Close-to-equilibrium regularity for reaction–diffusion systems

The close-to-equilibrium regularity of solutions to a class of reaction–diffusion systems is investigated. The considered systems typically arise from chemical reaction networks and satisfy a complex balanced condition. Under some restrictions on spatial dimensions (d≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\le 4$$\end{document}) and order of nonlinearities (μ=1+4/d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu = 1 + 4/d $$\end{document}), we show that if the initial data are close to a complex balanced equilibrium in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm, then classical solutions are shown global and converging exponentially to equilibrium in L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document}-norm. Possible extensions to higher dimensions and order of nonlinearities are also discussed. The results of this paper improve the recent work (Cáceres and Cañizo in Nonlinear Anal TMA 159:62–84, 2017).


Introduction and main results
This paper deals with global existence of classical solutions to a class of reactiondiffusion systems with initial data being close to an equilibrium. The considered systems typically arise from chemical reaction networks. More precisely, we consider N chemical substances S 1 , . . . , S N , which react in the R reactions where the r -th reaction has the form y r,1 S 1 + · · · + y r,N S N k r − → y r,1 S 1 + · · · + y r,N S N (1.1) for r = 1, . . . , R. Here y r = (y r,1 , . . . , y r,N ), y r = (y r,1 , . . . , y r,N ) ∈ ({0} ∪ [1, ∞)) N are stoichiometric coefficients, and k r > 0 is the reaction rate constant. We will assume naturally that there exists at least one i ∈ {1, . . . , N } such that y r,i = y r,i . With a slight abuse of notation we will rewrite reaction (1.1) as y r k r − → y r in which y r and y r are called complexes with y r is the reactant and y r is the product of the corresponding reaction. Denote by C = {y r , y r : r = 1, . . . , R} the set of all complexes. Note that each complex y ∈ C can be a reactant, a product or both (in possibly different reactions).
To set up reaction-diffusion system modelling (1.1) we assume that the reactions take place in a bounded vessel ⊂ R d with smooth boundary ∂ (e.g. C 2+ for > 0). For i = 1, . . . , N , denote by u i (x, t) the concentration S i at position x ∈ and at time t ≥ 0. Assume moreover that each substance S i diffuses at a positive constant rate d i > 0. The corresponding reaction-diffusion system for u = (u 1 , . . . , u N ) reads as for all i = 1, . . . , N , where ν is the outward normal on ∂ and the initial data u i,0 are nonnegative. Here the homogeneous Neumann boundary condition means that the system is closed. The reactions f i (u) are derived from all chemical reactions (1.1) using the law of mass action, that is for i = 1, . . . , N , k r (y r,i − y r,i )u y r with u y r = It is frequent that system (1.2) possesses a set of conservation laws. Denote by m = codim{(y r − y r ) r =1,...,R }. Then if m > 0 we can find a matrix Q ∈ R m×N such that Q f (u) = 0 for all u ∈ R N . Here f (u) = ( f 1 (u), . . . , f N (u)). This in combination with the homogeneous Neumann boundary condition of (1.2) leads (formally) to m conservation laws of the form Q u(t) = Q u 0 for all t > 0.
where u = (u 1 , . . . , u N ) with u i = 1 | | u i (x)dx (see Sect. 2 for more details). Global existence of solutions is one of the most important questions in studying reaction-diffusion systems (if not any other PDE models). Concerning system (1.2), (1.3), due to the linear diffusion and polynomial-type nonlinearities, the local existence of solutions follows from standard theory of parabolic systems (see e.g. [1]). The global existence of solutions to (1.2), (1.3) on the other hand is a subtle task due to the lack of suitable a priori estimates (note that the maximum principle fails to apply to reactiondiffusion systems except very special cases). This issue has been extensively addressed in the literature, see e.g. [2,3,6,10,12,14,15,23,25] and the survey [21].
Let us briefly mention the recent advances concerning the global existence of (1.2), (1.3).
