Stationary reaction-diffusion systems in L 1

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Stationary reaction-diffusion systems in L 1

El Haj Laamri, Michel Pierre

To cite this version:

El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in L 1. Mathematical Mod- els and Methods in Applied Sciences, World Scientific Publishing, 2018, 28 (11), pp.2161-2190.

�10.1142/S0218202518400110�. �hal-01748519�

Stationary reaction-diffusion systems in L

El Haj Laamri^∗, Michel Pierre^†, March 27, 2018

Abstract

We prove existence of solutions to stationarym×mreaction-diffusion systems where the data are inL¹ or inLLogL. We first give an abstract result where the “diffusions” are nonlinearm-accretive operators in L¹and the reactive terms are assumed to satisfymstructural inequalities. It implies that the situation is controlled by an associated cross-diffusion system and providesL¹-estimates on the reactive terms. Next we prove existence for specific systems modeling chemical reactions and which naturally satisfies less than mstructural (in)equalities. The main difficulty is also to obtain L¹-estimates on the nonlinear reactive terms.

Keywords: reaction-diffusion systems, nonlinear diffusion, cross-diffusion, global existence, porous media equation, weak solutions.

2010 MSC: 35K10, 35K40, 35K57

1 Introduction and main results

Our goal is to analyze the existence of solutions to stationary reaction-diffusion systems in L¹-spaces.

We first give a general abstract result for nonlinearities satisfying as many structural inequalities as the number of equations. This is done in the framework of abstract m-accretive operators inL¹ spaces for the diffusion part and the full system is controlled by a somehow “good”associated cross-diffusion system. We give several examples where this abstract result applies. Then, we provide a general result for more specific systems associated with general chemical reactions for which less structure holds for the nonlinearities.

We denote by (Ω, µ) a measured space whereµis a nonnegative measure on the set Ω withµ(Ω)<+∞.

We consider systems of the type (S)







∀i= 1, ..., m,

u_i ∈D(A_i)∩L¹(Ω, dµ)⁺, h_i(·, u)∈L¹(Ω, dµ), ui+Aiui =hi(·, u₁, ..., um) +fi(·) ∈ L¹(Ω, dµ),

(1) where for alli= 1, ..., m,

−A_i is a (possibly) nonlinear operator inL¹(Ω, dµ) (generally a diffusion operator in applications), defined onD(Ai)⊂L¹(Ω, dµ),

−hi : Ω×R^m→Ris a nonlinear “reactive” term,

−f_i ∈L¹(Ω, dµ)⁺[:={g∈L¹(Ω, dµ) ; g≥0, µ−a.e.}].

We are interested in nonnegative solutionsu= (u1, ..., um)∈L¹(Ω, dµ)^+m.

∗Universit´e de Lorraine, B.P. 239, 54506 Vandœuvre-l`es-Nancy, France, Email : El-Haj.Laamri@univ-lorraine.fr

†Univ Rennes, ENS Rennes, IRMAR, Campus de Ker Lann, 35170-Bruz, France. Email : michel.pierre@ens-rennes.fr

These systems are naturally associated with theevolution reaction-diffusion systems

∂_tu_i(t) +A_iu_i(t) =h_i(·, u(t)), t∈[0,+∞), i= 1, ..., m. (2) When approximating these evolution systems by an implicit time discretization scheme, we are led to solving the following set of equations, for all small time interval ∆t

( ∀i= 1, ..., m, n= 0,1, ..., uⁿ⁺¹_i ∈D(Ai),

uⁿ⁺¹_i −uⁿ_i

∆t +A_iuⁿ⁺¹_i =h_i(·, uⁿ⁺¹)or uⁿ⁺¹_i + (∆t)A_iuⁿ⁺¹_i = (∆t)h_i(·, uⁿ⁺¹_i ) +uⁿ_i.

This is exactly the system (S) with unknown u = uⁿ⁺¹, up to trivially changing (∆t)A_i into A_i and (x, u)7→[(∆t)hi(x, u) +uⁿ_i(x)]1≤i≤m into (x, u)7→[hi(x, u) +fi(x)]1≤i≤m.

The asymptotic steady states of the evolution system (2) are also the solutionsu= (u₁, ..., u_m) to the system {A_iu_i =h_i(·, u), i = 1, ..., m,} which is also essentially included in the system (S) up to a slight change of the operatorsAi.

The kind of operatorsAi we have in mind are :

−The Laplacianu7→ −∆uon a bounded open set Ω of R^N with various boundary conditions on∂Ω.

− More general elliptic operators u 7→ −PN i=1∂xi

PN

i=1aij(x)∂xju+biu

on the same Ω, with various boundary conditions as well.

−Nonlinear operators of porous media type like u 7→ −∆ϕ(u) where ϕ:R→ Ris an increasing function, again with different boundary conditions on∂Ω.

−Nonlinear operators ofp-Laplacian type likeu7→ −∆_pu:=−∇ · |∇u|^p−2∇u

wherep∈(1,+∞) and| · | is the euclidian norm inR^N.

−Classical perturbations and various associations of all of those.

All these operators will satisfy the assumption (A) below which means that each Ai is an m-accretive operator inL¹(Ω, dµ) whose resolvents are compact and preserve positivity, namely the following, where I denote the identity :

(A)















∀i= 1, ..., m,

Ai:D(Ai)⊂L¹(Ω, dµ)→L¹(Ω, dµ), and for allλ∈(0,+∞),

(A1) m−accretivity: I +λA_i is onto and (I+λA_i)⁻¹is nonexpansive, (A2) positivity:R

Ωsign⁻(u_i)Au_i≥0,∀u_i ∈D(A_i),

(A3) compactness: (I+λAi)⁻¹ :L¹(Ω, dµ)→L¹(Ω, dµ) is compact.

