Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction–diffusion systems

The convergence to equilibrium for renormalised solutions to nonlinear reaction–diffusion systems is studied. The considered reaction–diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, i.e. equilibrium states lying on the boundary of R+N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}_+^N$$\end{document}, then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite-dimensional inequality. This inequality is proven via a contradiction argument and thus not explicitly. An explicit method of proof, however, is provided for a specific application modelling a reversible enzyme reaction by exploiting the specific structure of the conservation laws. Our approach is also useful to study the trend to equilibrium for systems possessing boundary equilibria. More precisely, to show the convergence to equilibrium for systems with boundary equilibria, we establish a sufficient condition in terms of a modified finite-dimensional inequality along trajectories of the system. By assuming this condition, which roughly means that the system produces too much entropy to stay close to a boundary equilibrium for infinite time, the entropy method shows exponential convergence to equilibrium for renormalised solutions to complex balanced systems with boundary equilibria.


Introduction and main results
The large time behaviour of reaction-diffusion systems is a long standing and yet highly active topic in the analysis of partial differential equations. Classical methods include dynamical systems, invariant regions or linearisation methods. Recently, the so-called entropy method, a fully nonlinear approach, To specify a reaction-diffusion system modelling a chemical reaction network {S, C, R, K}, we assume the reactions to take place in a bounded vessel (or reactor) Ω ⊂ R n , where Ω is a bounded domain with Lipschitz boundary. We also assume (w.l.o.g. after a suitable rescaling of the space variable) that Ω has normalised volume, i.e.

