To cite this version: Michel Pierre, Takashi Suzuki, Haruki Umakoshi. Global-in-time behavior of weak solutions to reaction-diffusion systems with inhomogeneous Dirichlet boundary condition. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2017, 159, pp.393-407. �10.1016/j.na.2017.01.013�. �hal01387543�

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Global-in-time behavior of weak solutions to reaction-diffusion systems with inhomogeneous Dirichlet boundary condition Michel Pierre∗, Takashi Suzuki†, Haruki Umakoshi‡ October 25, 2016

Abstract We study reaction diffusion systems describing, in particular, the evolution of concentrations in general reversible chemical reactions. We concentrate on inhomogeneous Dirichlet boundary conditions. We first prove global existence of (very) weak solutions. Then, we prove that these although rather weak- solutions converge exponentially in L1 norm toward the homogeneous equilibrium. These results are proven by means of L2 duality arguments and through estimates provided by the nonincreasing entropy.

Keywords. reaction diffusion systems, Dirichlet conditions, global existence, asymptotic behavior, entropy, convergence to equilibrium. MSC(2010) 35K61, 35A01, 35B40, 35K57

1

Introduction

The purpose of the present paper is to study global existence and asymptotic behavior for reaction diffusion systems with inhomogeneous Dirichlet boundary conditions which include as a particular case the classical systems modeling reversible reaction processes for a set of chemical species Ai , 1 ≤ i ≤ n: α1 A1 + · · · + αn An β1 A1 + · · · + βn An , αi , βi ∈ N ∪ {0}.

(1)

According to the Mass Action law for the reactions and to Fick’s law for the diffusion, the concentrations at position x and time t of Ai , denoted by ui = ∗ ENS Rennes, IRMAR, UEB Campus de Ker Lann, 35170-Bruz, France. Email: [email protected] † Graduate School of Engineering Science, Department of System Innovation, Division of Mathematical Science, Osaka University. Email: [email protected] ‡ Graduate School of Engineering Science, Department of System Innovation, Division of Mathematical Science, Osaka University. Email: [email protected]

1

ui (x, t), satisfy the following evolution system n n Y Y α β uit − di ∆ui = (βi − αi ) uj j − uj j , 1 ≤ i ≤ n. j=1

j=1

We will consider more general systems, always with inhomogeneous Dirichlet boundary conditions, that is uit − di ∆ui = fi (u) in Q∞ = Ω × (0, ∞), 1 ≤ i ≤ n, ui (x, t) = gi (x, t) on Γ∞ = ∂Ω × (0, ∞), (2) ui (x, 0) = ui0 (x) in Ω, where Ω ⊂ RN is a bounded connected open subset with smooth boundary ∂Ω and di ∈ (0, ∞), 1 ≤ i ≤ n. The data u0 = (ui0 )1≤i≤n , g = (gi )1≤i≤n are assumed to be nonnegative. We will throughout assume that u0 ∈ L∞ (Ω) and, for simplicity, that g is smooth, for instance such that there exist Gi , i = 1, ..., n with Gi ∈ C 1 [0, ∞); C(Ω) ∩ C [0, ∞); C 2 (Ω) , (3) Gi = gi ≥ 0 on Γ∞ , ∂t Gi − di ∆Gi = 0 in Q∞ , Gi (·, 0) = gi (·, 0). In the system modeling (1) above, the functions fi are precisely given by n n Y Y α β uj j − uj j , ∀ u = (ui ) ∈ [0, ∞)n . (4) fi (u) = (βi − αi ) j=1

j=1

Although αi , βi are integers in the application to the chemical reaction (1), we will more generally assume that αi , βi ∈ [1, ∞) ∪ {0}. We will consider more general nonlinearities fi . Throughout the paper, they will satisfy fi : Rn → R is locally Lipschitz continuous for 1 ≤ i ≤ n.

(5)

Under this assumption, System (2) has a unique classical solution u local-intime. We will also throughout assume that the nonlinearity f = (fi )1≤i≤n is quasi-positive, which means fi (u1 , · · · , ui−1 , 0, ui+1 , · · · , un ) ≥ 0,

∀ 1 ≤ i ≤ n, ∀ u ∈ [0, ∞)n .

(6)

In this case, the solution u of System (2) is always nonnegative as far as it exists. Obviously (6) is satisfied by the particular f in (4). As for chemical systems of type (1), we will often assume that there exist ci > 0, 1 ≤ i ≤ n, such that n X

for u ∈ [0, ∞)n .

ci fi (u) = 0,

i=1

2

(7)

Existence of the ci in (1) is nothing but preservation of mass. It actually holds for fi as in (4) as soon as there exists i1 , i2 ∈ {1, ..., n} such that αi1 − βi1 > 0 and αi2 −βi2 < 0. Then after summing the equations in (2), equality (7) implies ∂t (c · u) − ∆(dc · u) = 0 for c = (ci )1≤i≤n > 0, dc = (ci di )1≤i≤n > 0. This guarantees several a priori estimates of the solution via duality arguments at least in the case of homogeneous Neumann boundary conditions ([4, 8, 19, 20]). Some of them may be extended to Dirichlet boundary conditions but not all. Actually some main estimates are missing for nonhomogeneous boundary conditions. We will consider the general system (2) with f = (fi )1≤i≤n satisfying (5), (6), (7) or even more generally the following (8) instead of (7): n X

ci fi (u) ≤ 0,

for all u ∈ [0, ∞)n .