(3)

HereI denotes the identity on L¹(Ω, dµ) and in (A₂), we define sign⁻(s) :=−sign⁺(−s), ∀s∈Rwhere sign⁺(s) := 1, ∀s∈[0,+∞), sign⁺(s) = 0, ∀s∈(−∞,0).

This implies immediately that (I+λA_i)⁻¹ L¹(Ω, dµ)⁺

⊂L¹(Ω, dµ)⁺ since then, forg ∈ L¹(Ω, dµ)⁺ and u_i := (I+λA_i)⁻¹g

0≥ Z

Ω

sign⁻(u_i)g dµ= Z

Ω

sign⁻(u_i)[u_i+λA_iu_i]dµ≥ Z

Ω

u⁻_i dµ ⇒u⁻_i = 0µ−a.e.

For simplicity, we consider only single-valued operatorsA_i, but everything would also work for multivalued m-accretive operators Ai.

The nonlinear reactive termshi will preserve positivity as well. We assume that, for alli= 1, ..., m:

(H1)











(1)hi : Ω×R^m→Rmeasurable ;

(2)ehi(·, R) := sup{|h_i(·, r)|;|r| ≤R} ∈L¹(Ω, dµ)⁺, ∀R∈[0,+∞) ; (3)r∈R^m7→h_i(·, r) is continuous ;

(4)quasipositivity: h_i(·, r₁, ..., ri−1,0, r_i+1, ..., r_m)≥0, ∀r∈[0,+∞)^m.

(4)

Moreover, the nonlinear reactive termshi we are considering are those for which mass conservation or, more generally, mass dissipation or at least mass control holds for the associated evolution system (2). It is the case when theh_i’s satisfy a relation like

m

X

i=1

h_i(·, r)≤0 for allr ∈[0,+∞)^m or more generally

(H2)

m

X

i=1

aihi(·, r)≤

m

X

i=1

biri+ω(·), ω∈L¹(Ω, dµ)⁺, for some 0≤bi < ai, i= 1, ..., m, r∈[0,+∞)^m. (5) As we will see, this structure naturally implies an a prioriL¹-estimate on the solutionsu_i of the system (S) (see Proposition 2.2), together with “standard operators”A_i by which we mean

(A_inf) a∞:= inf Z

Ω

A_iu_idµ, u_i∈D(A_i)∩L¹(Ω, dµ)⁺

>−∞ for each i= 1, ..., m. (6) This assumption essentially holds when 0∈D(Ai), i= 1, ..., m. It also holds with most diffusion operators with non homogeneous boundary conditions (see Section 3) except a few ones : this is discussed in Remark 3.4.

Actually, the main point will be to get a priori L¹-estimates even on the nonlinearities h_i(·, u) where u is solution of System (S). This will be satisfied if more structure is required on the nonlinear functionshi, namely (using the natural order inR^m) :

(M)







There exist two m×mmatrices M0, M1 where

M0 is invertible, with nonnegative entries and such that M₀h(·, r)≤M₁r+ Θ(·), ∀r ∈[0,+∞)^m, Θ∈L¹(Ω, dµ)^+m.

(7) Applying the matrix M0 to the system (S) leads to the following set of m inequalities for an associated cross-diffusion system:

M₀u+M₀Au=M₀h(·, u) +M₀f ≤M₁u+ Θ +M₀f, where we denoteAu:= (Aiui)1≤i≤m,h(·, u) := (h_i(·, u))1≤i≤m andf := (fi)1≤i≤m.

As proved in Proposition 2.3, this will imply an a priori estimate ofh_i(·, u), i= 1, ..., minL¹(Ω, dµ). More precisely, we will consider an approximate problem where the nonlinearities hi are replaced by truncated versions hⁿ_i. Then (M) will imply that hⁿ_i(·, uⁿ) is bounded in L¹(Ω, dµ) for the approximate solutions uⁿ. The compactness of uⁿ in L¹(Ω, dµ) will then follow. Thus up to a subsequence, uⁿ will converge in L¹(Ω, dµ)^m and µ-a.e. to someu and hⁿ_i(·, uⁿ) will also convergeµ-a.e. to hi(·, u) for alli. But this is not sufficient yet to pass to the limit in the system since one essentially needs the convergence of hⁿ_i(·, uⁿ) in L¹(Ω, dµ). It is the case if hⁿ_i(·, uⁿ) is uniformly integrable, by Vitali’s lemma. This will hold if we add the following technical assumption which is some kind of compatibility condition between the various operators Ai. It will be satisfied in the examples of Section 3.

(Φ)

( There existϕ: [0,+∞)^m→[0,+∞) continuous with lim

|r|→+∞ϕ(r) = +∞ andb∈(0,+∞)^m such that R

Ωsign⁺(ϕ(u)−k)b·M0Au dµ≥0, ∀u∈D(A)∩L¹(Ω, dµ)^+m, ∀k∈[0,+∞), D(A) := Π^m_i=1D(Ai).

(8) We now state our abstract result. It will be proved in Section 2 and applied to several explicit examples in Section 3. We refer to [11] where this kind of results and examples were already widely discussed and analyzed.

Theorem 1.1 Assume that(A),(H1),(H2),(A_inf),(M),(Φ) hold. Then the system (S) has a solution for allf ∈L¹(Ω, dµ)^+m.

