Asymptotic behaviour of reversible chemical reaction-diffusion equations

We investigate the asymptotic behavior of the a large class of reversible chemical reaction-diffusion equations with the same diffusion. In particular we prove the optimal rate in two cases : when there is no diffusion and in the classical"two-by-two"case.


Introduction
The self-ionization of water is the chemical reaction in which two water molecules react to produce a hydronium ion H 3 O + and a hydroxide ion OH − . It can be written as follow This is a classical example of a reversible reaction-diffusion chemical reaction between three species such that the water has to appear twice for the reaction.
We would like to investigate in this paper how fast reversible chemical reactions (as the self-ionization) tend to the equilibrium. To be more realistic a diffusion term will be added. Such term models the fact that particles have to move (to diffuse) before the reaction. If the reaction is not enough mixed, then the diffusion term will be more important in the equation.
The main model studied involves q > 1 species A i interacting together as follows where α i , β i ∈ N, with N denoting the set of nonnegative integers. We assume that for any 1 ≤ i ≤ q, α i − β i = 0 which correspond to the case of a reaction without catalyzer, as in the chemical equation presented in (1).
First we will introduce the differential equations that describe the evolution of the species A i for 1 ≤ i ≤ q in the relation (2). For some 0 ≤ i ≤ q we will denote by a i the concentration of the species A i . We assume existence of two non-negative real-valued rate functions which describe the evolution of the concentration a i in the two (reversible) reactions. In the first one we lose α i molecules of the specie A i and in the second one there is a gain of β i molecules of the same species A i . We get the opposite for the reverse reaction. Thus we can write where a = (a 1 , · · · , a q ). We assume the kinetics to be of mass action type, which means that where l is a positive constant called the rate constant of the reaction. This model was proposed by Waage and Guldberg in 1864. The mass action represents the probability of the reaction between all the species (α 1 , A i ) 1≤i≤q which are assumed to be independent. Let us denote by k > 0 the rate constant for the reverse reaction. Of course l and k could be different.
We will presume that the pot of the reaction is not mixed or not enough mixed. Then concentrations of species depend on the position in the pot and we have to add a diffusion term which depends also on the species. We obtain that the model has the following mathematical representation where for all i, L i is a diffusion operator.
We refer to [7,9] for a general introduction on chemical reaction-diffusion models.

Mathematical model
Throughout the entire paper we will assume that all species diffuse with the same speed, i.e. for all i ∈ {1, · · · , q}, L i = L, where L is general diffusion generator. (While this is a restrictive case -as generally this hypothesis is not realistic -assuming identical diffusion will allow us to obtain some optimal bounds.) In more detail we let Ω ⊂ R n (n 1) to be a bounded open and connected set and assume that the boundary ∂Ω of Ω, is C ∞ -smooth. Let a diffusion operator L be given by for any smooth functions f , with a i,j and b = (b i ) 1≤i≤n in C ∞ (Ω) and the matrix a(x) = (a i,j (x)) i,j symmetric and positive for all x ∈ Ω. (Generally the choice of coefficients a i,j and b i may depend on the domain Ω.) We will assume that there exists a probability measure µ on Ω, dµ = 1, which is L -invariant, i.e. for all functions f ∈ L 1 (dµ) in the domain of L Lf dµ = 0, The existence of an invariant measure is a quite standard problem and has a solution for a large class of generators L.
If we denote by a i (t, x), (t 0, x ∈ Ω), the concentrations of the species A i at time t in the position x, then the following reaction-diffusion system is satisfied: where k, l > 0 are rate constants of the reverse reaction. It is assumed that the initial conditions satisfy for all 1 ≤ i ≤ q, a 0 i 0 and a 0 i dµ > 0. The last equation in (5) represents the Neumann boundary conditions which is quite natural in the context of chemical reaction-diffusion.
Let us denote by D(Ω) the set of smooth functions f on Ω satisfying the Neumann boundary conditions, such that for all x ∈ ∂Ω, ∂f ∂ν (t, x) = 0.
Let (P t ) t be the semi-group associated to L, ( which is a linear operator defined for all functions f ∈ D(Ω) ), as follows Since µ is an invariant measure, so for all t 0, P t f dµ = f dµ. It could be the classical heat equation with boundary condition but also the heat equation with a drift part some example will be given in the example 1.
One of the main tools of the article is the spectral gap inequality (also called Poincaré inequality). We assume that the operator L satisfies a spectral gap inequality in D(Ω), that is : There exists C SG 0 such that for all smooth functions f ∈ D(Ω), which is equivalent to, for all functions f ∈ D(Ω) and t 0: In the general form, when L is given by (4), then for all function f with Neumann boundary conditions, where a(x) is the diffusion matrix of the generator L given in (4). The assumption of the spectral gap inequality is fundamental. First able it proves that the semigroup (P t ) t 0 is ergodic, that is for all f ∈ D(Ω), in L 2 (dµ). Moreover this inequality proves that the rate of convergence is exponential in L 2 (dµ), it is given by the inequality (7). See for instance chapter 2 of [2] of a review on semigroups tools and ergodic properties. One can consider the generator L given in (4) where b = 0 and a λId in Ω with λ > 0 in the sense of symmetric matrices. If a = Id then the associated semigroup is (P t ) t 0 is the classical heat semigroup with Neumann boundary conditions. The invariant measure is the Lebesgue measure and the spectral gap constant C SG , depends on the domain Ω.
But we can also consider a generator L = ∆ − ∇ψ∇ with ψ a smooth function on the domain Ω. The generator L is symmetric in L 2 (µ ψ ) where µ ψ = e −ψ dx/C is a probability measure in Ω with the normalization constant C. It satisfies a Spectral gap inequality with the dirichlet form − f Lf dµ ψ = |∇f | 2 dµ ψ for functions f satisfying the Neumann Boundary conditions.
We will first study the problem of a steady state of the differential system (5). Let v i (t, x) = λ i a i (t, x) for some constant λ i > 0, assuming that a i is solution of (5), then v i satisfies Let us define the set S = (z i ) 1≤i≤q , s.t. (10). Then for all (z i ) 1≤i≤q ∈ S, one gets In particular, due to the fact that µ is an invariant measure, The goal of this article is to understand the asymptotic behaviour of the reactiondiffusion problem in the spirit of [3,4], for a general but (diagonal) diffusion.
We will first define a Steady State of equation (9), in the following way standard in a chemical reversible reaction. Definition 1.1 A steady state of equation (9) with non-negative initial conditions v 0 i 1≤i≤q is a vector (s i ) 1≤i≤q ∈ (R + ) q such that for all (z i ) 1≤i≤q ∈ S:

Remark 1
We implicitly assume in the previous definition that a steady state is a vector (s i ) 1≤i≤q ∈ (R + ) q independent of x. In fact we don't know in the general case if there exists a steady state depending of the space variable x. But as it is proved in this paper, the solution converges to the one defined above.
Proof ⊳ The steady state (s i ) 1≤i≤n has to satisfy: We have to find t ∈ R, such that for all A simple computation gives that the function then ϕ is increasing and satisfies ϕ(a) = 0 and ϕ(b) = +∞. Thus there exists a unique t ∈ (a, b) such that ϕ(t) = 1, which ends the proof. ⊲ Due to the fact that all species are moving according to the same diffusion, one can reduce the problem as follow: for all 1 ≤ i, j ≤ q, one gets Let us fix 1 ≤ i ≤ q. Then the study of (9) is equivalent to the study of the following PDE with boundary conditions The existence problem in the general case when the operators L i depending on i are all different is a difficult problem. Some approaches one can find e.g. in [8,5,12,13,15] and also by discretization in [11,10].
Our contribution to the domain of reaction-diffusion equation is to prove, using the spectral gap inequality (or Poincaré inequality) and Markov semigroup tools, that the solution of reaction-diffusion equation (9) converges to the unique steady state associated to the initial condition. The goal of this paper is to explain how fast the solution converges to the steady state and to debate on the speed of convergence. In particular if the result obtained is far or close to the optimal one.
The main idea is to investigate a simple proof to explain the asymptotic behavior. Therefore, for simplicity, we will not focus on existence theorems, the optimal conditions of initial conditions and do not study the special case of a unbounded space Ω. Almost all results given here can be generalize to the entire space R n if a regular solution of the problem is given.
We will consider different cases. In Section 2 we will study the case without diffusion, that means that the concentration of different species do not depend on x ∈ Ω. We will prove that there exists a solution converging to the unique steady state with an exponential and optimal rate of convergence.
Then in Section 3 we study the classical "two-by-two" case treated in [4,6] by entropy methods. This case is interesting because we obtain the optimal rate of convergence and it gives tools to understand a more general case. We finish in Section 4 with the general case We prove that under the existence of a non-negative solution and under the assumption that α i β i = 0, we get that solution converges with an exponential rate to the steady state. In this case the rate may be not optimal.