|Ω| = 1.
We denote by c(x, t) = (c 1 (x, t), . . . , c N (x, t)) the vector of concentrations where c i (x, t) is the concentration of S i at time t > 0 and position x ∈ Ω. Each substance S i is assumed to diffuse in Ω with a strictly positive diffusion coefficient d i > 0. The corresponding reaction-diffusion system then reads as ∂ ∂t where k r > 0 denotes the reaction rate constant of the rth reaction. Finally, system (1) is subjected to non-negative initial data c 0 (x) ≥ 0 (by which we mean c 0 (x) := (c 1,0 (x), .., c N,0 (x)) and c i,0 (x) ≥ 0 for i = 1, .., N and x ∈ Ω), and homogeneous Neumann boundary conditions c(0, x) = c 0 (x) for x ∈ Ω, and ∇c · ν = 0 for (x, t) ∈ ∂Ω × R + , ZAMP Convergence to equilibrium for reaction-diffusion systems Page 3 of 30 54 where ν := ν(x) is the outward normal unit vector at point x ∈ ∂Ω. Many chemical reaction networks exhibit mass conservation laws. For system (1)-(3) we define the Wegscheider matrix W = [(y r − y r ) r=1,...,NR ] ∈ R NR×N (see, e.g. [61, page 859]) and denote by m = codim(W ). Then, if m > 0, there exists a (non-unique) matrix Q ∈ R m×N whose rows are formed by linear independent (left-zero) eigenvectors of W . It follows from (2) that R(c) ∈ range(W ) and thus Q R(c) = 0 for all c ∈ R N + .
Therefore, it follows from (1) that ∂ t (Q c) − QDΔc = Q R(c) = 0, and hence, due to the homogeneous Neumann boundary condition, the co-dimension of the Wegscheider's matrix W leads to m (linearly independent) mass conservation laws of the following form where c = (c 1 , . . . , c N ) and c i = Ω c i dx (after recalling that |Ω| = 1), and M is called an initial mass vector, which depends on the choice of Q. By changing the signs of some rows of Q if necessary, we can always consider (w.l.o.g.) a matrix Q such that the initial mass vector M is non-negative, i.e. M ∈ R m + . To state the main results of this paper, we need the following definitions concerning equilibria of chemical reaction networks.
where {s : y s = y} denotes the set of all reactions y s ks>0 − −−→ y s with fixed product complex y s = y ∈ C. • c ∞ is called a boundary detailed/complex balanced equilibrium (or shortly a boundary equilibrium) if c ∞ is a detailed/complex balanced equilibrium and c ∞ ∈ ∂R N + . • It follows directly from the above definitions that c ∞ is a detailed balanced equilibrium =⇒ c ∞ is a complex balanced equilibrium =⇒ c ∞ is an equilibrium, but the reverse is in general not true. A chemical reaction network is called complex balanced if for each strictly positive mass vector M ∈ R m >0 it possesses a strictly positive (i.e. not a boundary) complex balanced equilibrium.
The concept of detailed balance goes back as far as Boltzmann for modelling collisions in kinetic gas theory and for proving the H-theorem for Boltzmann's equation [4]. It was then applied to chemical kinetics by Wegscheider [59]. The complex balanced condition was also considered by Boltzmann [5] under the name semi-detailed balanced condition or cyclic balanced condition and was systematically used by Horn, Jackson and Feinberg in the seventies for chemical reaction network theory, see, e.g. [21,40,42]. It is well known that if a chemical reaction network [such as modelled by system (1)-(3)] has one complex balanced equilibrium, then all other possible equilibria (independently of the initial mass vector) are necessarily also complex balanced, see, e.g. [40,42]. Moreover, for every fixed positive initial mass vector M ∈ R m >0 , there exists a unique complex balanced equilibrium c ∞ ∈ R N >0 satisfying the mass conservation laws determined by the initial mass vector M ∈ R m >0 . Note that (possibly infinitely) many boundary equilibria may exist as well.
Throughout this paper, we will refer to this strictly positive equilibrium as the complex balanced equilibrium, while all other equilibria are simply called boundary equilibria. Moreover, we will consider positive initial mass vectors M ∈ R m >0 in order to ensure that any considered complex balanced network features a positive complex balanced equilibrium c ∞ ∈ R N >0 . Note that all our results hold equally true for non-negative initial mass vectors M ∈ R m + as long as there exists a unique positive complex balanced equilibrium c ∞ ∈ R N >0 , which will typically (but not always) be the case. This paper aims to prove exponential convergence to equilibrium of solutions to the nonlinear reactiondiffusion system (1)-(3) under the assumption the considered chemical reaction network is complex balanced.
The method of proof is the so-called entropy method. The main idea of the entropy method is to qualitatively exploit the decay of a suitable entropy (e.g. convex Lyapunov) functional E[f ] along a trajectory f of an evolution process: where D[f ] is called entropy production functional or also entropy dissipation functional in cases when E[f ] is physically an energy functional. The latter is the case for nonlinear complex balanced reactiondiffusion systems of the form (1)-(3), where the following logarithmic relative free energy functional constitutes a suitable entropy functional. Note that for nonlinear reaction-diffusion systems, the above logarithmic relative entropy is the only generally existing Lyapunov functional, while for linear complex balanced systems, other generalised relative entropy functionals do also exist, see, e.g. [25,48]. The following explicit form of the entropy dissipation functional D(c) associated to (7) along the flow of system (1)-(3) was derived in [15]: where Ψ : The entropy method applies to general evolution processes, which are well behaved in the sense that This condition holds true for the system (1)- (3), where D(c) = 0 is satisfied by all constant states which balance the reactions of the complex balanced network. Thus, provided no boundary equilibria exist, taking into account all conservation laws uniquely identifies the complex balanced equilibrium c ∞ . Given such a well-behaved evolution process, the entropy method aims to quantify the decay of the entropy functional E[f ] in terms of the relative entropy towards the equilibrium state. More precisely, the goal is an entropy-entropy production estimate (which is a functional inequality independent of the flow of the evolution process) of the form where Φ(x) ≥ 0 and Φ(x) = 0 ⇔ x = 0. More specifically for system (1)-(3), the first key result of this paper is to prove the following entropy-entropy dissipation estimate (or rather free energy-free energy dissipation estimate) for some constant λ > 0. Assuming that such a functional inequality is proven and that a suitable concept of solutions to systems (1)-(3) satisfies a weak entropy-entropy dissipation law [i.e. an integrated version of the formal relation (8)] of the form for almost all 0 ≤ s < t, then a Gronwall argument implies exponential convergence to equilibrium first in relative entropy, i.e.
and consequently in L 1 -norm, thanks to a Csiszár-Kullback-Pinsker type inequality, see Lemma 2.2.
In the first main results of this paper, we prove for general, complex balanced reaction-diffusion systems (1)-(3) without boundary equilibria, that any so-called renormalised solution (which is the only existing solution concept for such a general class of nonlinear reaction-diffusion systems, see Theorem 2.1) converges exponentially to the complex balanced equilibrium with a rate which can be explicitly estimated in terms of the systems' parameters and a constant obtained from a finite-dimensional inequality with mass conservation constraints. More precisely, our first main theorem reads as Here c c ∞ Finally, as a consequence of functional inequality (9), any renormalised solution c(x, t) to (1)-(3) (see Theorem 2.1) associated with initial data c 0 satisfying Q c 0 = M and E(c 0 |c ∞ ) < +∞, converges exponentially to c ∞ in L 1 -norm with the rate λ/2, that is for almost all t > 0, (12) where C CKP is the constant in a Csiszár-Kullback-Pinsker type inequality (see Lemma 2.2).

Remark 1.
Note that while an explicit bound for H 1 in (11) can certainly be obtained near the equilibrium c ∞ via Taylor expansion, such bounds far from equilibrium are highly nontrivial and an open problem due to the non-convexity of the involved nonlinear terms. Moreover, an additionally difficulty stems from the lack of a constructive approach to characterise and exploit the matrix Q.