As already explained, the assumption (M) means that m independent inequalities hold between the m nonlinear functionshi. But many systems come with less thanmsuch relations. The following 2×2 system satisfies only one relation of type (H2) and not (M):







u1−d1∆u1= (β1−α1)[u^α₁¹u^α₂²−u^β₁¹u^β₂²] +f1 =:h1(u1, u2) +f1, u₂−d₂∆u₂= (β₂−α₂)[u^β₁¹u^β₂²−u^α₁¹u^α₂²] +f₂ =:h₂(u₁, u₂) +f₂,

∂_νu₁ = 0 =∂_νu₂ on ∂Ω,

(9) where for i= 1,2, d_i ∈(0,+∞), α_i, β_i ∈ {0} ∪[1,+∞), f_i ∈ L¹(Ω)⁺,Ω ⊂R^N equipped with the Lebesgue measure. This kind of nonlinearity appears in reversible chemical reactions with two species, namely

α1A1+α2A2 β1A1+β2A2. (10)

Here (β1−α1)(β2−α2)<0. The nonlinearityh1 and h2 are quasipositive but satisfy the only one relation γ₂h₁+γ₁h₂ = 0, γ_i := |β_i−α_i|. It turns out that existence indeed holds for system (9) whenf_i ∈L¹(Ω)⁺ for all i, as we prove it here. Actually we prove existence for a general class of such “chemical systems”

when fi ∈L¹(Ω)⁺ and also filogfi ∈L¹(Ω). This is the second main result of this paper. Let us consider the system

(CHS)







For alli= 1, ..., m, ui−di∆ui = (βi−αi)

k1Π^m_k=1u^α_k^k−k2Π^m_k=1u^β_k^k

+fi,

∂νui= 0 on∂Ω,

(11)

wherek1, k2 ∈(0,+∞) and, for all i= 1, ..., m,di ∈(0,+∞), α_i, βi∈ {0} ∪[1,+∞) andfi ∈L¹(Ω)⁺ where Ω is bounded regular open subset ofR^N and where







I :=

i∈ {1, ..., m}; αi−βi >0 , J :=

j∈ {1, ..., m}; βj−αj >0 satisfy :

I 6=∅, J 6=∅, I∪J ={1, ..., m}.

(12) We denote by|I|(resp. |J|) the number of elements ofI (resp. J).

Theorem 1.2 Assume thatf_i ∈L¹(Ω)⁺, f_ilogf_i∈L¹(Ω) for alli= 1, ..., m. Assume also that |I| ≤2 (or

|J| ≤ 2). Then there exists a nonnegative solution u ∈ W^2,1(Ω)^+m of (CHS) with hi(u) ∈ L¹(Ω) for all i= 1, ..., m. If moreoverm= 2, then the same result holds iff_i ∈L¹(Ω)⁺, i= 1,2 only.

Remark 1.3 System (11) arises when modeling a general reversible chemical reaction with m species, according to the mass action law and to Fick’s linear diffusion. In fact, it also contains systems written more generally as

v_i−d_i∆v_i=λ_i

K₁Π^m_k=1v_k^αk−K₂Π^m_k=1v_k^βk +fe_i,

whereλ_i∈R, λ_i(β_i−α_i)>0, i= 1, ..., m. Indeed, we may go back to the exact writing of (11) by setting u_i:= (β_i−α_i)v_i/λ_i, k₁ :=K₁Π_k[λ_k/(β_k−α_k)]^αk, k₂:=K₂Π_k[λ_k/(β_k−α_k)]^βk, f_i := (β_i−α_i)fe_i/λ_i. On the other hand, it does not include systems like

u₁−d₁∆u₁= +[u³₁u²₂−u²₁u³₂] +f₁ (=u²₁u²₂[u₁−u₂] +f₁), u2−d2∆u2=−[u³₁u²₂−u²₁u³₂] +f2.

Here the conditionλ_i(β_i−α_i)>0 is not satisfied.

Remark 1.4 Note that besides the 2×2 system (9), Theorem 1.2 contains as particular cases some favorite systems of the literature like

h_i(u) = (−1)ⁱ[u^α₁¹u^α₃³ −u^β₂²], i= 1,2,3, α₁, α₃, β₂ ∈[1,+∞), hi(u) = (−1)ⁱ[u1u3−u2u4], i= 1,2,3,4.

We may use Remark 1.3 to write them as in (11). Analysis of systems of this kind may be found for instance in [12], [7], [18], [13], [9], [5], [6], etc.

In fact Theorem 1.2 applies to quite general systems like, for instance, those obtained by multiplying the above 3×3 (resp. 4×4) example byu^σ₁¹u^σ₂²u^σ₃³ (resp. u^σ₁¹u^σ₂²u^σ₃³u^σ₄⁴) whereσ_i∈[0,+∞). They lead to the following nonlinearities

hi(u) =λi[Π^m_k=1u^α_k^k−Π^m_k=1u^β_k^k],

where m = 3 (resp. m = 4) and λi(βi −αi) > 0. Actually, Theorem 1.2 applies also to these same nonlinearities whenm= 5. Indeed, since I ∪J ={1,2,3,4,5}, it follows that I orJ contains at most two elements.

We believe that the result of Theorem 1.2 holds even if|I|and |J|>2. But it is not clear how to extend our main Lemma 4.2 to the general case. We leave this as an open problem.