(8)

i=1

We will sometimes also assume that |f (u)| ≤ C(1 + |u|γ ), γ ∈ (1, ∞).

(9)

as it is the case in example (4). The goal of this paper is to provide several global existence results for System (2) and to prove exponential asymptotic stability of these global solutions when βi i f is as in (4) and gi = si with Πsα i = Πsi . As expected in these systems, we will deal with different definitions of solutions, and in particular: 1) ”Classical solutions” when the fi (u) ∈ L∞ (QT ) for all T ∈ (0, ∞) in which case the solutions have classical derivatives and the equation is to be understood in a classical sense. 2) ”Weak solutions” as defined next. 3) ”Very weak solutions” as used in Theorem 2. In this paper, first, we show the existence of weak global-in-time solutions for the system (2) when the diffusion rates d1 , d2 , ..., dn are ”quasi-uniform” in the sense of (13) below (see Theorem 1). These solutions may be even classical if the diffusion rates are even closer (see Remark 1). Next we prove in Theorem 2 the convergence of approximate solutions no matter the values of the di , this for a very general system with dissipating entropy and including (2) with fi as in (4). The limit is some kind of ”very weak solution” for which some properties of ”renormalized solution” could be proved (see Remark 2). We prove in Theorem 3 that all these ”solutions” are asymptotically exponentially stable for the specific system (2), (4) when the data are compatible with stationary solutions in the sense of (19). Definition 1 (weak solution) We say that u = (u1 , ..., un ) is a weak solution to (2) if the following conditions are satisfied for all T ∈ (0, ∞) where QT = Ω × (0, T ) and ΓT = ∂Ω × (0, T ): 3

(i) ui ∈ C([0, ∞); L1 (Ω)), fi (u) ∈ L1 (QT ), 2 (ii) For any S ϕ : Ω × [0, T ] → R with continuous ϕ, ∂t ϕ, ∇x ϕ, ∇x ϕ and ϕ = 0 on ΓT (Ω × {T }) it holds that ZZ Z −ui ϕt − di ui ∆ϕ dxdt = ui0 (x)ϕ(x, 0) dx QT Ω ZZ ZZ + fi (u)ϕ dxdt − gi ∂ν ϕ dSdt, 1 ≤ i ≤ n. QT

ΓT

To state our result, let us introduce a = min di , i

b = max di , where 0 < a ≤ b < +∞. i

Let furthermore Cm,q ∈ (0, ∞) be the best constant in the estimate k∆vkLq (QT ) ≤ Cm,q kF kLq (QT )

(10)

where v : QT → R is the solution of the backward heat equation with homogeneous Dirichlet boundary condition: −(vt + m∆v) = F ≥ 0 in QT ,

v = 0 on ΓT ,

v(x, T ) = 0 in Ω.

(11)

For instance by Corollary 7.31 in [16] or Theorem 6.2 in [25], inequality (10) is valid for each q ∈ (1, ∞) (see also Lemma 2.1 in [4]). As a standard approximation of Problem (2), we will consider the solution uk = (uk1 , ..., ukn ) of for 1 ≤ i ≤ n, fi (uk ) ukit − di ∆uki = 1+k−1 P in Q∞ , (12) n k j=1 |fj (u )| k k ui = gi on Γ∞ , ui (·, 0) = ui0 ≥ 0 in Ω. Since the nonlinearity is uniformly bounded (by k), there exists a global-in-time classical and nonnegative solution uk = (uki )1≤i≤n ≥ 0, 1 ≤ i ≤ n, for each k. Theorem 1 Assume (5), (6), (8), (9). If moreover b−a C a+b ,γ 0 < 1, 2 2

(13)

then, a subsequence of the solutions (uk )k≥0 of (12) converges in Lγ (QT )n and C([0, T ]; L1 (Ω)n ) for all T > 0. Moreover, any limit of such converging subsequences is a weak solution of System (2). If γ = 2, then (13) is satisfied for all 0 < a ≤ b < +∞. Remark 1 We may even obtain classical solutions in Theorem 12 if b − a is smaller than in (13). This is the case if (b − a) γ C a+b ,q0 < 1 where q 0 < . 2 2 γ − 2/(N + 2) 4

(14)