Remark 2.
We also remark that the results of Theorem 1.1 can be directly extended to space-dependent diffusion coefficients with positive upper and lower bounds. In particular, for all i = 1, . . . , N, the operators Theorem 1.1 comprises in our opinion the most general equilibration result for complex balanced reaction-diffusion systems, which is currently feasible. It generalises previous results on the exponential convergence to equilibrium for reaction-diffusion systems, partially in terms of considering complex balanced instead of detailed balanced systems, partially in terms of applying to renormalised solutions rather than weak or classical solutions, and partially that the obtained convergence rate λ is explicitly stated in terms of the key constant H 1 .
At this point, we review some previous results concerning the large time behaviour of reaction-diffusion systems arising from chemical reaction networks: • The first results on the entropy methods for nonlinear reaction-diffusion systems trace back to works of Gröger, Glitzky and Hünlich [30][31][32][33][34], where the authors consider electro-chemical drift-diffusionrecombination models. However, the proof of associated entropy-entropy dissipation estimate was based on a contradiction argument in combination with a compactness method, thus provided only convergence to equilibrium in space dimension two and without explicit control of the rate of convergence. • The first quantitative results providing convergence to equilibrium with explicit constants were obtained in [11,12], which considered prototypical nonlinear reactions of the form 2S 1 S 2 , S 1 + S 2 S 3 or S 1 + S 2 S 3 + S 4 . Various generalisations were treated in [17,18,24]. Note that all these works consider special cases of (1)-(3).
• For detailed balanced systems without boundary equilibria, a first general approach to prove exponential convergence to equilibrium for (1)-(3) was presented in [45]. The inspired key idea of [45] was to prove an entropy-entropy dissipation estimate via a suitable convexification argument (of the non-convex sum of reaction terms in (8)). The disadvantage, however, is that except in special cases (e.g. 2S 1 S 2 ) the convexification argument seems not to allow for explicit estimates on the rate of convergence. The results of [45] were extended in [15] to complex balanced systems thanks to the derivation of the entropy dissipation (8).
• In a recent work [26], we proposed a constructive approach to show exponential convergence to equilibrium for general detailed balanced reaction-diffusion systems, which allows to obtain explicit bounds on the rates of convergence in contrast to the convexification argument of [45]. The applicability of the constructive approach was demonstrated for two typical example systems: i) a reversible reaction of arbitrary many chemical substances α 1 S 1 + · · · + α I S I β 1 B 1 + · · · + β J B J ( * ) and ii) a reversible enzyme reaction S 1 +S 2 S 3 S 4 +S 5 . This approach is also applicable to complex balanced systems as demonstrated in [15] for a cyclic reaction Also in [26], we provided an If-Theorem that for any detailed balanced systems, under the assumption of a finite-dimensional inequality [like (11)] and a technical non-degeneracy assumption on the entropy dissipation, then the solutions converge exponentially to the positive equilibrium with explicit rates. In this paper, we are able to remove these technical assumptions as well as generalise the result to complex balanced systems. It is also worth mentioning that the reversible reaction with arbitrary chemical substances ( * ) was also recently treated in the paper [52].
Altogether, these previous results prove either exponential convergence for general systems at the price of a lack of explicitness of convergence rates, or they showed explicit rates of convergence for some special classes of reaction-diffusion systems.
The results of Theorem 1.1 improve the previous results in several directions: i. We prove the functional inequality (9) explicitly up to the finite-dimensional inequality (11). More precisely, Theorem 1.1 states that the constant λ in (9) scales with the minimum of λ 1 (derived from the diffusion coefficients and the Logarithmic Sobolev Inequality) and the constant H 1 from (11) times the structural constant K 2 /K 1 with K 1 and K 2 given in (18) and (29). We note that the idea of proving (9) by using a finite-dimensional version was already considered in [45]. However, the approach therein lacks explicitness due to the use of the convexification argument. ii. We provide a general result of exponential convergence to equilibrium for complex balanced systems without boundary equilibria. In particular, the rate of convergence is explicitly controlled in terms the constant H 1 of the finite-dimensional inequality (11) (and other explicit parameters). It is emphasised that although the constant H 1 is not explicit in general, we believe it is possible to explicitly estimate H 1 in any concrete system once the mass conservation laws are explicitly known (see Sect. 2.2 for such a system arising from reversible enzyme reactions). iii. Another important advantage of Theorem 1.1 and our method of proof is its role in a potential strategy to consider systems with boundary equilibria. This leads to the second main result of this paper, which is discussed in the following paragraphs.
It is important to point out that the entropy-entropy dissipation inequality (9), and consequently the finite-dimensional inequality (11), cannot hold for general systems with boundary equilibria: If a solution trajectory of such a system should approach a boundary equilibrium, then the entropy dissipation D(c) tends to zero while the relative entropy to the complex balanced equilibrium E(c|c ∞ ) remains positive, see, e.g. [15,26] for the details. Consequently, an entropy-entropy dissipation estimate of the form (9) cannot hold.
This structural difficulty is already encountered in complex balanced reaction networks in the ODE setting, i.e. by considering the solution u(t), which satisfies the ODE system where R(u) is defined as (2) with u in place of c. There is an extensive literature concerning the large time asymptotics of complex balanced systems of the form (13). Indeed, it is proven that the unique strictly positive complex balanced equilibrium of an ODE reaction network is locally stable (cf. [42]). Moreover, it is conjectured that the positive complex balanced equilibrium is in fact globally stable, i.e. it is the unique global attractor for the dynamical system given by the ODE network (with the exception of initial data starting on ∂R N + ). This statement is usually called the Global Attractor Conjecture (GAC) and has remained one of the most important open problems in the theory of chemical reaction networks, see, e.g. [1,8,36,46] and the references therein. A recently proposed proof of this conjecture in the ODE setting is currently under verification [9].
For reaction-diffusion systems of the form (1)-(3), it was pointed out in [15,Remark 3.6] that if the boundary equilibria are unstable in the sense that solution trajectories cannot stay too close to those equilibria (in L 1 -norm distance) for too long, then the convergence to the complex balanced equilibrium follows via a contradiction argument. However, proving such an instability for boundary equilibria is usually a subtle issue, in particular in the PDE setting (1)-(3). In this paper, by using elements of the proof of Theorem 1.1, we establish a weaker condition entailing instability of boundary equilibria and convergence to the complex balanced equilibrium. More precisely, our condition is based on a quantitative estimate that solution trajectories do not converge to a boundary equilibrium "too fast" (if it should converge at all), see Theorem 1.2. To explain this approach further, we remark at first that our proof of deriving the entropy-entropy dissipation inequality (9) from the finite-dimensional inequality (11) is independent of the presence of boundary equilibria. Thus, instead of trying (or rather failing) to prove (11) as a pure functional inequality, we look for a generalisation with a time-dependent coefficient H 1 (t) along the trajectories of solutions, where H 1 (t) may tend to zero in case a solution trajectory would converge to a boundary equilibrium. Therefore, we look for a modified entropy-entropy dissipation inequality along solutions c(x, t) of (1)-(3) of the following form [which is no longer a pure functional inequality like (9)] with (18) and (29). Intuitively, the timedependent function λ(t) (which may decay to 0 as t → ∞) gives a lower bound for the entropy dissipation D(c(t)) or equivalently for the convergence of a trajectory towards a boundary equilibrium (where  (10) that By using this (so far non-exponential) convergence to the complex balanced equilibrium c ∞ , we obtain the L 1 -instability of the boundary equilibria. In return, this instability allows to show an entropy-entropy dissipation estimate of the form (9) on a reduced domain of states, which is strictly bounded away from the boundary equilibria. Thus, we recover exponential convergence to the complex balanced equilibrium c ∞ after a sufficiently large time. Our second main result reads as follows: Then, the renormalised solution c(x, t) converges exponentially to the positive complex balanced equilibrium c ∞ in the L 1 -norm with a rate, which can be explicitly computed in terms of the function H 1 , the domain Ω, the diffusion matrix D, the stoichiometric coefficients y ∈ C, the initial mass M, the complex balanced equilibrium c ∞ and the reaction rate constants k r .
The main progress of Theorem 1.2 is that the question of convergence to equilibrium for complex balanced reaction-diffusion systems with boundary equilibria is reduced to proving the finite-dimensional inequality (16). Moreover, if the function H 1 (t) is explicitly computable (i.e. for some specific systems), ZAMP Convergence to equilibrium for reaction-diffusion systems Page 9 of 30 54 then the rate of equilibration of the renormalised solution c(x, t) to (1)-(3) can also be computed explicitly. However, proving (16) for general systems with boundary equilibria remains a difficult problem since it requires suitable estimates on renormalised solutions, more precisely, on the behaviour of the L 1 -norm of renormalised solutions near the boundary ∂R N >0 , which is already a hard problem for ODE systems with boundary equilibria. Nevertheless, we will show in Sect. 3.2 how to apply Theorem 1.2 to specific systems.