Remark 1.5 Theorem 1.2 provides a solution u such thathi(u)∈L¹(Ω). It is easy to write down explicit examples where the nonlinear terms Πku^α_k^k,Πku^β_k^k are not separately in L¹(Ω). For instance, let N > 4 and let us introduce the following functionσ wherer :=|x|, x∈Ω :=B(0,1)⊂R^N, b ∈(N/2, N −2), c∈ (0,+∞):

σ(r) :=r^−b+br+c,∀0< r≤1,

⇒σ−∆σ=f, f(r) :=c+r^−b+br+b(N−2−b)r^−(b+2)−b(N−1)r⁻¹, σ⁰(1) = 0,

wherec is chosen large enough so thatf ≥0. Note thatf ∈L^p(Ω), p∈[1, N/(b+ 2)). Let us now consider the solution (which exists by Theorem 1.2) of the system











u₁, u₂ ∈W^2,1(Ω), u²₂−u²₁ ∈L¹(Ω), u1−∆u1 =u²₂−u²₁+f /2,

u2−∆u2 =−[u²₂−u²₁] +f /2,

∂_νu₁=∂_νu₂= 0on ∂Ω.

Then (u₁+u₂)−∆(u₁+u₂) =f, ∂_ν(u₁+u₂) = 0 on∂Ω. Thusu₁+u₂ =σ. Butu₁, u₂ cannot be inL²(Ω) sinceσ is not in L²(Ω) by the choice ofb > N/2.

Remark 1.6 It is classical that an entropy structure holds in System (11) since

m

X

i=1

[logu_i+µ_i]h_i(u) =−[log k₁Π^m_k=1u^α_k^k

−log k₂Π^m_k=1u^β_k^k

][k₁Π^m_k=1u^α_k^k −k₂Π^m_k=1u^β_k^k]≤0,

when choosingµ_i:= [logk₁−logk₂]/[m(α_i−β_i)]. Under the assumptions of Theorem 1.2, and in particular theLLogLassumption on the f_i, we have the estimate

Z

Ω m

X

i=1

[logu_i+µ_i]u_i+d_i|∇u_i|² u_i ≤

Z

Ω m

X

i=1

[logu_i+µ_i]f_i ≤ Z

Ω m

X

i=1

f_ilogf_i+ (µ_i−1)f_i+u_i, where, for the last inequality, we use the Young’s inequality (75) withr=f_i, s= logu_i.

But we will not use here this stucture in the proof of Theorem 1.2. Our strategy will consist in proving that the nonlinearityhi(u) is a priori bounded in L¹(Ω). Then adequate compactness arguments allow us to pass to the limit in the approximate system. Note that the entropy inequality provides the extra information that∇√

u_i∈L²(Ω) for the solutions obtained in Theorem 1.2 whenf_ilogf_i ∈L¹(Ω), i= 1, ..., m.

Theorem 1.1 will be proved in Section 2, examples of applications are given in Section 3 and the proof of Theorem 1.2 is given in Section 4.

2 Proof of Theorem 1.1

Sinceµis fixed, we will most of the time more simply writeL¹(Ω), L¹(Ω)⁺instead ofL¹(Ω, dµ), L¹(Ω, dµ)⁺. We will use the natural order in R^m namely [r ≤ rˆ ⇔ r_i ≤ rˆ_i, i = 1, ..., m] and we also denote r⁺ :=

(r₁⁺, ..., r_m⁺).

Lemma 2.1 Let f = (f1, ..., fm)∈L¹(Ω)^+m. We set hⁿ_i(x, r) := h_i(x, r)

1 +n⁻¹

m

X

j=1

|h_j(x, r)|

, ∀i= 1, ..., m, r ∈R^m, x∈Ω. (13)

Assume that (A) and (H1) hold. Then the following approximate system (Sⁿ)

∀i= 1, ..., m, uⁿ_i ∈D(A_i)∩L¹(Ω)⁺,

uⁿ_i +Aiuⁿ_i =hⁿ_i(·, uⁿ) +fi(·) in L¹(Ω, dµ), (14) has a nonnegative solution uⁿ= (uⁿ₁,· · · , uⁿ_m).

Proof. We consider the mappingT :v∈L¹(Ω)^m 7→w∈L¹(Ω)^m wherew= (w1, ..., wm) is the solution of ∀i= 1, ..., m, w_i ∈D(A_i),

wi+Aiwi =hⁿ_i(·, v⁺) +fi(·) in L¹(Ω, dµ), (15)

where v⁺ := (v₁⁺, ..., v⁺_m). The mapping T is well defined since |hⁿ_i(x, r)| ≤ n for all (x, r) ∈ Ω×R^m and f ∈L¹(Ω)^m so that the solution is given by

w_i = (I +A_i)⁻¹ hⁿ_i(·, v⁺) +f_i

, i= 1, ..., m.

Moreover khⁿ_i(·, v) +f_ik_L1(Ω)≤n µ(Ω) +kf_ik_L1(Ω)=:M_i. Let Ki:= (I+Ai)⁻¹

{g∈L¹(Ω) ; kgk_L1(Ω)≤Mi}

, K:=K1×...×Km⊂L¹(Ω)^m. By assumption (A) and in particular (A3), K is a compact set of L¹(Ω)^m and K ⊃ T L¹(Ω)^m

. On the other hand, we easily check that T is continuous. Thus T is a compact operator from L¹(Ω)^m into the compact setK ⊂L¹(Ω)^m. By Schauder’s fixed point theorem (see e.g. [21]),T has a fixed point uⁿ. This means thatuⁿ is solution of (Sⁿ).

To prove the nonnegativity property ofuⁿ, we multiply thei-th equation bysign⁻(uⁿ_i) and integrate to obtain, using

Z

Ω

sign⁻(u_i)A_iu_i≥0 (see (A2) ), Z

Ω

(uⁿ_i)⁻dµ≤ Z

Ω

sign⁻(uⁿ_i)hⁿ_i(·,(uⁿ)⁺) +sign⁻(uⁿ_i)fi

dµ.

