−λτfεTγ(˜u−ε)dxdτ = 0.
From the choice of λ in (4.2), the nonnegativity of m and fε and using H¨older’s inequality, one gets
d dt
Z
Ω
Sγ(˜u−ε(t, x))dx+a 2
Z
Ω|∇Tγ(˜u−ε(t, x))|2 dx
≤ kKk2L∞(QT)
2a Z
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
Z
Ω
Sγ(˜u−ε(t, x))dx≤ kKk2L∞(QT)
a Z
Ω
Sγ(˜u−ε(t, x))dx.
Since the datau0,ε is nonnegative, we deduce that ˜u−ε = 0.
4.2. A priori estimates. In this section, we are concerned with a priori estimates satisfied by the sequence (˜uε)0<ε≤1 which lead to compactness properties. 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 that the solutions of (4.3)satisfy
k˜uεkLp(QT)+k˜uεkL∞(0,T;L1(Ω))≤α1. (4.5) Assuming condition (CL1) or condition (CL2) to hold, there exists α2 > 0 not depending onε (0< ε≤1), such that
sup
γ≥0
Z
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
sup
γ≥0
Z
Bγ|∂xu˜ε(t, x)|2dxdt≤α3+α4
Z T
0
Z
Ω|u˜ε(t, x)|2dxdt. (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 < 43 when N = 1. Then, there exists C >0 depending onα1, α2, α3, α4, meas(Ω), T and q such that
ku˜εkLq(0,T;W1,q(Ω))≤C, 0< ε≤1. (4.8)
Now, we are interested in the nonlinear term h(t, x,u˜ε). We have
Proposition 1. Let (˜uε)0<ε≤1 be the sequence of solutions of (4.3). Then the sequence (h(t, x,u˜ε))0<ε≤1 satisfies
γlim→∞ sup
0<ε≤1
Z
{|u˜ε|≥γ}h(t, x,u˜ε) 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 the sequence (˜uε)0<ε≤1 is bounded in Lq(0, T;W1,q(Ω)), for 1≤q < N+2N+1 when N ≥ 2, and for 1 ≤ q < 43 when N = 1. In view of the equation satisfied by ˜uε this implies that ∂tu˜ε is bounded in L1(0, T;¡
W1,q(Ω)¢0
) +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 inLq(0, T;W1,q(Ω)), such that asε−→0
˜
uε−→u˜ strongly in Lq(QT) and a.e. in QT,
∇˜uε−→ ∇˜u weakly in Lq(QT), h(t, x,u˜ε)−→h(t, x,u)˜ a.e. in QT,
(4.10) The convergence property of h(t, x,u˜ε) in (4.10) is too weak. It can be improved:
Lemma 3. The sequence (h(t, x,u˜ε))0<ε≤1 converges to h(t, x,u)˜ almost everywhere in QT and strongly in L1(QT).
A proof is found in Subsection 6.3. This result is similar to those obtained in [15] in the context of elliptic problems.
Since h(t, x,u˜ε) = k(t, x)e(p−1)λt|˜uε(t, x)|p−1u˜ε(t, x) and by Lemma 3, it is clear that the sequence (˜uε)0<ε≤1 converges strongly inLp(QT) forp >1.
We complete the properties of the sequence ˜uεwith the following two results.
Lemma 4. The sequence (∇˜uε)0<ε≤1 converges to ∇˜u a.e. inQT as εgoes to zero.
Lemma 5. The sequence(˜uε)0<ε≤1is a Cauchy sequence inC([0, T];L1(Ω)).
The proof of Lemma 4 and Lemma 5 are found in Subsection 6.4 and Subsection 6.5 respectively. Now using the following weak formulation
− Z T
0
Z
Ω
uεϕtdxdt+ Z T
0
ϕ(0, x)u0,ε(x)dx +
Z T
0
Z
Ω
(A(t, x)∇uε+uεK(t, x))· ∇ϕ dxdt+
Z T
0
Z
Ω
(m(t, x)−b(t, x))uεϕ dxdt
+ Z T
0
Z
Ω
k(t, x)upεϕ dxdt= Z T
0
Z
Ω
fεϕ dxdt, (4.11)
withϕ∈Cc1([0, T)×Ω), we can letε−→0 and obtain a weak solution.