• In [15], by De Giorgi's method (1.2) was proved to have global classical solutions in one and two dimensions with the nonlinearities of order three and two, respectively (which means d = 1 and μ = 3 or d = 2 and μ = 2). These results have been re-proved (and slightly improved) in [25] using simpler arguments. It was also proved in [3], again by De Giorgi's method, that if the order of nonlinearities is strictly subquadratic (i.e. μ < 2), then classical solutions exist globally in any dimension. See also [14] for quadratic systems in heterostructure. • In [2] and later in [12] system (1.2) in higher dimensions and higher order of nonlinearities are shown to possess global classical solutions provided the diffusion coefficients d 1 , . . . , d N are "close to each other", i.e.
• Concerning weaker notions of solutions, it was shown in [6,19] that (1.2) with quadratic nonlinearities has global weak solutions in any dimensions, by using a duality method. An even weaker solution called "renormalised solution" was proved global in [13] for any dimension and any order of nonlinearities.
We remark that despite recent above-mentioned advances, the global existence of classical solutions to (1.2) is widely open. For example, it remains unknown whether the "four-species" system modelling in dimension three (and higher) possesses global classical solutions or not (without assuming the closeness of the diffusion coefficients).
Recently, Cáceres and Cañizo in [4] investigated a different regime of (1.2) socalled close-to-equilibrium regularity. The main idea is that when the solution stays in a small neighbourhood of an equilibrium, then it is possible that the linear part is dominating the behaviour of the system, and this helps consequently to obtain more regularity and hence the global existence of classical solutions. The authors proved in [4] that for dimension d ≤ 4, under the assumption that the nonlinearities are at most quadratic and (1.2) satisfies a detailed balanced condition (see Definition 2.1), if initial data are close to an equilibrium in L 2 -norm, then the classical solutions exist globally and converge exponentially to the equilibrium in L ∞ -norm.
The aim of this paper is to extend the results in [4] to systems with complex balance condition and with higher order of nonlinearities.
Our main result reads as follows.
the classical solution to (1.2) exists globally, and converges to u ∞ in L ∞ -norm expo- with C > 0 and λ > 0 are constants. Theorem 1.1 improves the results of [4] in the following senses: • Firstly, it allows to treat systems with higher-order nonlinearities, in particular in one and two dimensions. The main idea which leads to this improvement is to utilise the Gagliardo-Nirenberg inequality, especially in one and two dimensions, which helps to treat nonlinearities of order higher than two. Note that the idea was also used in the recent paper [25] to prove the global existence of classical solutions to reaction-diffusion systems in one and two dimensions (without the assumption of initial data being close to equilibrium). • Secondly, it applies to complex balanced systems which are more general than detailed balanced systems in [4] (see Definition 2.1). One of the main steps in proving Theorem 1.1 is obtaining a spectral gap for the linearised operator. This step was done in [4] thanks to a natural Lyapunov functional of (1.2) inherited from the detailed balanced condition. Extending this to systems with complex balance condition requires nontrivial calculations (see Lemma 3.3).
Last but not least, Theorem 1.1 successfully treats systems with boundary equilibria, i.e. equilibria belonging to ∂R N + (see Remark 2.3). It is remarked that for general complex balanced system (1.2) without boundary equilibria, it was proved in [5,11,18] that any renormalised solution converges exponentially in L 1 -norm to equilibrium. The situation is quite different with the occurrence of boundary equilibria, since in such cases the methods in these aforementioned works do not apply. It is however conjectured that for any complex balanced system the strictly positive equilibrium is the only attracting point. This is now named Global Attractor Conjecture and remains as one of the most important open questions in chemical reaction networks. Theorem (1.1) shows that despite boundary equilibria, any solution with initial data being close Vol. 18 (2018) Close-to-equilibrium regularity for reaction-diffusion systems 849 enough to any strictly positive complex balanced equilibrium in L 2 -norm, converges exponentially to that equilibrium. The results of this paper hence are also the extensions of local stability of complex balanced systems in ODE settings (see [17,24]). Let us briefly describe the method used in this paper. First, we consider the linearised system of (1.2) around a strictly positive complex balanced equilibrium. By utilising the complex balanced condition, we obtain a spectral gap for the linearised system. Due to the restrictions on dimensions and order of nonlinearities, it is then shown that the linear part is dominating in the behaviour of (1.2) in the close-to-equilibrium regime that means the L 2 -norm of solution converges exponentially to the equilibrium. After that, by utilising the Gagliardo-Nirenberg inequality and smoothing effect of the heat operator, we get the global existence of classical solution whose L ∞ -norm grows at most polynomially in time, which in a combination with the exponential convergence in L 2 -norm leads to the global convergence of solutions to the complex balanced equilibrium in L ∞ -norm.