Indeed, in this case we obtain (see Remark 6 after the proof of Theorem 1) that uk is bounded in Lq (QT ) where q > (N + 2)γ/2. Going back to the equation (2) and using (9), we deduce that uk is bounded in L∞ (QT ) and the solution at the limit is classical. The result of Theorem 1 does not provide global existence for the system modeling the chemical reaction (1) when the αi , βi are quite larger than 2 and when the di are not close enough to each other. Actually, this is known as a rather difficult and open question. It was significantly analyzed in the case of Neumann boundary conditions in [12]: there the solutions of the approximate System (12) are proved to converge a.e. up to a subsequence and the limit is a renormalized solution in the spirit of [9], but with an adequate definition for this kind of systems as introduced in [12]. Here, we are able to prove a similar convergence result in the case of nonhomogeneous Dirichlet boundary conditions, no matter the values of the di . The situation is not so easy since it does not lead to a priori estimates as good as with Neumann boundary conditions, but they nevertheless provide good enough compactness properties for the approximate solutions, at least locaaly inside Ω. As in [12], they strongly rely on the entropy inequality valid for System (2) with f as in (4), namely n X (log ui )fi (u) ≤ 0. (15) i=1

Theorem 2 Assume (5), (6), (15). Then a subsequence of the solution (uk )k≥0 of (12) converges in L2 (QT )m for all T > 0. Remark 2 If the di are close enough so that (13) (resp. (14)) is satisfied, then the limit obtained in Theorem 2 is a weak (resp. a classical) solution P of (2). For general di ’s, using truncations as in (39) and the functions Tr (uki +η j6=i ukj ), we could prove that the limit is a renormalized solution inside QT in the following sense inspired from [12]. We denote Ψ := {ψ ∈ C 2 (Rn ; (0, ∞))+ with ∂i ψ compactly supported for 1 ≤ i ≤ n} where ∂i ψ(u) is a notation for the derivative of ui ∈ R → ψ(u1 , ..., ui , ..., um ). Starting formally from ∂t ui − di ∆ui = fi (u), we have for all ψ ∈ Ψ X X ∂t ψ(u) = ∂i ψ(u)∂t ui = ∂i ψ(u)[di ∆ui + fi (u)]. (16) i

i

And this may be rewritten ( ∂t ψ(u) =

X i

di [∇ · (∂i ψ(u)∇ui ) −

) X

∂j ∂i ψ(u)∇uj ∇ui ] + ∂i ψ(u)fi (u) .

(17)

j

This equation may be understood in the sense of distributions in QT as soon as χ[ui ≤r] ∇ui ∈ L2loc (QT ) for all r ∈ (0, ∞), T > 0, 1 ≤ i ≤ n. 5

(18)

Indeed, since ∂i ψ is compactly supported for 1 ≤ i ≤ n, we then have ∂i ψ(u)fi (u) ∈ L∞ (QT ), ∂i ψ(u)∇ui ∈ L2loc (QT ), ∂j ∂i ψ(u)∇uj ∇ui ∈ L1loc (QT ).

And as proved later in (34), the estimate (18) will indeed hold here. Note that the estimate is local inside Ω and it is not clear how to extend it up to the boundary except in some cases (see Remark 7). Since our goal here is to mainly concentrate on the asymptotic behavior of the solutions, and since we do not need to know (17) for doing so, we will not prove it here. Actually, it is an interesting point to see that we can control the asymptotic behavior of the ”very weak solutions” without knowing much about them. Thus a main result of this paper is the exponential stability of the limit ”solutions” of (2)-(4) in the case when n Y

gj (x, t) ≡ sj > 0,

j=1

α

sj j =

n Y

β

sj j .

(19)

j=1

Then ui = si > 0, 1 ≤ i ≤ n, is a spatially homogeneous stationary state of (2)-(4). Notation. k · kp , 1 ≤ p ≤ ∞ will denote the standard Lp norm on Ω. Theorem 3 Assume f is given by (4) with (19). Then, the approximate solutions (uk )k≥0 of (12) lie in a compact set of L2 (QT ) for all T > 0. There exist positive constants C1 , C2 such that, for any limit u of converging subsequences kui (·, t) − si k1 < C1 exp(−C2 t), for all t ≥ 0, 1 ≤ i ≤ n.

(20)

Several existence results of global-in-time solutions and their asymptotic behavior have been known for the reaction diffusion system associated with (1), particularly, when the boundary condition is of homogeneous Neumann type. β3 1 α2 First, when n = 3 with f1 = −uα 1 u2 + u3 = f2 = −f3 , existence results of global classical solutions are proved in [14] in particular when β3 > α1 + α2 and for some other particular situations. Exponential convergence towards the stationary solutions is proved in [10] for these fi for all α1 , α2 , β3 ≥ 1 (see also [6] for other results with n = 3). When n = 4 and fi = (−1)i (u1 u3 − u2 u4 ), weak solutions exist globally in time for any space dimension N (see [8]). Furthermore, classical solutions exist globally in time if N ≤ 2 (see [4], [13]) or in any dimension if the diffusion coefficients are quasi-uniform in the sense of (13) (see [4]). Exponential asymptotic stability for the L1 -norm is proved in [5], [7]. For the general system (2),(4), weak (resp. classical) solutions exist globally in time when the diffusion coefficients are quasi-uniform in the sense of (13) (resp. (14)) (see [4]). Finally, global renormalized solutions are proved to exist in [12] for rather general systems with general diffusions and Neumann type of boundary conditions. And their asymptotic behavior is analyzed in [21]. 6