Remark 3.
(Towards a Global Attractor Conjecture for reaction-diffusion systems). It is worthwhile to remark on the key assumption Secondly, note that since (16) constitutes a finite-dimensional inequality for the spatial averages c(t), one could conjecture to prove (16) by assuming the Global Attractor Conjecture for the corresponding ODE system (13), see [9] for a proof under review of the GAC for complex balanced ODE systems.
Indeed at a time t 1 > 0, consider the spatial averages c(t 1 ) as initial data u(t 1 ) = c(t 1 ) of (13). Then, the ODE Global Attractor Conjecture for (13) should imply (via a contradiction argument) the existence for u to be the solution of (13), since the ODE system (13) shares the same complex balanced equilibria as the PDE system. Moreover, formal estimates seem to suggest that it is possible to establish bounds on H 1 (t) via H ODE 1 (t) on a sufficiently small time interval (t 1 , t 2 ) provided that there is a good comparison between of the evolution of the ODE system u(t) and the evolution of the PDE system c(t) via its spatial averages c(t). Next at time t 2 , one restarts the ODE evolution (13) with a second set of initial data u(t 2 ) = c(t 2 ) and uses that also this ODE system satisfies the GAC and yields another function H ODE 2 (t) on a time interval (t 2 , t 3 ) and so forth. Assuming that the evolution of c(t) converges sufficiently fast to these family of related ODE solutions, is seems possible to prove a statement like ODE GAC implies GAC for the PDE systems.
However, the problem of deriving good convergence estimates on the difference between the ODE system (13) and the evolution of the spatial averages c(t) seems to be (at least) as hard as understanding directly the evolution of c(t). First, the non-convexity of R(u) prevents any direct comparison between Moreover, the evolution of the difference c(t) − u(t) is not non-negative and does not seem to feature an entropy functional. Hence, it seems that in order to derive estimates on the difference c(t) − u(t), one is brought back to understanding the equilibration of c(t), which is the problem to solve at first.
Notations: Throughout this paper, we will use the following set of convenient notations: • Capital letters for square roots of corresponding normal letter, that is • For two vectors y = (y 1 , . . . , y N ) and z = (z 1 , . . . , z N ) in R N with z i = 0 for all i = 1, . . . , N, we write • For a function f : R → R and a vector y ∈ R N , we denote by For example, Organisation of the paper: In Sect. 2, we present the proof of Theorem 1.1 and show its application to a reversible enzyme reaction. The proof of Theorem 1.2 and applications to networks with boundary equilibria will be presented in Sect. 3. Moreover, we assume that system (1) is complex balanced and thus possesses the entropy dissipation structure (7) and (8).