Case without diffusion
Assume that concentrations of species do not depend on x ∈ Ω. It is the case the pot used for the chemical reaction is mixed constantly so that its contents remain spatially homogeneous. This case is important because we can solve it explicitly and it gives tools to study the general case in the later sections. We believe that results of this section are not new, we give here the proof to keep the article self contained and also because we did not find appropriate reference.
The chemical reaction, without diffusion is given by the following system, for all i ∈ {1, · · · , q}, where k i is defined in (10), with initial conditions v i (0) > 0 for 1 ≤ i ≤ q. Equivalently, using the same method as for (12), for some 1 ≤ i ≤ q fixed, Theorem 2.1 Let the initial conditions (v j (0)) 1≤j≤q be positive. Then equation (13) has a unique solution defined on [0, +∞), which satisfies for all 1 ≤ i ≤ q, where K is a constant depending on initial conditions, the steady state (s i ) 1≤i≤q is defined in Lemma 1.2 and Moreover the constant C is the optimal rate of convergence.
is empty one can use the negative part. Assume for simplicity that j 0 = 1. Then the reaction equation becomes By the definition of j 0 one have, Getting a positive solution (v i ) 1≤i≤q) of (13) is equivalent to getting a solution v 1 of (16) which is defined by then it satisfies the following inequality which proves that F ′ (s 1 ) = Q(s 1 ) < 0 and then Q(X) < 0 for all X ∈ (0, M ). Function F is locally a Lipschitz function, thus by Cauchy-Lipschitz's Theorem, there exists a unique maximal solution starting at v 1 (0) of the equation for some T > 0. One has Q(X) < 0 for all X ∈ (0, M ), which implies that if v 1 (0) > s 1 , then v 1 is non-increasing and moreover v 1 (t) s 1 for all t ∈ [0, T [, while if v 1 (0) < s 1 , then v 1 is non-decreasing and moreover v 1 (t) ≤ s 1 for all t ∈ [0, T [, and if v 1 (0) = s 1 , then v 1 (t) = s 1 for all t ∈ [0, T [. Then one gets that T = +∞ and for all t 0, v 1 (t) ∈ (0, M ).
Links between v i and v 1 gives the last inequalities for any v i . Moreover, equality (18) proves that the constant C is the optimal rate. ⊲ 3 The "two-by-two" case The goal of this section is to investigate the asymptotic behaviour of a chemical reaction of particular type A + B ⇋ C + D.
We will assume that all species are moving according to the same generator and the speed of the two reactions are the same, for instance equal to 1.
This case was treated with a general diffusion in [4,6] by using entropy method, but the optimal rate of convergence was not obtained.
Let us denote by a, b, c and d concentrations of A, B, C and D in the domain Ω. In this case the functions a, b, c and d are solutions of the following system on Ω, with non-negative initial conditions a 0 , b 0 , c 0 and d 0 in D(Ω), such that a 0 dµ > 0 and the same for b 0 , c 0 and d 0 . We recall that the boundary conditions are included in the definition of the domain D(Ω).
If a, b, c, d are solutions, then a + c = P t (a 0 + c 0 ), a + d = P t (a 0 + d 0 ) and a − b = P t (a 0 − b 0 ) which gives that the function a is solution of the linear equation Since the solution is given by a linear solution there exists a classical non-negative solution of the problem. One can see for example [1,14] for a proof . Some remarks on the existence are given in Section 4.
We also obtain the estimation useful for the asymptotic behaviour.
Theorem 3.2 Assume that the semigroup (P t ) t 0 satisfies a spectral gap inequality (7) with respect to the invariant probability measure µ. Let a 0 , b 0 , c 0 and d 0 be non-negative initial conditions satisfying a 0 , b 0 , c 0 , d 0 ∈ L 4 (dµ).
C SG denotes the constant in the spectral gap inequality (7). Let s a is the steady state associated to the initial condition a 0 , if M a+b+c+d = 1 8C SG , then the solution a of (20) satisfies, for all t 0, and if M a+b+c+d = 1 8C SG then for all t 0, The same inequality holds for b, c and d associated to s b , s c and s d .
If the initial conditions satisfy M a+b+c+d < 1 8C SG , then the rate of the convergence is optimal.
Let us start with a general estimate. Lemma 3.3 Assume that the semigroup (P t ) t 0 satisfies a spectral gap inequality (7) with respect to the invariant probability measure µ, then for all functions f ∈ L 4 (µ) and all t 0, (23) Proof ⊳ Setf = f − f dµ, then using semigroup properties and the Cauchy-Schwartz inequality applied to (P t ) t 0 , one gets, dµ.
If we set F = Pt 2 (f ), then one has which gives by definition of F , We apply twice inequality (7) to F and to f to obtain which implies (23). ⊲ Proof of Theorem 3.2 ⊳ Is this case, the steady state is the following limit, the limit can be seen in L 4 (dµ). Let us denote by M a+c = (a 0 + c 0 )dµ and define similarly M a+d and M a+b+c+d . One has, d dt then by (20) and (24) one obtains d dt Let us consider the last term:

First spectral gap inequality gives
Since the semigroup (P t ) t 0 is contractive : d To finish, Lemma 3.3 gives , which implies for the last term of (25): For the other term one gets Then we obtain where Integration of the last differential inequality which finished the proof of (22). If M a+b+c+d < 1 8C SG , then the rate becomes e −M a+b+c+d t , one can check that M a+b+c+d is equal to the constant C defined in (15) which is optimal. ⊲ Remark 2 In the case of a linear equation of diffusion, ∂ t u = Lu where L is given by (4), the optimal rate of convergence in L 2 (dµ) is given by the spectral gap constant, which is the constant C SG in inequality (6). This rate is independent of the initial condition. For reaction-diffusion equations, Theorem 2.1 and Theorem 3.2 prove that the optimal rate of convergence strongly depends on the initial conditions, which is natural for a chemical reaction, the reaction will converges quickly if the species are more concentrated at the beginning.

Remark 3
The result obtained in Theorem 3.2 can be of course generalizes in the case when the generator L is just an operator satisfying a spectral gap inequality on the domain considered Ω. For example one can consider the case of a fractional Laplacian, a p-Laplacian,..., and others. (A problem may remain to prove the existence of a nonnegative solution in some of these cases.)

Remark 4
We recall also that while one can find in the literature some results on long time behaviour of solutions, (as for example nice bounds in [15]), generally it is a challenge to obtain the optimal bounds. Contrary to Theorem 3.2, the rate obtained in [6] for a general diffusion always depends on the spectral gap constant.