Inhomogeneous Dirichlet boundary condition are studied in [11]. They are concerned with the case n = 3, α1 = β2 = α3 = 1, and β1 = α2 = β3 = 0: ut − d1 ∆u = −u + vw in Ω × (0, T ) v − d ∆v = u − vw in Ω × (0, T ) 2 t wt − d3 ∆w = −u + vw in Ω × (0, T ) u(x, t) = a, v(x, t) = b, w(x, t) = c on ∂Ω × (0, T ) u(x, 0) = u (x) > 0, v(x, 0) = v (x) > 0, w(x, 0) = w (x) > 0 in Ω. 0 0 0 (21) If a, b, c are positive constants satisfying a = bc, there exists a classical solution (u, v, w) = (u(·, t), v(·, t), w(·, t)) global-in-time and it holds that lim (u(·, t), v(·, t), w(·, t)) = (a, b, c)

t→∞

in C ν (Ω)

where 1 < ν < 2. However, asymptotic behavior of the global-in-time ”weak” solution has not been studied for the general case of (2)-(4).

2

Proof of Theorem 1

We first show an estimate on the solution of a parabolic differential inequality. It is similar to Proposition 1.1 of [4], but with nonhomogeneous Dirichlet boundary conditions. α Given α ∈ (0, 1), we take M = M (x, t) ∈ C α, 2 (Ω × (0, T ]) satisfying 0 < a ≤ M (t, x) ≤ b < ∞,

(x, t) ∈ QT .

We consider the parabolic differential inequality ut − ∆(M u) ≤ 0 in QT , u = g on ΓT , u(·, 0) = u0 (x) ≥ 0 in Ω.

(22)

(23)

We will estimate kukLp (QT ) for p ∈ [2, ∞), under the assumption C a+b ,p0 · 2

b−a < 1, 2

1 1 + 0 = 1. p p

(24)

where Cm,q ∈ (0, ∞) stands for the best constant in the parabolic regularity (10)-(11). 2 so that (24) is always satisfied for p0 = 2 Remark 3 We have C a+b ,2 ≤ a+b 2 and for all 0 < a < b < ∞. Indeed, multiplying (11) by −∆v leads to ZZ ZZ vt ∆v + m(∆v)2 = −F ∆v ≤ kF kL2 (QT ) k∆vkL2 (QT ) . QT

We then use

QT

RR QT

vt ∆v =

R 1 2

Ω

|∇v(x, 0)|2 ≥ 0 to deduce Cm,2 ≤ 1/m.

7

Remark 4 It is interesting to notice that the condition (24) is ”open” with respect to p0 in the sense that if (24) holds with p0 , then it holds with (p + )0 for − small enough. Indeed, the Cm,q has the property: Cm,q := lim inf η→0+ Cm,q−η ≤ Cm,q . To see it, let qη satisfy 1 1 1 1 + i.e. qη = q − ηq/(2q − η). = qη 2 q q−η By the Riesz-Thorin interpolation theorem (see e.g. [18], chapter 2) applied to the mapping F 7→ ∆v in (11), we have 1/2

1/2 − 1/2 − − Cm,qη ≤ Cm,q Cm,q−η ⇒ Cm,q ≤ Cm,q (Cm,q )1/2 ⇒ Cm,q ≤ Cm,q .

Notation. For the boundary ΓT , we will use dS, ν, ∂ν to denote respectively the surface element, the exterior unit normal and the exterior normal derivative. For the solution of equation (11), we can define the best constant Em,q,T ∈ (0, ∞) for the inequality ZZ |∂ν v| dSdt ≤ Em,q,T kF kLq (QT ) , q > 1, (25) ΓT

using the trace embedding Wq1 (Ω) ,→ L1 (∂Ω). α

Proposition 4 Let u ≥ 0 be a classical solution to (23) with M ∈ C α, 2 (Ω × (0, T ]) satisfying (22) and (24). Then it holds that 1 ˜ a+b 0 · b · kgkL∞ (Γ ) kukLp (QT ) ≤ (1 + bDa,b,p0 ) T p ku0 kp + E T ,p ,T

(26)

2

for p ∈ [2, ∞), where Da,b,p0 =

C a+b ,p0 2

1 − C a+b ,p0 · 2

˜a,b,p0 ,T = E a+b 0 , E ,p ,T b−a 2

2

1+

b−a Da,b,p0 2

.