5. Sketch of the proof of Theorem 1.
We introduce ˆσ is measurable onQT, continuous with respect tou,v and w.
ˆ
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), whereσis replaced by ˆσ. We introduce the following smooth approximations of the data u0, v0, w0 and f, 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
kZεkL1(QT)≤ kZkL1(QT), Zε→Z inL1(QT), asε→0;
kZ0,εkL1(Ω)≤ kZ0kL1(Ω), Z0,ε →Z0 inL1(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))0), of solutions of (S1)-(S2)-(S3), where u0, v0, w0 and f, g, h are replaced by u0,ε, v0,ε, w0,ε and fε, gε, hε respectively, and σ is replaced by ˆσ. Letλ > 0 satisfies
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 fori= 2,3.
(5.2) We will often write ˆσ(t, x,·,·,·) = ˆσ(·,·,·) and ri(t, x,·,·,·) = ri(·,·,·) for i = 1,2,3 when no confusion can arise. Set uε = eλtu˜ε, vε = eλt˜vε and wε=eλtw˜ε; then ˜uε, ˜vε and ˜wε satisfies
Z T
0
< ∂tu˜ε, ϕ > dt+ Z T
0
Z
Ω
A1∇u˜ε· ∇ϕ dxdt+ Z T
0
Z
Ω
˜
uεK1· ∇ϕ dxdt
+ Z T
0
Z
Ω
e−λtσ(eˆ λtu˜ε, eλtv˜ε, eλtw˜ε)ϕ dxdt+ Z T
0
Z
Ω
(λ+m−b)˜uεϕ dxdt
− Z T
0
Z
Ω
bw˜εϕdxdt+ Z T
0
Z
Ω
r1,λ(˜uε,˜vε,w˜ε)ϕdxdt= Z T
0
Z
Ω
e−λtfεϕ dxdt, (5.3) Z T
0
< ∂tv˜ε, ψ > dt+ Z T
0
Z
Ω
A2∇˜vε· ∇ψ dxdt+ Z T
0
Z
Ω
˜
vεK2· ∇ψ dxdt
+ Z T
0
Z
Ω
(m+λ+α) ˜vεψ dxdt−π Z T
0
Z
Ω
e−λtσ(eˆ λtu˜ε, eλt˜vε, eλtw˜ε)ψ dxdt +
Z T
0
Z
Ω
r2,λ(˜uε,v˜ε,w˜ε)ψ dxdt= Z T
0
Z
Ω
e−λtgεψ dxdt, (5.4) Z T
0
< ∂tw˜ε, χ > dt+ Z T
0
Z
Ω
A3∇w˜ε· ∇χ dxdt+ Z T
0
Z
Ω
˜
wεK3· ∇χ dxdt
+ Z T
0
Z
Ω
(λ+m) ˜wεχ dxdt−(1−π) Z T
0
Z
Ω
ˆ
σ(eλtu˜ε, eλtv˜ε, eλtw˜ε)χ dxdt +
Z T 0
Z
Ω
r3,λ(˜uε,v˜ε,w˜ε)χ dxdt= Z T
0
Z
Ω
e−λthεχ dxdt, (5.5) for all ϕ, ψ, χ∈L2(0, T;H1(Ω))∩L∞(QT).
Hereinri,λ(t, x,u˜ε,v˜ε,w˜ε) =e(p−1)λtri(t, x,u˜ε,˜vε,w˜ε) for i= 1,2,3.
5.1. Nonnegativity.
Lemma 6. The solution(˜uε,˜vε,w˜ε) is nonnegative.