Proof of
Then, for any non-negative initial data c 0 : Ω → R N having finite relative entropy E(c 0 |c ∞ ) < +∞, there exists a global renormalised solution c( √ c i ∈ L 2 loc (R + ; H 1 (Ω)) and for any smooth function ξ : R N + → R with compactly supported derivative ∇ξ and every ψ ∈ for almost every T > 0.

Moreover, any renormalised solution c(x, t) to (1)-(3) satisfies the weak entropy-entropy dissipation law
for almost all t ≥ s ≥ 0, and the mass conservation laws, i.e. Q c(t) = Q c 0 for a.a. t > 0.

Preliminary estimates
We present in this part some useful preliminary estimates which are needed for the sequel proofs.
The following Csiszár-Kullback-Pinsker type inequality shows that convergence to equilibrium in relative entropy implies convergence to equilibrium in L 1 -norm. For its proof, even in more general settings, we refer the reader to, e.g. [3,11,12,15].

Lemma 2.3. (Additivity of relative entropy). For all measurable
The proof follows from direct computations; hence, we omit it here. Lemma 2.3 allows to prove the entropy-entropy dissipation inequality (9) by estimating E(c|c) and E(c|c ∞ ) separately. The first part E(c|c) can be easily controlled by D(c) thanks to the Logarithmic Sobolev inequality as in the following

the best constant in the Logarithmic Sobolev inequality.
Proof. By using the Logarithmic Sobolev inequality, see, e.g. [38] Ω Thanks to Lemmas 2.3 and 2.4, the remaining part of this section is dedicated to control the second part E(c|c ∞ ) of the relative entropy E(c|c ∞ ). Note that such a control has to quantify the system behaviour of the reacting concentrations c as well as the conservation laws Q c(t) = M = Q c 0 . Therefore, the control of E(c|c ∞ ) is much more challenging and technical.
We first show that E(c|c ∞ ) is bounded above by the right hand side of the finite-dimensional inequality (11).

Lemma 2.5. For any measurable
for an explicit constant K 1 > 0 depending on K and c ∞ [see (18)].
Proof. First, by using the elementary inequalities Next, we introduce the function which is continuous on [0, ∞) with the extensions Φ(0) = lim z→0 Φ(z) = 1 and Φ(1) = lim z→1 Φ(z) = 2, and monotone increasing. By using now the bound c i ≤ K, we can estimate By using Lemmas 2.3, 2.4 and 2.5, where the latter establishes the right hand side of (11), our proof of Theorem 1.1 still requires (i) to control the left hand side of the finite-dimensional inequality (11) in terms of D(c), and (ii) to prove (11). These will be done in Lemmas 2.6, 2.7 and Lemma 2.8, respectively.
As the first step, we observe that the entropy dissipation D(c) is a combination of the diffusion and reaction processes of the system (1), see (8). The reaction term seems hard to control due to the nonconvex nonlinearities (with arbitrary high-order polynomials) and the very low regularity of renormalised solutions. In fact, we will not show that the entropy dissipation D(c) is bounded for renormalised solutions, but only that it constitutes an upper bound even while potentially unbounded. We will prove in the following lemma that, with the help of the diffusion terms, the reaction part is bounded below by "reactions of averaged concentrations". Herein, we recall the convention of square roots C i = √ c i and for an explicit constant K 3 > 0 [see (25)].
Proof. By using the identity ∇ √ c i = ∇c i /(2 √ c i ) and the elementary inequality Ψ(x, y) = x log(x/y) − To prove (19), we use similar arguments to [15,22,26]. Fix a constant L > 0. The proof uses a domain decomposition corresponding to the deviation of C i around the averages C i , i.e. by denoting . . , N} and S c = Ω\S. We will see that on S the reaction part is crucial, while the diffusion part is sufficient on S c .
On the set S, by using the bounds (17)], as well as Taylor expansion of terms like