Remark 5 Note that, according to Remark 3, Da,b,2 < +∞ so that any u satisfying (23) is bounded in L2 (QT ) for all T > 0 with a bound depending on ku0 kL2 (Ω)n , kgkL∞ (ΓT )n . To prove Proposition 4, we begin with a parabolic estimate for the dual problem ψt + M ∆ψ = −Θ in QT ,

ψ(T, x) = 0 in Ω,

ψ(x, t) = 0 on ΓT ,

(27)

C0∞ (QT ).

where Θ ∈ This inequality will be proved similarly as in Lemma 2.2 of [4] concerning homogeneous Neumann boundary condition. α

Lemma 5 For M = M (x, t) ∈ C α, 2 (Ω × (0, T ]) satisfying (22) and (24) and 1 < p0 ≤ 2, the following holds for the solution ψ of (27) k∆ψkLp0 (QT ) ≤ Da,b,p0 kΘkLp0 (QT ) (28) kψ(·, 0)kp0 ≤ (1 + bDa,b,p0 )T 1/p kΘkLp0 (QT )

8

Proof: By the standard theory (e.g. Corollary 7.31 and Theorem 7.32 in [16]) or Theorem 6.2 in [25]), Problem (27) admits a unique classical solution ψ = ψ(x, t). We write (27) as a+b a+b ψt + ∆ψ = − M ∆ψ − Θ. (29) 2 2 Then (10) implies k∆ψkLp0 (QT )

≤ ≤

a+b

C a+b ,p0 − M ∆ψ − Θ

p0 2 2 L (QT ) b−a C a+b ,p0 k∆ψkLp0 (QT ) + kΘkLp0 (QT ) 2 2

b−a ∞ where we used: k a+b 2 − M kL (QT ) ≤ 2 . Therefore, b−a 1 − C a+b ,p0 k∆ψkLp0 (QT ) ≤ C a+b ,p0 kΘkLp0 (QT ) . 2 2 2

This is the first inequality of (28) (we use (24) here). The second inequality is RT derived from −ψ(0) = 0 ψt (·, t)dt and (27) which imply kψ(0)kp0 ≤ T 1/p kψt kLp0 (QT ) ≤ T 1/p [b k∆ψkLp0 (QT ) + kΘkLp0 (QT ) ]. Proof of Proposition 4: If 0 ≤ Θ ∈ C0∞ (QT ), the classical solution to (27) satisfies ψ = ψ(x, t) ≥ 0. Then both u and ψ are nonnegative and we obtain Z Z d uψ dx = ut ψ + uψt dx dt Ω ZΩ ≤ [∆(M u)]ψ + u(−M ∆ψ − Θ) dx. Ω

Since

Z

Z [∆(M u)]ψ − (M u)∆ψ dx = −

Ω

gM ∂ν ψ dS, ∂Ω

it holds that d dt

Z

Z uψ dx ≤ −

Ω

Z uΘ dx −

Ω

gM ∂ν ψ dS.

(30)

∂Ω

Here we use (25), (29), and (28) to conclude ZZ b−a |∂ν ψ| dSdt ≤ E a+b ,p0 ,T 1 + Da,b,p0 kΘkLp0 (ΩT ) . 2 2 ΓT

9

(31)

Inequalities (30) and (31) imply ZZ ZZ uΘ dxdt ≤ ku0 kp kψ(·, 0)kp0 + kgkL∞ (ΓT ) · b |∂ν ψ| dSdt Q ΓT o nT 1 ˜a,b,p0 ,T bkgkL∞ (Γ ) kΘk p0 (32) ≤ (1 + bDa,b,p0 )T p ku0 kp + E L (QT ) . T Inequality (32), valid to any 0 ≤ Θ ∈ C0∞ (QT ), implies (26) by duality since u ≥ 0. Proof of Theorem 1: Let us consider the global regular solution uk of the approximate problem (12). Recalling (8), let v k = c · uk ,

h = c · g,

v0 = c · u0 ,

where c = (ci ), uk = (uki ), g = (gi ) and u0 = (ui0 ). Let also dc := (di ci ). Then (8) implies vtk − ∆(M v k ) ≤ 0 in QT v k (x, t) = h(x, t) on ΓT ,

v k (x, 0) = v0 (x) ≥ 0 in Ω

with M = M (x, t) = dc · uk /c · uk which satisfies (22) with a = mini di , b = maxi di . According to Remark 4, the assumption (13) implies that b−a 2 C a+b ,(γ+)0 < 1 2

for some > 0. By Proposition 4, kv k kLγ+ (QT ) ≤ CT for all T > 0. It follows by (9) that fi (uk ) is bounded in L1+η (QT ) for η = /γ > 0. We then may use the L1 -compactness property of the heat operator saying (see e.g. [2], [1]) that the mapping (w0 , F ) ∈ L1 (Ω) × L1 (QT ) → w ∈ L1 (QT ) is compact where w is the solution of wt − m∆w = F in QT , w = 0 on ΓT , w(·, 0) = w0 .