The restriction d ≤ 4 and corresponding μ = 1 + 4 d are due to the fact that small L 2 -initial data lead only to control of L 2 -norm of solutions, which is only enough with the current techniques to control the higher norm under the mentioned assumptions on dimension d and the order of nonlinearity μ. The extension to arbitrary dimension seems to need more delicate analysis instead of using only the boundedness in L 2 (see e.g. [20] for a system which has solutions being bounded in L 2 but blowing up in L ∞ !). For extending this work to higher dimension d ≥ 5 and μ ≥ 3, it is naturally expected that the initial data should be close to equilibrium in L p -norm for some p > 2 depending on d and μ. Still the main obstacle is to obtain a global bound in time of L p -norm (or L p+ε -norm) of the solution. We discuss this extension in more details in Sect. 5.
Notations Throughout this paper, we denote by · p the norm in L p ( ) for 1 ≤ p ≤ ∞. For any other Banach space X then · X is used for its norm. The inner product in L 2 ( ) is written as ·, · . For any T > t 0 ≥ 0 we denote by Q t 0 ,T = × (t 0 , T ) and L p (Q t 0 ,T ) = L p (t 0 , T ; L p ( )). When t 0 = 0 we simply write Q T = Q 0,T .
The rest of the paper is organised as follows: in Sect. 2 we recall basic concepts concerning complex balanced chemical reaction networks. Section 3 deals with the linearised system around a strictly positive complex balanced equilibrium and proved that the linearised system has a spectral gap. Section 4 is devoted for the proof of Theorem 1.1. Finally, we discuss in the last section possible extensions to higher dimensions.

Complex balanced chemical reaction networks
In this section, we gather the basic concepts of chemical reaction networks with complex balanced condition. For more details, the interested reader is referred to [8,9] for the ODE setting and also [5,11] for the PDE setting. Recalling that we consider N chemical substances S 1 , S 2 , . . . , S N reacting in R reactions of the form J. Evol. Equ. y r,1 S 1 + · · · + y r,N S N k r − → y r,1 S 1 + · · · + y r,1 S N or shortly y r k r − → y r where y r , y r ∈ N N are stoichiometric coefficients. Under the assumptions of no out-flux and using the law of mass action, one arrives at reaction-diffusion system (1.2). Since the system under consideration is closed, there are often conservation of masses. Indeed, denote by W = (y r − y r ) r =1,...,R ∈ R N ×R the Wegscheider's matrix. Let m = codim(W ). In case m > 0, there exists a matrix Q ∈ R m×N whose rows are basis of ker(W ). Since we have, thanks to the homogeneous Neumann boundary condition, formally is the vector of averages. In the case m = 0, then system (1.2) has no conservation laws. Note that we can change signs of some rows of Q if necessary to have M positive (componentwise). Therefore, from now on, we always consider positive initial mass.
There are two important classes of (chemical) equilibrium for (1.2): detailed balanced equilibrium and complex balanced equilibrium. It is straightforward that u ∞ is a detailed balanced equilibrium ⇒ u ∞ is a complex balanced equilibrium, but the reverse is in general not true.
If system (1.2) admits a strictly positive complex balanced equilibrium, then it is called a complex balanced system. Note that condition (2.3) does not give a unique Vol. 18 (2018) Close-to-equilibrium regularity for reaction-diffusion systems 851 complex balanced equilibrium but a manifold of equilibria instead. The uniqueness of (positive) complex balanced equilibrium is nevertheless determined via the set of conservation laws (2.2).
PROPOSITION 2.2. (Uniqueness of positive equilibrium) [8,9] Assume that (1.2) is complex balanced. Then for each positive initial mass vector 0 = M ∈ R m + in (2.2), there exists a unique strictly positive complex balanced equilibrium u ∞ ∈ (0, ∞) N to (1.2). REMARK 2.3. The strictly positive complex balanced equilibrium is uniquely determined through the initial mass vector M rather the initial data. That means it is possible to have the same complex balanced equilibrium for different initial data as long as they have the same initial mass.