by Jensen's inequality and
with On the other hand, on S c , by using the lower bound |δ i | ≥ L for some 1 ≤ i ≤ N , the upper bound  (24) where |y| = N i=1 y i for any y ∈ C. Note that all the constants β 1 , β 2 , β 3 , β 4 defined in (22) and (24) are independent of S and S c . By combining (21) and (23), we can estimate with any γ ∈ (0, 1) and the Poincaré inequality where β 1 , β 2 , β 3 and β 4 are defined in (22) and (24).

Remark 4.
The constant L in Lemma 2.6 can be chosen arbitrary. One certainly can choose L in order to optimise (i.e. maximise) the constant K 3 in (25). This may help to improve the rate of convergence. However, due to the multistage proof of the entropy-entropy dissipation inequality, the estimated rates are not optimal. Now we are able to control the left hand side of (11) by D(c).

Lemma 2.7.
For any measurable c : Ω → R N + satisfying E(c|c ∞ ) ≤ K and Q c = M, there exists an explicit constant K 2 > 0 [see (29)] such that Remark 5. Lemma 2.7 is a crucial step in proving Theorem 1.1. As mentioned in the introduction, we are here able to remove a technical assumption on cases when the L 1 -norm of the concentrations approaches the boundary ∂R N + , which was needed in [26,Theorem 1.4]. The key observation is the remainder estimate (27). Note that this idea was also used in [39,Lemma A.5] for energy-reactiondiffusion systems.

Proof. By denoting
which leads to and consequently Note that Q(C i ) ≥ 0 and that Similarly to (21), we use Taylor expansion and ansatz (26) to get in which the constant H i is estimated as Hence, it follows from Lemma 2.6 and (28) and the Poincaré inequality that for any θ ∈ (0, 1) by choosing θ = min 1 The last step now is to prove the finite-dimensional inequality (11). Let us recall that until this point, we have not used the fact that the system under consideration possesses no boundary equilibria. This fact turns out to be very useful when dealing with systems having boundary equilibria (see Sect. 3).

Remark 6.
We remark here that while all the constants in previous lemmas can be explicitly estimated, the constant H 1 in (11) (as established in the this lemma) is in general not explicit since the proof utilises a contradiction argument. However, we believe that for any concrete system, where the conservation laws are explicitly known, H 1 can be computed explicitly via only elementary calculations (see Sect. 2.2). Estimating H 1 for general systems is a subtle issue since the structure of conservation laws, which is crucial for an explicit estimate, is unclear in general and remains thus an open problem.
Proof. Observe that the right hand side of (11) equals zero if and only if c = c ∞ . Therefore, we first prove that the left hand side of (11) can only be zero when c ≡ c ∞ . Indeed, assuming that the left hand side of (11) is zero, then we have Thus, for any y ∈ C we have That means that c is a complex balanced equilibrium. Since the chemical reaction network has no other complex balanced equilibrium than c ∞ , we obtain the desired claim that c ≡ c ∞ . Now define where Σ K,M = {c ∈ [0, K] N : Q c = M} and K is the constant in Lemma 2.5, i.e. in the estimate c i ≤ K for all i = 1, . . . , N, which is implied from E(c|c ∞ ) < +∞. Since either sides of (11) equal zero if and only if c = c ∞ and the fact that the denominator of the above fraction is bounded above, we deduce that H 1 can possibly only be zero if and only if Ξ = 0 where Ξ is defined by It is obvious that Ξ ≥ 0. Now assume by contradiction that Ξ = 0. By linearising both the nominator and denominator around c ∞ , and by setting σ = c − c ∞ and η = σ c∞ = σ1 c1,∞ , . . . , σN cN,∞ , we obtain ZAMP Convergence to equilibrium for reaction-diffusion systems Page 17 of 30 54 Note that η is the same vector for all r = 1, . . . , N R in the numerator. Note moreover that both numerator and denominator are of homogeneity two. We can thus rescale and normalise η w.l.o.g. and only consider η on the unit ball, that is |η| = 1. Moreover, Ξ = 0 if and only if the nominator is zero: which is only possible when η ∈ ker(W ), where we recall that W is the Wegscheider matrix Recall that m = codim(W ) = dim(ker(W )) is the number of conservation laws. If m = 0 and the system (1)-(2) does not have a conservation law and equivalently ker(W ) = {0}, then it follows that η = 0, which is a contradiction to |η| = 1. If m > 0, then by using η ∈ ker(W ) and the fact that the rows of Q form a basis of ker(W ), it follows that η = Q γ with some γ ∈ R m . Since Q σ = Q (c−c ∞ ) = M−M = 0, we obtain (by recalling η = σ c∞ ) which implies γ = 0 since Q has full rank. Thus, η = 0 which again contradicts with |η| = 1.
In conclusion, we have proved that Ξ > 0, which implies the existence of a constant H 1 > 0 and hence completes the proof.
We can now begin the Proof of Theorem 1.1. From Lemmas 2.5, 2.7 and 2.8 we get which in combination with Lemma 2.4 leads to the desired estimate (9). Next, thanks to Theorem 2.1, any renormalised solution satisfies the conservation laws and the weak entropy-entropy dissipation law (10). Hence, we can apply a variant version of Gronwall's inequality (see, e.g. [18] or [60]) to get the exponential decay for almost all t > 0. This convergence in a combination with the Csiszár-Kullback-Pinsker type inequality in Lemma 2.2 leads to the claimed convergence to equilibrium (12).