(33)

Applying this here to m = di , F = fi (uk )/(1 + k −1

X

|fj (uk )|), w = uki − Gi , w0 = uki0 − gi (0),

j

where Gi is defined in (3), we deduce that uk lies in a compact set of L1 (QT )m . Up to a subsequence, we may assume that, for all T > 0, uk converges in L1 (QT )m and a.e. to some u which, by Fatou’s Lemma, belongs to L1 (QT )m . It implies that fi (uk ) converges a.e. to fi (u) for all i. Since fi (uk )/(1 + P −1 k 1+η k (QT ), we deduce by Egorov’s theorem that j |fj (u )|) is bounded in L 1 the convergence holds also in L (QT ). Now we may pass to the limit in ZZ Z k k −ui ϕt − di ui ∆ϕ dxdt = uki0 ϕ dx QT

ZZ + QT

Ω

fi (uk ) P ϕ dxdt − −1 k 1+k j |fj (u )| 10

ZZ gi ∂ν ϕ dSdt, ΓT

1 ≤ i ≤ n,

for all ϕ as in Definition 1. To conclude that u is a weak solution, we only need to check that u ∈ C([0, ∞); L1 (Ω)n ). This follows from the L1 -contraction property of the heat operator, namely kuki (t) − upi (t)k1 ≤ kuki0 − upi0 k1 +

t

Z

k 0

fi (uk ) fi (up ) P P − k1 dt. 1 + k−1 j |fj (uk )| 1 + p−1 j |fj (up )|

This proves that uk converges in L∞ ([0, T ] : L1 (Ω)n ) and the limit is therefore continuous from [0, ∞) into L1 (Ω). Remark 6 If we replace (13) by the (stronger) assumption (14), then by Proposition 4, and the same proof as above, uk is bounded in Lq (QT ) for q > (N + 2)γ/2. This implies by (9) that fi (uk ) is bounded in Ls (QT ) for some s > (N + 2)/N . And it is well-known (see e.g. [15]) that uki is then bounded in L∞ (QT ) and so is fi (uk ). The limit of uk is then a classical solution of (2).

3

Proof of Theorem 2

Let us first prove the following estimate for the solution uk of (12). For δ > 0, we denote Ωδ = {x ∈ Ω; d(x, ∂Ω > δ}. Then ZZ |∇uki |2 ≤ Cδ r for all r ∈ [0, ∞), 1 ≤ i ≤ n, k ∈ N. (34) [uk i ≤r]∩Ωδ

In the following computation, for simplicity, we drop the us introduce wi := ui log ui + 1 − ui ≥ 0. We have ∂t

X i

wi − ∆

X i

di wi =

X

log ui fi (u) −

X

i

k

in the notation. Let

X √ √ 4di |∇ ui |2 , (35) 4di |∇ ui |2 ≤ −

i

i

the last inequality coming from the assumption (15). Let ϕ be the first eigenfunction of the Dirichlet-Laplacian on the open connected set Ω, namely −∆ϕ = λ1 ϕ in Ω, ϕ = 0 on ∂Ω, kϕk∞ = 1, ϕ > 0 on Ω.

(36)

Multiplying the previous inequality by ϕ and integrating on QT gives RR P P R √ ϕ i wi (T ) + QT ϕ i di (λ1 wi + 4|∇ ui |2 ) ΩR RR P P ≤ Ω ϕ i wi (0) − ΓT ∂ν ϕ i di (gi log gi + 1 − gi ). We deduce that for some C = C(maxi {kgi k∞ , kui0 log ui0 k1 }, k∂ν ϕk1 ) < ∞ Z Z √ |∇ui |2 max 4ϕ|∇ ui |2 = max ϕ ≤ C, (37) i i ui QT QT and the estimate (34) follows with Cδ = C/ minx∈Ωδ ϕ(x).

The same inequality (35) implies also the following L2 -estimate: max kuki log uki kL2 (QT ) ≤ C = C(max{kgi k∞ , kui0 log(ui0 )k2 , T, a, b) < +∞. i

i

11

(38)

Proof. Indeed, (35) implies X X X X ∂t ( wi ) − ∆(M wi ) ≤ 0, M := di w i / wi . i

i

i

i

Thanks to wi ≥ 0, we have a = mini di ≤ M ≤ b =Pmaxi di . Thus Proposition 4 applied with p = 2 (see Remark 5) implies that k i wi kL2 (QT ) ≤ C where C is as in (38). Now, using again the nonnegativity of the wi and the fact that s log s is bounded from a bove for s large by 2[s log s + 1 − s], estimate (38) follows. Proof of Theorem 2. Let us prove the convergence of uk in L2 (QT ) for all T > 0. We will first prove that uk converges a.e. on Q∞ . Then the L2 (QT ) convergence will follow from the estimate (38). For all r ∈ (0, ∞), we introduce Tr ∈ C 2 ([0, ∞); [0, ∞)) with 0 ≤ Tr0 (s) ≤ 1 and Tr00 (s) ≤ 0 for all s ∈ [0, ∞), (39) Tr (s) = s for s ∈ [0, r], Tr0 (s) = 0 for s ∈ [2r, ∞). P k Let now vik = uki + ηUik where Uik = j6=i uj , η > 0. We also denote P −1 Fi = fi /(1 + k j |fj |). Then ( ∂t Tr (vikh) − di ∆Tr (vik ) = i P (40) Tr0 (vik ) Fi (uk ) + η j6=i [Fj (uk ) + (dj − di )∆ukj ] − di Tr00 (vik )|∇vik |2 . Let us analyze each of the terms in the right-hand side of this equality. We will repeatedly use that [vik ≤ 2r] ⇒ [uki ≤ 2r] and [ukj ≤ 2r/η] ∀j 6= i. In the following, r is fixed arbitrarily in (0, ∞) and η > 0 is fixed small enough (to be made precise later). By the choice of Tr and of v k X Tr0 (vik )[Fi (uk ) + η Fj (uk )] is bounded in L∞ (QT ) independently of k. (41) j6=i