By the nonnegativity of f_i and the quasipositivity of h_i assumed in (H1), the integral on the right is nonpositive so that

Z

Ω

(uⁿ_i)⁻dµ≤0 oruⁿ_i ≥0µ−a.e..

We will now progressively obtain estimates onuⁿindependently ofn. We start with the control of the total mass of uⁿ.

Proposition 2.2 Assume that (A), (H1), (H2), (A_inf) hold. Then there exists C ∈(0,+∞) independent of n such that the solutionuⁿ of the approximate system (Sⁿ) satisfies

1≤i≤mmax kuⁿ_ik_L1(Ω)≤C.

Proof. According to (H2), we multiply each equation byai and we add them to obtain

m

X

i=1

ai(uⁿ_i +Aiuⁿ_i) =

m

X

i=1

ai[hⁿ_i(·, u_n) +fi]≤

m

X

i=1

[biuⁿ_i +aifi].

We use (A_inf) to obtain

n

X

i=1

(ai−bi) Z

Ω

uⁿ_idµ≤ −a∞ m

X

i=1

ai+

m

X

i=1

ai

Z

Ω

fidµ.

Usinga_i > b_i for all iyields the estimate of Proposition 2.2.

We now prove that the nonlinearities are bounded inL¹ independently ofn.

Proposition 2.3 Assume that (A), (H1), (H2), (A_inf), (M) hold. Then there exists C ∈(0,+∞) inde- pendent of nsuch that the solution uⁿ of the approximate system(Sⁿ) satisfies

1≤i≤mmax khⁿ_i(·, uⁿ)k_L1(Ω) ≤C.

Proof. According to the assumption (M), let us multiply the system (Sⁿ) by the matrix M0. As before, we denote

Auⁿ:= (A1uⁿ, ..., Amuⁿ), hⁿ(·, uⁿ) := (hⁿ₁(·, uⁿ), ..., hⁿ_m(·, uⁿ)), f := (f1, ..., fm).

Then

M₀uⁿ+M₀Auⁿ=M₀hⁿ(·, uⁿ) +M₀f ≤M₁uⁿ+ Θ +M₀f. (16) Since the entries of M0 are nonnegative, and thanks to (A_inf) and the nonnegativity of uⁿ, there exists C∞∈[0,+∞)^m such that

Z

Ω

[M0uⁿ+M0Auⁿ]dµ≥ −C∞ which implies Z

Ω

[M0hⁿ(·, uⁿ) +M0f]dµ≥ −C∞. We deduce

Z

Ω

[M1uⁿ+ Θ−M0hⁿ(·, uⁿ)]dµ≤C∞+ Z

Ω

[M1uⁿ+ Θ +M0f]dµ≤D∞∈(0,+∞)^m,

the last inequality using also Proposition 2.2. As the functionM₁uⁿ+ Θ−M₀hⁿ(·, uⁿ) is nonnegative, these inequalities provide a bound for its L¹(Ω)^m-norm. It follows that M₀hⁿ(·, uⁿ) is also bounded in L¹(Ω)^m independently of n. SinceM0 is invertible, we deduce thathⁿ(·, uⁿ) is itself bounded in L¹(Ω)^m: indeed if

| · |denotes the euclidian norm inR^m andk · k the induced norm on the m×m matrices, we may write

|hⁿ(·, uⁿ)|=|M₀⁻¹M0hⁿ(·, uⁿ)| ≤ kM₀⁻¹k|M₀hⁿ(·, uⁿ)|.

Whence Proposition 2.3.

Proposition 2.4 Assume that (A), (H1), (H2), (A_inf), (M), (Φ) hold. Then, if uⁿ is the solution of the approximate system (Sⁿ), {hⁿ(., uⁿ)}_n is uniformly integrable in Ω which means that, for all ε > 0, there exists δ >0 such that for all measurable set E ⊂Ω

µ(E)< δ ⇒ Z

E n

X

i=1

|hⁿ_i(·, uⁿ)|dµ≤ε for alln.

Proof. By Proposition 2.3, we already know that hⁿ(·, uⁿ) is bounded in L¹(Ω)^m. Let us prove that the extra condition (Φ) implies that it is even uniformly integrable.

First, since uⁿ_i = (I +Ai)⁻¹(hⁿ(·, uⁿ) +fi), the compactness condition (A3) in (3) implies that, for all i= 1, ..., m,uⁿ_i belongs to a compact set ofL¹(Ω).

Let us show thatM0hⁿ(·, uⁿ) is uniformly integrable. SinceM0 is invertible, this will imply thathⁿ(·, uⁿ) is itself uniformly integrable and end the proof of Proposition 2.4.

We will successively prove that (M0hⁿ(·, uⁿ))⁺ and (M0hⁿ(·, uⁿ))⁻ are uniformly integrable.

Note that by the definition (13), hⁿ(·, uⁿ) ≤ h(·, uⁿ). We may write (recall that the entries of M₀ are nonnegative)

M0hⁿ(·, uⁿ)≤M0h(·, uⁿ)≤M1uⁿ+ Θ ⇒(M0hⁿ(·, uⁿ))⁺≤(M1uⁿ)⁺+ Θ. (17) Sinceuⁿ is in a compact set ofL¹(Ω)^m, so is (M1uⁿ)⁺and it is in particular uniformly integrable. We then deduce from the last inequality that (M0hⁿ(·, uⁿ))⁺ is uniformly integrable.