Applications to reversible enzyme reactions
Theorem 1.1 shows that any renormalised solution of complex balanced reaction-diffusion systems without boundary equilibria converges exponentially to equilibrium with a constant and a rate, which can be explicitly estimated up to the finite-dimensional inequality (11). Proving (11) with an explicit constant H 1 seems to be a difficult task in full generality due to the non-convex nonlinear reaction terms and the non-explicit structure of conservation laws, i.e. due to the fact that we have no explicit structure of the constraints imposed by the matrix Q .
In this section, however, we will show that for a specific system, where the conservation laws are explicitly known, we can prove inequality (11) with an explicit constant H 1 by using elementary estimates. Hence, we obtain convergence to equilibrium for (1) with explicit bounds for the convergence rates and constants in a highly relevant model of enzyme reactions.
For notational convenience, we use a change of variables and rewrite the finite-dimensional inequality (11) in a form, which is easier to handle in the specific case at hand. By denoting for μ i ∈ [−1, +∞) and μ = (μ 1 , . . . , μ N ), inequality (11) rewrites as follow: where μ satisfies the following constraint inherited from the mass conservation laws and where we recall the convention c ∞ (μ 2 + 2μ) = (c i,∞ (μ 2 i + 2μ i )) i=1,...,N . We apply our approach to a reversible variant of the famous Michaelis-Menten enzyme reaction For the sake of clarity, we shall assume k 1 = k 2 = k 3 = k 4 = 1, but we emphasise that the subsequent analysis can be equally carried out for general k i > 0, i = 1, . . . , 4, without additional technical difficulties. The corresponding mass action reaction-diffusion system reads as with homogeneous Neumann boundary conditions ∇c i · ν = 0 on ∂Ω and non-negative initial data . . , 4, in which Ω is a bounded domain with sufficiently smooth boundary (e.g. ∂Ω ∈ C 2+ with > 0) and normalised volume |Ω| = 1. The large time behaviour of various reaction-diffusion models of reversible enzyme kinetics has also been recently studied in, e.g. [16,Section 8] or [26]. It is easy to check that there are two linear independent mass conservation laws for (34) and that the matrix Q can be chosen as It is straightforward to check that this equilibrium is a complex balanced equilibrium (and even a detailed balanced equilibrium) and that system (34) possesses no boundary equilibria. The existence of global renormalised solution to (34) follows immediately from Theorem 2.1. Moreover, since the nonlinearities in (34) are quadratic, it is well known (see, e.g. [49,50]) that (34) has a global weak solution. Moreover, thanks to the special structure of (34), we show in the following that these weak solutions are in fact strong solutions and grow at most polynomially in time.