Proof. Using the definition of ˆσ, we have Z T
0
Z
Ω
e−λtˆσ(eλtu˜ε, eλtv˜ε, eλtw˜ε)Tγ( ˜wε−)dxdt≥0, Z T
0
Z
Ω
e−λtσ(eˆ λtu˜ε, eλt˜vε, eλtw˜ε)Tγ(˜vε−)dxdt≥0, Z T
0
Z
Ω
e−λtσ(eˆ λtu˜ε, eλtv˜ε, eλtw˜ε)Tγ(˜u−ε)dxdt= 0,
and Z T
0
Z
Ω
r3,λ(˜uε,v˜ε,w˜ε)Tγ( ˜wε−)dxdt≤0, Z T
0
Z
Ω
r2,λ(˜uε,v˜ε,w˜ε)Tγ(˜vε−)dxdt≤0,
Z T
0
Z
Ω
r1,λ(˜uε,˜vε,w˜ε)Tγ(˜u−ε)dxdt≤0.
Let us substitute in first χ = −Tγ( ˜w−ε) in (5.5) where ˜w−ε = sup(−w˜ε,0).
Sincehεis nonnegative, the choice ofλin (5.2), and using H¨older’s inequality and (4.4), one gets
d dt
Z
Ω
Sγ( ˜w−ε(t, x))dx≤ kK3k2L∞(QT)
a
Z
Ω
Sγ( ˜w−ε(t, x))dx.
Since the dataw0,ε is nonnegative, we deduce thatw−ε = 0.
Next, substituteϕ=−Tγ(u−ε) in (5.3). Sincefε and wε are nonnegative, the choice ofλin (5.2) and using H¨older’s inequality and (4.4), one gets
d dt
Z
Ω
Sγ(˜u−ε(t, x))dx≤ kK1k2L∞(QT)
a
Z
Ω
Sγ(˜u−ε(t, x))dx, Since the datau0,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 not depending on εsuch that the sequences (˜uε,˜vε,w˜ε)0<ε≤1 satisfies
ku˜ε+ ˜vε+ ˜wεkL∞(0,T;L1(Ω)) ≤c1, (5.6) ku˜ε+ ˜vε+ ˜wεkLp(QT)+kσ(eλtu˜ε, eλtv˜ε, eλtw˜ε)kL1(QT)≤c2. (5.7) Assuming condition (CN1) or condition (CN2), there exist c3 such that
sup
γ≥0
( Z
{γ≤|u˜ε|<γ+1}|∇u˜ε|2 dxdt+ Z
{γ≤|˜vε|<γ+1}|∇v˜ε|2 dxdt +
Z
{γ≤|w˜ε|<γ+1}|∇w˜ε|2 dxdt)≤c3.0< ε≤1. (5.8) Assuming condition (CN3), there exist c4 and c5 such that
sup
γ≥0
Z
{γ≤|z˜ε|<γ+1}|∂xz˜ε|2 dxdt≤c4+c5(kz˜εk2L2(QT)), 0< ε≤1; (5.9) herein z˜ε= ˜uε, ˜vε, w˜ε.
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 inLq(0, T;W1,q(Ω)). Sim- ilarly to the scalar case,∂tu˜ε, ∂tv˜ε, ∂tw˜εis bounded inL1(0, T; (W1,q(Ω))0) + L1(QT), therefore, possibly at the cost of extracting subsequences denoted
(˜uε,˜vε,w˜ε), see e.g. [25], we can assume that there exist ˜u, ˜v and ˜w in Lq(0, T;W1,q(Ω)) such that as εgoes to 0
˜
uε−→u˜ strongly inLq(QT) and a.e. inQT,
˜
vε−→v˜ strongly inLq(QT) and a.e. in QT,
˜
wε−→w˜ strongly inLq(QT) and a.e. inQT,
σ(eλtu˜ε, eλtv˜ε, eλtw˜ε)−→σ(eλtu, e˜ λtv, e˜ λtw) a.e. in˜ QT, r1,λ(t, x,u˜ε,v˜ε,w˜ε)−→r1,λ(t, x,u,˜ v,˜ w) a.e. in˜ QT, r2,λ(t, x,u˜ε,v˜ε,w˜ε)−→r2,λ(t, x,u,˜ v,˜ w) a.e. in˜ QT, r3,λ(t, x,u˜ε,v˜ε,w˜ε)−→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. Having in mind for using Vitali’s Theorems and pass to the limit. Let us analyze the behaviour of the nonlinear terms when (2.6) holds.