To control (M0hⁿ(·, uⁿ))⁻, we go back to (16) and, using (17), we rewrite it as follows

M0uⁿ+M0Auⁿ+ (M0hⁿ(·, uⁿ))⁻ = (M0hⁿ(·, uⁿ))⁺+M0f ≤(M1uⁿ))⁺+ Θ +M0f. (18) According to (Φ), we multiply this inequality by sign⁺(ϕ(uⁿ) −k)b. By (Φ) on one hand and by the nonnegativity of uⁿ, b and of the entries of M₀ on the other hand, we have

Z

Ω

sign⁺(ϕ(uⁿ)−k)b·M0Auⁿdµ≥0, Z

Ω

sign⁺(ϕ(uⁿ)−k)b·M0uⁿdµ≥0, ∀k∈[0,+∞).

Combining with (18), we deduce Z

Ω

sign⁺(ϕ(uⁿ)−k)b·(M₀hⁿ(·, uⁿ))⁻dµ≤ Z

Ω

sign⁺(ϕ(uⁿ)−k)b·

(M₁uⁿ))⁺+ Θ +M₀f

dµ. (19) Since lim

|r|→+∞ϕ(r) = +∞, for all k ∈ [0,+∞), there exists R(k) such that [|r| ≥R(k)] ⊂ [ϕ(r)≥k] and therefore

[|uⁿ| ≥R(k)]⊂[ϕ(uⁿ)≥k], µ−a.e.. (20)

LetE ⊂Ω be a measurable set. Then Z

E

b·(M₀hⁿ(·, uⁿ))⁻dµ≤ Z

E∩[|uⁿ|≤R(k)]

b·(M₀hⁿ(·, uⁿ))⁻dµ+ Z

E∩[|uⁿ|≥R(k)]

b·(M₀hⁿ(·, uⁿ))⁻dµ=:I₋ⁿ+I₊ⁿ. If we denoteϕR:= max{ϕ(r);|r| ≤R}, thenR ∈[0,+∞)7→ϕR∈[0,+∞) is a nondecreasing function such that [ϕ(r)≥k]⊂[|r| ≥ψ(k)] where ψ(k) := infϕ⁻¹_R ([k,+∞)). By (20) and (19) and [ϕ(uⁿ)≥k]⊂[|uⁿ| ≥ ψ(k)], we have

I₊ⁿ ≤ Z

[ϕ(uⁿ)≥k]

b·

(M1uⁿ)⁺+ Θ +M0f dµ≤

Z

[|uⁿ|≥ψ(k)]

b·

(M1uⁿ)⁺+ Θ +M0f

dµ. (21)

Sinceuⁿlies in a compact set of L¹(Ω)^m and lim

k→+∞ψ(k) = +∞, then lim

k→+∞µ([|uⁿ| ≥ψ(k)]) = 0 uniformly inn. Thus, givenε∈(0,1), there existsk=kε large enough so that

I₊ⁿ ≤ε/2 for all n. (22)

We can also controlI₋ⁿ as follows. We remark that (see assumption (2) in (H1))

|uⁿ| ≤R(k_ε) ⇒ |hⁿ_i(·, uⁿ)| ≤ |h_i(·, uⁿ)| ≤eh_i(·, R(k_ε)).

This implies that for someB ∈(0,+∞), I₋ⁿ =

Z

E∩[|uⁿ|≤R(k)]

b·(M₀hⁿ(·, uⁿ))⁻dµ≤ Z

E

B

m

X

i=1

eh_i(·, R(k_ε)) (23)

We may chooseδ small enough (independent of n) such that µ(E)≤δ ⇒

Z

E m

X

i=1

eh_i(·, R(k_ε))< ε/2B.

Combining withI₊ⁿ ≤ε/2 proved above (see (22)), we deduce thatb·(M₀hⁿ(·, uⁿ))⁻ is uniformly integrable.

Since b∈ (0,+∞)^m, this implies that (M0hⁿ(·, uⁿ))⁻ is itself uniformly integrable. We already know that (M₀hⁿ(·, uⁿ))⁺ is uniformly integrable. Thus so isM₀hⁿ(·, uⁿ) and this ends the proof of Proposition 2.4.

Proof of Theorem 1.1. We consider the solutionuⁿof the approximate system (Sⁿ) built in Lemma 2.1.

By Propositions 2.2, 2.3, 2.4, up to a subsequence asn→+∞, we may assume thatuⁿ converges inL¹(Ω)^m andµ-a.e. to someu∈L¹(Ω)^+m. Moreover, by definition ofhⁿ_i (see (13) ) and the continuity property ofh_i assumed in (H1),hⁿ_i(·, uⁿ) convergesµ-a.e. toh(·, u). Moreoverhⁿ(·, uⁿ) is uniformly integrable. By Vitali’s Lemma (see e.g. [20] or [8]),hⁿ(·, uⁿ) converges also inL¹(Ω)^mtoh(·, u). Sinceuⁿ_i = (I+A_i)⁻¹(hⁿ(·, uⁿ)+f_i) this implies thatu_i = (I+A⁻¹_i (h(·, u) +f_i) which means thatu is solution of the limit system (S) and this

ends the proof of Theorem 1.1.

3 Examples

In all examples below, Ω is a bounded open subset ofR^N with regular boundary and equipped with the Lebesgue measure.

3.1 Examples with linear diffusions and homogeneous boundary conditions

We start with a simple example associated with the Laplacian operator with homogeneous boundary conditions.