Remark 7.
Note that the L ∞ -bounds of Proposition 2.9 are sufficient to apply standard parabolic bootstrap arguments and show that c(x, t) is indeed a classical solution (or even smooth if ∂Ω is smooth) and thus unique.
Proof of Proposition 2.9. The proof relies on duality estimates and comparison principle arguments for scalar parabolic equations, which exploit the special structure of (34). In this proof we always denote by C T a general constant depending polynomially on T > 0. First, it follows from (34) that By a classical duality estimate (see, e.g. [51]) and by denoting L 2 (Q T ) = L 2 (0, T ; L 2 (Ω)), we have Moreover, (34) is quasi-positive in the sense of, e.g. [50] and thus preserves non-negativity of weak solutions c 1 , . . . , c 4 from non-negative initial data. This implies Next, we apply Therefore, it follows from (37) and c 3 ∈ L 2 (Q T ) in particular that and On the other hand, by another duality estimate (see, e.g. [53,Lemma 33.3]), it follows from that the regularity and the polynomial dependence of C T on T are transferred from c 1 to c 3 , which implies that By repeating this procedure, we obtain after finitely many steps that c 3 L q (QT ) ≤ C T with q ≥ N +2 2 . Then, (37) implies for all r ∈ [1, ∞), which yields in return c 3 L r (QT ) ≤ C T for all r ∈ [1, ∞). Hence, after one application of the classical smoothing effect of heat operator, the proof of the Proposition is completed.  (36): where the constant C and the rate λ can be explicitly estimated in terms of Ω, the equilibrium c ∞ , initial masses M 13 and M 234 and the diffusion coefficients. Moreover, if the initial data c 0 belong to L ∞ (Ω) 4 , then (34) has a unique global classical solution, which converges exponentially to c ∞ in any L p -norm for for all t ≥ 0, with explicit constant C and rate λ .
Proof. Since the system satisfies the complex balanced condition and possesses no boundary equilibria, Theorem 1.1 implies immediately that any renormalised solution converges exponentially to the equilibrium defined in (36). It remains to bound of convergence rate explicitly. Thanks to Theorem 1.1 and (32) that means to compute explicitly a constant H enzyme 1 > 0 in the finite-dimensional inequality satisfying the following constraints, which are equivalent to the mass conservation laws (35): From (40a) and by observing that (μ 1 + 2), (μ 3 + 2) ≥ 1, it follows directly that μ 1 and μ 3 must have different signs, which leads to the following two cases: (i) Consider μ 1 ≥ 0 and μ 3 ≤ 0: First, we have From (40b) and μ 3 ≤ 0, we infer that at least either μ 2 ≥ 0 or μ 4 ≥ 0, which leads to two subcases: (ia) Suppose μ 2 ≥ 0. Then, since μ 1 μ 3 ≤ 0. Similarly, and hence completes the proof of explicit convergence of renormalised solutions to equilibrium for (34). Concerning the L p convergence (38), we interpolate Proposition 2.9 and have for θ ∈ (0, 1) and p = 1 for a constant C and any λ < λθ 2 .

Proof of Theorem 1.2
Proof of Theorem 1.2. We already mentioned in the proof of Lemma 2.8, that the validity of Lemmas 2.3, 2.4, 2.5 and 2.7 is independent of the presence or absence of boundary equilibria. We recall here the key estimates of the Lemmas for the sake of readability: the additivity of the relative entropy allows to control the term E(c(t)|c(t)) via the Logarithmic Sobolev inequality in terms of the entropy dissipation, i.e.

E(c(t)|c ∞ ) = E(c(t)|c(t)) + E(c(t)|c ∞ ) and λ 1 E(c(t)|c(t)) ≤ D(c(t)).
The second term E(c(t)|c ∞ ) satisfies the upper bound while the entropy dissipation obeys the lower bound These two estimates are connected by assumption (16), and we obtain all together

D(c(t)) ≥ λ(t)E(c(t)|c ∞ )
with λ(t) = 1 2 min λ 1 ; K 2 H 1 (t) K 1 . Thus, the trajectory c(t) converges to c ∞ in relative entropy and, consequently, in L 1 -norm due to the Csiszár-Kullback-Pinsker type inequality in Lemma 2.2. Therefore, after some finite time T > 0, the solution trajectory will always stay outside of any small enough neighbourhood of all boundary equilibria. It then follows from [15,Remark 3.6] that the solution converges exponentially to the positive complex balanced equilibrium.

Application to a specific system possessing boundary equilibria
In order to show convergence to equilibrium for renormalised solutions c(x, t) of complex balanced reaction-diffusion systems with boundary equilibria, we have to verify (16) as stated in Theorem 1.2.
Similarly to Sect. 2.2, it will be convenient to change variables in the finite-dimensional inequality (16). By setting c i (t) = c i,∞ (1 + μ i (t)) 2 , for i = 1, . . . , N, inequality (16) becomes where μ(t) = (μ 1 (t), . . . , μ N (t)) and the function H 1 (t) is required to satisfy Proving (54) for general complex balanced systems would yield a proof of the Global Attractor Conjecture (GAC), which is a very interesting, yet challenging open problem. Our aim in this section is to study a typical class of complex balanced systems with boundary equilibria, in which proving (54) for renormalised solutions is a possible approach to answer the GAC in the associated PDE setting. More precisely, we consider here the reaction-diffusion system modelling the following reaction network with arbitrary α ≥ 1 and k 1 , k 2 , k 3 > 0. The special case α = 1 was investigated in [15]. Here, we study the entire range α ≥ 1 in order to show the robustness of our arguments. The above network (C) is considered in a bounded domain Ω ⊂ R n with smooth boundary ∂Ω (e.g. C 2+ for any > 0) and normalised volume, i.e. |Ω| = 1. The corresponding mass action reaction-diffusion system reads as x ∈ Ω, t > 0, ∂ t c 2 − d 2 Δc 2 = k 1 αc 1 + k 2 c α 2 c 3 − k 3 (α + 1)c α+1