Proposition 3. The sequences
(σ(eλtu˜ε, eλtv˜ε, eλtw˜ε))0<ε≤1, (r1,λ(˜uε,v˜ε,w˜ε))0<ε≤1, (r2,λ(˜uε,˜vε,w˜ε))0<ε≤1, (r3,λ(˜uε,v˜ε,w˜ε))0<ε≤1
satisfy
γlim→∞ sup
0<ε≤1
³ Z
{|u˜ε|≥γ}
£r1,λ(˜uε,v˜ε,w˜ε) +σ(eλtu˜ε, eλtv˜ε, eλtw˜ε
¤dxdt
´
= 0, (5.11) and
σ(eλtu˜ε, eλt˜vε, eλtw˜ε)−→σ(eλtu, e˜ λt˜v, eλtw),˜ (5.12) ri,λ(˜uε,˜vε,w˜ε)−→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 (∇w˜ε)0<ε≤1 con- verge to ∇˜u, ∇˜v and ∇w˜ almost everywhere in QT as εgoes to zero.
Lemma 8. The sequences (˜uε)0<ε≤1, (˜vε)0<ε≤1 and ( ˜wε)0<ε≤1 are Cauchy sequences in C([0, T];L1(Ω)).
The proof of Lemma 7 and Lemma 8 are similar to the proofs of Lemma 4 and Lemma 5, when the termsri,λ fori= 1,2,3 andfε−bw˜ε−σ,π σ+gε,
(1−π) σ +hε are replaced by h and fε respectively in (4.3). Finally, by passing to the limit ε−→0 in the following weak formulation
− Z T
0
Z
Ω
uεϕtdt− Z
Ω
ϕ(0, x)u0,ε(x)dx+ Z T
0
Z
Ω
(A1∇uε+uεK1)· ∇ϕ dxdt +
Z T
0
Z
Ω
m(t, x)uεϕ dxdt− Z T
0
Z
Ω
b(t, x)(uε+wε)ϕ dxdt +
Z T
0
Z
Ω
σ(t, x, uε, vε, wε)ϕ dxdt+ Z T
0
Z
Ω
r1(t, x, uε, vε, wε)ϕ dxdt
= Z T
0
Z
Ω
fεϕ dxdt,
− Z T
0
Z
Ω
vεψtdt− Z
Ω
ψ(0, x)v0,ε(x)dx+ Z T
0
Z
Ω
(A2∇vε+vε K2)· ∇ψ dxdt +
Z T
0
Z
Ω
(m(t, x) +α(t, x))vεψ dxdt−π Z T
0
Z
Ω
σ(t, x, uε, vε, wε)ψ dxdt +
Z T
0
Z
Ω
r2(t, x, uε, vε, wε)ψ dxdt= Z T
0
Z
Ω
gεψ dxdt,
− Z T
0
Z
Ω
wεχtdt− Z
Ω
χ(0, x)w0,ε(x)dx+ Z T
0
Z
Ω
(A3∇wε+wεK3)· ∇χ dxdt +
Z T
0
Z
Ω
m(t, x)wεχ dxdt−(1−π) Z T
0
Z
Ω
σ(t, x, uε, vε, wε)χ dxdt +
Z T
0
Z
Ω
r3(t, x, uε, vε, wε)χ dxdt= Z T
0
Z
Ω
hεχ dxdt,
withϕ, ψ, χ∈Cc1([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 and
˜
uε ∈L2(0, T;H1(Ω)), one hasTγ(˜uε)∈L2(0, T;H1(Ω)), see [6], moreover,
∇Tγ(˜uε) =1|u˜ε|≤γ∇˜uε.
Thus, we choose ϕ=T1(˜uε) as a test function in (4.3); after integration by parts and using H¨older’s inequality, it follows that
Z
Ω
S1(˜uε)(t, x)dx+ Z t
0
Z
Ω
h(τ, x,u˜ε)T1(˜uε)dxdτ