We also have, with QδT := (0, T ) × Ωδ , (recall that Tr00 vanishes outside [0, 2r]) X kTr00 (vik )1/2 ∇vik kL2 (QδT ) ≤ C kχ[uki ≤2r] ∇uki kL2 (QδT ) + kχ[ukj ≤2r/η] ∇ukj kL2 (QδT ) . j6=i

Thus, by (34) Tr00 (vik )|∇vik |2 is bounded in L1 (QδT ) ∀ δ > 0. Now we write Tr0 (vik )∆ukj = ∇ · (Tr0 (vik )∇ukj ) − Tr00 (vik )∇vik ∇ukj , 12

(42)

kT 0 (vik )∇ukj kL2 (QδT ) ≤ Ckχ[ukj ≤2r/η] ∇ukj kL2 (QδT ) , kTr00 (vik )∇vik ∇ukj kL1 (QδT ≤ CkTr00 (vik )1/2 ∇vik kL2 (QδT ) kχ[ukj ≤2r/η] ∇ukj kL2 (QδT ) . We deduce, by using (34) again, that Tr0 (vik )∆ukj is bounded in L2 (0, T ; H −1 (Ωδ )) + L1 (0, T ; L1 (Ωδ )) ∀ δ > 0. (43) Let ψ ∈ C0∞ (Ω)+ . We deduce from (41), (42), (43) and equation (40) that ∂t (ψTr (vik )) − di ∆(ψTr (vik )) is bounded in L1 (QT ) + L2 (0, T ; H −1 (Ω)). Since moreoverψTr (vik ) is bounded by rkψk∞ and vanishes on ΓT , it follows that ψTr (vik ) k≥0 lies in a compact set of L1 (QT ): to see this, we may use the L1 compactness as stated for w in (33) and the fact that, if in (33), F is bounded in L1 (0, T ; H −1 (Ω)) and w0 = 0 then w lies in a compact set of L2 (QT ) (see e.g. [17], Th´eor`eme 5.1). Using a diagonal extraction process, we can deduce that there exists a subsequence of vik (still denoted vik ) such that Tr (vik ) converges a.e. on Q∞ for all r ∈ (0, ∞) (here we fix η small enough as indicated below). This implies that vik converges a.e. on Q∞ itself. Indeed, let us denote by wr the pointwise limit of Tr (vik ) and let Kr = [wr < r]. For (x, t) ∈ Kr , Tr (vik (x, t)) = vik (x, t) for k large enough so that vik (x, t) converges to wr (x, t). Therefore vik converges a.e. on Kr . On the other hand, since, thanks to (38), vik is bounded in L1 (QT ) and we have (using Fatou’s lemma for ⇒) Z Z Z k k +∞ > C ≥ 2vi ≥ Tr (vi ) ⇒ C ≥ wr ≥ r|[wr ≥ r] ∩ QT |. QT

QT

QT

Thus limr→∞ |[wr ≤ r] ∩ QT | = 0 and ∪r∈(0,∞) [w < r] ∩ QT = QT a.e.. Whence the a.e. convergence of vik on Q∞ . Since the n × n matrix A with 1 on its diagonal and η elsewhere is invertible for η small, and since v k = Auk , it follows that uki converges also a.e. on Q∞ . We now use the fact that uki log uki is bounded in L2 (QT ) by (38). Together with the a.e. convergence this implies the convergence of uki in L2 (QT ) by Egorov’s theorem again.

4

Proof of Theorem 3

Here, fi is defined by (4) and gi = si so that the approximate problem (12) becomes ukit − di ∆uki = (βi − αi )

R(uk ) P 1 + k −1 j |fj (uk )|

uki = si on ΓT , uki (x, 0) = uki0 (x) in Ω, n n Y Y α β where R(u) = uj j − uj j . j=1

j=1

13

in QT

1 ≤ i ≤ n,

P P Here (9) holds with γ = max{ i αi , i βi }. Moreover (15) is satisfied since n X

fi (u) log ui = R(u)

i=1

n n n X Y Y i (βi − αi ) log ui = R(u){log uβi i − log uα i } ≤ 0. (44) i=1

i=1

i=1

Therefore we can apply the results of Theorem 2. But, we will have more estimates here due to the choice of the si . Let us introduce L(uki , si ) ≡ uki (log uki − log si ) + si − uki ≥ 0,

(45)

the nonnegativity coming from ξ ≥ log ξ + 1 for ξ > 0 and applied to ξ = si /uki . Lemma 6 ζ k (x, t) =

n X

L(uki (x, t), si ),

ζ0 (x) =

i=1

it holds that (

n X

L(ui0 (x), si )

(46)

i=1

Pn P |∇uk |2 ζtk − i=1 di ∆L(uki , si ) + i di uki ≤ 0 in QT i ζ k = 0 = ∂ν ζ k on ΓT , ζ k (·, 0) = ζ0 ≥ 0 in Ω.