Study of the general case
Let us consider now the general case for q 1, Assume now that Ω ∈ R n is bounded domain.
Let consider non-negative initial conditions v 0 i on Ω, for all 1 ≤ i ≤ q. A weak solution of (9) on the time interval I is q measurable functions (v i ) 1≤i≤q such that for all t ∈ I, v i (·, t) ∈ L 1 (Ω), G(v 1 (·, t), · · · , v q (·, t)) ∈ L 1 (Ω), t 0 G(v 1 (·, s), · · · , v q (·, s)) L 1 ds < +∞ and moreover for all 1 ≤ i ≤ q, x ∈ R n and t > 0, which satisfies also for all x ∈ Ω, v i (0, x) = v 0 i (x). The result given here is a direct application of Rothe [14,Theorem 4]. The main interest of this result is to see why the solution remains non-negative and is defined on [0, ∞) which generally can be quite surprising for a fully nonlinear parabolic equation; (see also [15] for arguments based on a comparison principle).
Then, for any non-negative bounded and measurable initial condition v 0 i 1≤i≤q , there exists a non-negative weak solution of the system (9).
Proof ⊳ We will give here just a sketch of the proof and refer to [14] and references therein to get more informations.
Let us see how solutions are bounded and non-negative. The idea is to solve a different problem : As for equation (11), for all 1 ≤ i ≤ q we note by C i,1 (t, x) = and consider the PDE wherē and for x ∈ R, (x) + = max {x, 0}. By [14], (27) has an optimal and weak solution. Let us see why the solution is bounded and non-negative. Letv 0 1 be a non-negative bounded initial condition and let v 1 a weak solution of (27) (with the same definition as for equation (9), replacing G by F in (26)).
Then (v i ) 1≤i≤q is a solution of whereḠ Let us multiply (29) by −(v i ) − := min {v i , 0}. After integration, we obtain d dt where we put 0 at the position i. Since for all j, (v j ) + 0 then it is not difficult to see that in all cases k i (β i − α i )Ḡ(v 1 , · · · , 0, · · · ,v q ) 0. Which gives that d dt Since at time t = 0, ((v 0 i ) − ) 2 dµ = 0 then for all t 0,v i (t) 0 almost everywhere. Assume that β 1 − α 1 > 0. Then, since the solutions are non-negative, we get the following global estimate of the solution, If β 1 − α 1 > 0 does not hold, we use i 0 instead j 0 to get the same result with j 0 . The last estimate proves thatv 1 is bounded and then the solution is defined on [0, ∞).
The same method as above in (30) proves that for all 0 ≤ i ≤ q,v i 0. This implies thatḠ(v 1 , · · · ,v q ) = G(v 1 , · · · ,v q ) and then (v i ) 1≤i≤q is also a non-negative weak solution of (9) which finished the proof of the existence. ⊲ Remark 5 This restriction on parameters (α i , β i ) is natural in the context of a chemical reaction by the the principle of conservation of mass by Lavoisier.
The following corollary is a direct consequence of (11).
Let (v i ) 1≤i≤q be a solution of (9). Then for all 1 ≤ i ≤ q such that β i − α i > 0 one gets for all t 0 and x ∈ Ω : and for all 1 ≤ i ≤ q such that β i − α i < 0 : In particular when initial conditions are bounded, solutions of (5) are also bounded with an explicit upper bound.
Theorem 4.3 Assume that the semigroup (P t ) t 0 satisfies a spectral gap inequality (7) with respect to the invariant probability measure µ. Assume also that for all 1 ≤ i ≤ q, α i β i = 0.
Let v 0 i 1≤i≤q be a non-negative bounded initial condition. We assume furthermore that for all 1 ≤ i ≤ q, v 0 i dµ > 0. Let (s i ) 1≤i≤q be the steady state given by Lemma 1.2. Then for all 1 ≤ i ≤ q, one gets where a > 0 depends on α i , β i and C SG , and M, K > 0 depend on the initial conditions.
Proof ⊳ The idea is almost the same as in Theorem 3.2 except that we do not obtain the optimal rate. Assume that β 1 − α 1 > 0, the opposite case could be treated in the similar way. Equation (12) applied for j = 1 reads and functions C i,1 (t, x) are defined in (11). By the ergodicity properties of the semigroup, equation (8), implies that in L 2 (dµ), Denote by F ∞ (y) the limit of F (t, x, y) when t goes to infinity. Note that F ∞ does not depend on x ∈ Ω. Then, one gets since v 1 Lv 1 dµ ≤ 0, d dt Note that F ∞ is equal to the polynomial function F defined in (17) where in the definition of C j,i in (14), v i (0) is replaced by the mean value of initial conditions v 0 i dµ. Let us set and for its limit as t goes to ∞ As it was shown in the proof in Theorem 2.1, F ∞ (X) = (X−s 1 )Q(X) with Q(X) < 0 for all X ∈ (0, M ∞ 1 ). Now since for all 1 ≤ i ≤ q, α i β i = 0 then s 1 is a simple root of the polynomial function F ∞ . It implies that that Q(0) < 0 and Q(M ∞ 1 ) < 0. Then by continuity of Q there exist ǫ, η > 0 such that Q(X) ≤ −ǫ for all X ∈ [0, M ∞ 1 + η]. For the second term in (32), we get and then for some constant K depending on initial conditions Corollary 4.2 implies that M 1 (t, ·) v 1 and then Markov inequality gives (M 1 (t, ·)) . (33) Since for q measurable functions g i ∈ L 2 (µ) one has Var µ (g i ) , so the last term of (33) gives where K ′ is an another constant depending on initial conditions. Let us note F (t, x, y) = γ i=1 K t,i,x y i and F ∞ (y) = γ i=1 K ∞,i y i where we note The first term of (32) gives Let us consider one of them, with 1 ≤ i ≤ q, one has v 2i 1 (K t,i,x − K ∞,i ) 2 dµ ≤ v 4i 1 dµ (K t,i,x − K ∞,i ) 4 dµ.
There exist some sets Γ i,j and ∆ i and constants µ i,j,k ∈ R, γ i,j,k ∈ N such that K t,i,x = j∈∆ i k∈Γ i,j µ i,j,k (C i,1 (t, x)) γ i,j,k .
Then for some constant K i > 0, p i,j,k 2 and q i,j,k > 0, Since the initial conditions are bounded and p i,j,k 2, one gets for some another constant K and then spectral gap inequality gives for some K ′ , Thus we have proved that there exits γ i > 0 and R i > 0 depending on initial conditions and C SG such that All of these estimates give for some α > 0 depending on α i , β i and C SG and R > 0 depending on initial conditions the following bound If ϕ(t) = (a 1 − s 1 ) 2 dµ, the equation (32) becomes, by the previous equation and (34), ϕ ′ (t) ≤ −k 1 (β 1 − α 1 )ǫϕ(t) + Re −αt + K ′ e − 1 CSG t , which finishes the proof. ⊲

Remark 6
• One can generalize the last theorem in the following way without assuming that for all i, α i β i = 0. Let us consider i + and j − such that Assume only that α i + β i + = α j − β j − = 0. Then the computation for the species i + and j − are the same as in the proof of Theorem 4.3.
• On can also generalize in assuming that the initial conditions (v 0 i ) 1≤i≤q are in L q (dµ) for some q > 1 instead of L ∞ (dµ). In that case the proof will be more technical.