Corollary 3.1 Assume the nonlinearity h = (h₁, ..., h_m) satisfies (H1),(H2),(M). Let d_i ∈(0,+∞), f_i ∈ L¹(Ω)⁺, i= 1, ..., m. Then the following system has a solution







for alli= 1, ..., m;

ui ∈W₀^1,1(Ω)⁺, hi(·, u)∈L¹(Ω), u_i−d_i∆u_i=h_i(·, u) +f_i.

(24)

Proof. We consider the operators

D(Ai) :={u∈W₀^1,1(Ω) ; ∆u∈L¹(Ω)},

Aiu:=di∆u. (25)

It is classical that these operatorsA_isatisfy the three conditions of (A) in (3) (see e.g. [4]). Since 0∈D(A_i), as already noticed just after (6), (Ainf) is also satisfied. And for (Φ), if we denote M0 = [mij]1≤i,j≤m, we chooseϕ(r) :=Pm

j=1(Pm

i=1mij)djrj, b= (1, ...,1)^t∈(0,+∞)^m. Note thatPm

i=1mij >0 for allj= 1, ..., m sincem_ij ≥0 andM₀ is invertible. In particular, lim

|r|→+∞ϕ(r) = +∞). Then, ifu∈D(A), k∈(0,+∞), Z

Ω

sign⁺(ϕ(u)−k)b·M₀Au=− Z

Ω

sign⁺





m

X

i,j=1

m_ijd_ju_j−k



∆





m

X

i,j=1

m_ijd_ju_j



≥0. (26)

Then we may apply Theorem 1.1.

Remark 3.2 As examples of functions h, we may for instance choose







m= 2, α_i, β_i ∈ {0} ∪[0,+∞), i= 1,2, λ∈(0,1) h1(·, u₁, u2) =λu^α₁¹u^α₂² −u^β₁¹u^β₂²,

h2(·, u₁, u2) =−u^α₁¹u^α₂² +u^β₁¹u^β₂².

We easily check that (H1) holds and that (H2) is satisfied witha_i= 1, b_i= 0 for all iand ω = 0 (in other words h1+h2≤0). Moreover (M) is satisfied withM1 = 0,Θ = (0,0)^t and

M₀ =

1 1 1 λ

.

Note that forλ= 1,M0 is not invertible so that only one relation h1+h2 ≤0 holds and Theorem 1.1 does

not apply. This kind of systems is considered in Theorem 1.2.

•Here is another example of a nonlinearity h= (h1, h2, h3) which satisfies the corollary.

m= 3, αi∈[1,+∞), i= 1,2, h₁ =u₃−u^α₁¹u^α₂² =h₂=−h₃.

The corresponding evolution problem is studied in [19] forN ≤5, [15] for any dimensionN withα1 =α2 = 1, and in [12] where (α1, α2)∈[1,+∞)².

Here (H1) is obviously satisfied and so is (H2) witha_i = 1, i= 1,2, a₃ = 2, b_i = 0, i= 1,2,3, ω = 0. Then (M) is satisfied with Θ = (0,0,0)^tand

M₀=





1 0 0 0 1 0 1 1 2



, M₁ =





0 0 1 0 0 1 0 0 0



.

A main point here is that the dependence in u3 is linear. When it is superlinear, the system does not fit any more into the scope of Theorem 1.1. It is however analyzed in Theorem 1.2.

3.2 About more general linear diffusions

Let us now make some comments on the following example where diffusions are more general than in Corollary 3.1.



















ui ∈W₀^1,1(Ω), hi(u1, u2)∈L¹(Ω), i= 1,2, u₁−

N

X

i,j=1

a_ij∂_xixju₁ =h₁(u₁, u₂) +f₁, u₂−

N

X

i,j=1

b_ij∂_xixju₂ =h₂(u₁, u₂) +f₂,

(27)

where

a_ij, b_ij ∈R,

N

X

i,j=1

a_ijξ_iξ_j,

N

X

i,j=1

b_ijξ_iξ_j ≥α|ξ|², α∈(0,+∞), ∀ξ= (ξ₁, ..., ξ_N)∈R^N. (28) We consider the operators











D(A₁)[resp.D(A₂)] :=n

v∈W₀^1,1(Ω), PN

i,j=1a_ij∂_xixjv[resp. PN

i,j=1b_ij∂_xixjv]∈L¹(Ω)o , A₁v:=

N

X

i,j=1

a_ij∂_xixjv, A₂v:=

N

X

i,j=1

b_ij∂_xixjv.

It is easy to see that the assumptions (A), (A_inf) are satisfied. Thus if (H1),(H2),(M) are satisfied like in Corollary 3.1, then for the approximate solutionsuⁿ of this system, as defined in the proof of Theorem 1.1, it follows thatuⁿ, hⁿ(·, uⁿ) are bounded inL¹(Ω)² anduⁿlies in a compact set ofL¹(Ω)². However, the extra condition (Φ) is not satisfied in general so that it is not clear whetherhⁿ(·, uⁿ) is uniformly integrable.

Actually, we have to choose here a different strategy with does not seem to be generalized to the abstract setting of Section 2. It consists in looking at the equation satisfied by Tk(uⁿ₁ +ηuⁿ₂) where Tk is a cut-off function. It is then easy to pass to the limit in the nonlinear terms of the truncated approximate system since they are multiplied by T_k⁰(uⁿ₁ +ηuⁿ₂) which vanishes for uⁿ_i large for all i. Therefore a.e. convergence is sufficient to pass to the limit. The difficulty is then to prove precise estimates independent ofn, in terms ofη, in order to control the other terms as it done in the parabolic case (see [16], [17], [13]). This approach through cut-off functionsT_k is precisely developed in Section 4 to prove Theorem 1.2 and we refer the reader

to this other approach without giving more details here.