(47)

Proof: We have ukj = ukj (x, t) > 0 for t > 0, and then it follows that ∂t L(uki , si ) − di ∆L(uki , si ) X |∇uki |2 = (log uki − log si )(ukit − di ∆uki ) − di uki i ( ) k k 2 X R(u ) |∇u | i P = (log uki − log si )(βi − αi ) − di . k −1 k )| 1 + k |f (u u j j i i P k k We already P checked that i log ui (βi − αi )R(u ) ≤ 0 (see (44)). On the other hand, i log si (βi − αi ) = 0 due to the assumption (19). The first estimate of (47) follows. The boundary conditions follow from uki = si at the boundary so that L(uki , si ) = 0, ∂ν L(uki , si ) = ∂ν uki (log uki − log si ) = 0. Now we use the argument of [24], taking a ball Ω0 such that Ω ⊂ Ω0 . Let λ1 > 0 and ϕ = ϕ(x), kϕk∞ = 1 be the first eigenvalue and the associated eigenfunction, respectively: −∆ϕ = λ1 ϕ, ϕ > 0 in Ω0 ,

ϕ = 0 on ∂Ω0 .

Let δϕ = inf Ω ϕ > 0. Lemma 7 It holds that Z Z δϕ ζ k (x, t) dx ≤ ζ k (x, 0) dx · e−aλ1 t Ω

Ω

where a = mini di . 14

(48)

Proof: Thanks to the inequality and the boundary conditions in (47), we have Z Z XZ X |∇uk |2 d i ≤ ∆ϕ di L(uki , si )dx. (49) ζ k ϕ dx + ϕ di k dt Ω u Ω Ω i i i In particular d dt

Z

n Z X

ζ k ϕ dx ≤ −λ1

Ω

i=1

di L(uki , si )ϕ dx ≤ −λ1 a

Ω

Z

ζ k ϕ dx.

Ω

Whence (48).

Here we use an elementary inequality. Lemma 8 (Cziszar-Kullback) For any measurable functions f : Ω 7→ [0, ∞), g : Ω 7→ (0, ∞), it holds that 2

Z |f − g| dx

3 Ω

Z

f [f log − f + g] dx . g Ω

Z

≤

(2f + 4g) dx Ω

(50)

Proof: Since 3|ξ − 1|2 ≤ (2ξ + 4)(ξ log ξ − ξ + 1), for all

ξ>0

it follows by choosing ξ = f (x)/g(x) (assuming f (x) < ∞) and taking the square root that √

1/2 1/2 f (x) 3|f (x) − g(x)| ≤ 2f (x) + 4g(x) f (x) log − f (x) + g(x) . g(x)

If the right-hand side of (50) is infinite, then the inequality holds. Therefore, we may assume that it is finite which implies that f and g are finite a.e. We integrate the above pointwise inequality over Ω and apply Schwarz’ inequality to the second integral to obtain (50). Proof of Theorem 3: By Lemma 7 and Lemma 8 applied with f = uki and g = si , we obtain ( R R kuki (·, t) − si k21 ≤ 13 · Ω 2uki (x, t) + 4si dx · Ω L(uki , si )dx R (51) 1 t) ≤ exp(−aλ · Ω 2uki (x, t) + 4si dx · kζ0 k1 . 3δϕ This inequality implies also that (after taking the square root) 1/2 kuki k1 ≤ si + (kζ0 k1 /3δϕ )−1/2 2kuki k1 + 4si |Ω| . This implies that sup kuki (·, t)k1 < +∞.

k∈N,t≥0

15

(52)

thus we may deduce from (51) that kuki (·, t) − si k21 ≤ C exp(−aλ1 t) with C > 0 independent of k. Then, Theorem 2 and Fatou’s Lemma imply (20). p Remark 7 Going back to the estimate (49), we seepthat uki is bounded in √ L2 (0, T ; H 1 (Ω)) for all i = 1, ..., n. In other words, uki − gi is bounded in √ √ ui − gi ∈ L2 (0, T ; H01 (Ω)). Therefore, for any limit u of uk in Theorem 3, L2 (0, T ; H01 (Ω)). In other words, the nonhomogeneous Dirichlet boundary condition is kept at the limit.

Acknowledgements This work has partially been supported by JSPS Grand-in-Aid for Scientific Reserach (A)26247013 (B)15KT0016, and JSPS Core-to-Core program Advanced Research Networks.

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