3.3 Examples with linear diffusions and Robin-type boundary conditions

Now we analyze what happens for systems like (24) when the boundary conditions are different. If we replace the homogeneous boundary conditions of (24) by homogeneous Neumann boundary conditions, then the result is exactly the same. On the other hand, the situation is quite more complicated if the boundary conditions are of different type for each of the u_i’s. This is actually connected with the content of the assumption (Φ). For simplicity, we do it only for a 2×2-system. Given a nonlinearity h satisfying (H1),(H2),(M), we consider the following system with general Robin-type boundary conditions.















i= 1,2,

ui ∈W^1,1(Ω)⁺, hi(u1, u2)∈L¹(Ω), ui−di∆ui=hi(u1, u2) +fi, λ_iu_i+ (1−λ_i)∂_νu_i =ψ_ion ∂Ω,

λ_i ∈[0,1], d_i∈(0,+∞), ψ_i ∈[0,+∞), f_i ∈L¹(Ω)⁺.

(29)

Corollary 3.3 Let (f1, f2) ∈ L¹(Ω)⁺ ×L¹(Ω)⁺. Assume the nonlinearity h satisfies (H1),(H2),(M).

Assume moreover [0≤λ₁, λ₂<1] or [λ₁ =λ₂= 1 andψ₁ =ψ₂= 0]. Then the system (29) has a solution.

Remark 3.4 It is known that the case when one of the λ_i is equal to 1 is different (see the analysis in [14]). For instance, finite time blow up may occur for the associated evolution problem when the boundary conditions are u1 = 1, ∂νu2 = 0 (see [2], [3]). Then the operator A1 does not satisfy (A_inf) as easily seen by considering (as in [14]) the following simple example: u_σ(x) := cosh(σx)/cosh(σ) on Ω := (−1,1) with σ∈(0,+∞). Thenuσ ≥0 anduσ = 1 on ∂Ω. But−R

Ωu⁰⁰_σ =u⁰_σ(−1)−u⁰_σ(1)→ −∞asσ→+∞. Here we consider only the cases that directly fall into the scope of Theorem 1.1. A few other cases could be treated directly likeλ₁ =λ₂ and positive data ψ₁, ψ₂ or also λ₁ = 06=λ₂ and ψ₁= 0.

Proof of Corollary 3.3. Here we define

D(B_i) ={u∈W^2,1(Ω); λ_iu_i+ (1−λ_i)∂_νu_i =ψ_i on ∂Ω}, B_iu:=−d_i∆u_i.

Then the closureAi of Bi inL¹(Ω) satisfies the assumption (A) (see [4]).

•Let us first assume 0≤λi<1, i= 1,2. Then, forui ∈D(Bi) Z

Ω

Biui=− Z

Ω

di∆ui =− Z

Ω

di∂νui =di

Z

Ω

(1−λi)⁻¹(λiui−ψi)≥ −d_i(1−λi)⁻¹ Z

Ω

ψi.

This remains valid for Ai by closure. Thus the assumption (Ainf) is satisfied. For the condition (Φ), we come back to (26) with the sameϕ, b. Letpq(r) be a standard approximation of sign⁺(r−k) like

p_q(r) = 0, ∀r ∈(−∞, k−1/q]; p_q(r) =q(r−k) + 1, ∀r ∈[k−1/q, k]; p_q(r) = 1, ∀r ∈[k,+∞). (30) Then, foruj ∈D(Bj), j= 1,2 and forV :=P₂

i,j=1mijdjuj

Z

Ω

p_q(ϕ(u)−k)b·M₀Bu= Z

Ω

p⁰_q(V)|∇V|²− Z

∂Ω

p_q(V)∂_νV ≥ − Z

∂Ω

p_q(V)∂_νV.

− Z

∂Ω

p_q(V)∂_νV = Z

∂Ω

p_q(V)

2

X

i,j=1

m_ijd_j(1−λ_j)⁻¹(λ_ju_j −ψ_j).

Ifψ_j = 0 for allj= 1, ..., m, then −R

∂Ωp_q(V)∂_νV ≥0 and lettingq→+∞, we obtain thatB and therefore Asatisfies the condition (Φ). When ψj 6= 0 for some j, we only obtain

Z

Ω

sign⁺(ϕ(u)−k)b·M₀Bu≥ − Z

∂Ω

sign⁺(V −k)

m

X

i,j=1

m_ijd_j(1−λ_j)⁻¹ψ_j =:−η(V, k).

The point is that we can slightly modify the proof of Theorem 1.1 to get the same conclusion with this weaker estimate from below. Indeed, in the present case, we have to add the termη(V_n, k) in the inequality (19) whereVn=P2

i,j=1mijdjuⁿ_j. Sinceη(Vn, k) tends to 0 ask→+∞uniformly in n, the rest of the proof remains unchanged if we choosek_ε large enough so thatη(V, k_ε)< εalso.

•Assume now that λ1 =λ2= 1. Then we make the same choice as in (26) and we see that

− Z

Ω

sign⁺(ϕ(u)−k)b·M₀Au≥ − Z

∂Ω

sign⁺(

2

X

i,j=1

m_ijd_jψ_j−k)∂_ν(

2

X

i,j=1

m_ijd_ju_j)≥0, sinceu_j ≥0, u_j = 0 on∂Ω imply that∂_νu_j ≤0 on∂Ω.

For the same reason, (Ainf) holds since− Z

Ω

∆uj =− Z

∂Ω

∂νuj ≥0.