It is remarked that though (1.2) possesses a unique strictly positive complex balanced equilibrium, it may have possibly many so-called boundary equilibrium u * that means u * ∈ ∂R N + and u * satisfies condition (2.3). For complex balanced systems without boundary equilibria it was proved that any solution converges exponentially to the strictly positive complex balanced equilibrium, see [5,11]. If a complex balanced system possesses boundary equilibria, then the large time behaviour of solutions is in general unclear. The Global Attractor Conjecture asserts that despite boundary equilibria, the strictly positive one is still the only attracting point. This conjecture however still remains unsolved in full generality.
The class of complex balanced systems is an important class in chemical reaction network theory and thus has been studied extensively in the last decades. We emphasise that most of the existing works dealt with complex balanced systems in ODE settings, while the PDE settings are much less investigated. We refer the interested reader to recent works [5,11] for studies of complex balanced systems in PDE settings.

Spectral gap for the linearised operator
This section shows that if (1.2) is complex balanced, then its linearised system around any strictly positive complex balanced equilibrium converges exponentially to that equilibrium. Denote by u ∞ a strictly complex balanced equilibrium to (1.2), i.e. u ∞ ∈ (0, ∞) N and u ∞ satisfies (2.3). We first write down the linearised system of ( . . . , v N ) we obtain the linearised system around the equilibrium u ∞ as follow with LEMMA 3.1. Assume that u ∞ is a (strictly positive) complex balanced equilibrium. Then we have the following identity Proof. We proof the identity by comparing the coefficients and the coefficient in the right-hand side is This can be proved similarly to (3.7), so we omit it here.
REMARK 3.2. The proof of Lemma 3.1 has to utilise complex balanced condition (2.3) and hence is a nontrivial extension from the case of detailed balanced condition systems in [4].
The next crucial lemma is the main part of this section. It shows that the linearised operator has a spectral gap. This results mostly from identity (3.3) in Lemma 3.1 and conservation laws (2.2) of (1.2). Note that we have Q u(t) = M = Q u ∞ , thus Q v(t) = 0 for all t > 0.
recalling that ·, · is the inner product in L 2 ( ).
Proof. By using Lemma 3.1 and integration by parts we have

Firstly, by the Poincaré inequality
On the other hand, by Jensen's inequality we get, We will now prove that there exists β > 0 such that for all We prove (3.9) by a contradiction argument. Assume that Since both the nominator and denominator have the homogeneity of order two, we can assume w.l.o.g. that v = 1 with · denotes the Euclidean norm in R N . Thanks to this assumption, the denominator is bounded from above. If (3.10) holds, then there Recalling the Wegscheider's matrix W = (y r − y r ) r =1,...,R , then this implies that where D = diag(u 1,∞ , . . . , u N ,∞ ). Now in which (·, ·) denotes the inner product in R m ; hence, ξ = 0 since the matrix D 1/2 Q has full rank. But ξ = 0 leads to v = 0 which contradicts with v = 1. Therefore (3.9) is proved. Finally, a combination of (3.8) and (3.9) gives us the desired estimate of this lemma.
Since system (1.2)-(1.3) has diagonal linear diffusion matrix and the nonlinearities are polynomial, locally Lipschitz, the local existence of classical solution is well known from literature. By classical solution to (1.2)    This property has a simple physical interpretation: if a chemical substance S i has zero concentration, then it cannot be consumed in corresponding reaction (1.1).
Thanks to this corollary we can shift the initial time to t 0 > 0 to make use of the fact that u i ∈ C([t 0 , T max ); H 2 ( )) and still keep the closeness to equilibrium u ∞ when the initial data are close to u ∞ .
System (1.2) is rewritten as for a constant C > 0.
We write where we have used f i (u ∞ ) = R r =1 k r (y r,i − y r,i )u y r ∞ = 0 in the last step. By Taylor's expansion we have (v j + u j,∞ ) y r, j = u y r, j j∞ + y r, j u y r, j −1 j,∞ v j + 1 y r, j >1 O(|v j | y r, j ) Vol. 18 (2018) Close-to-equilibrium regularity for reaction-diffusion systems 857 where 1 y r, j >1 = 1 if y r, j > 1 and 1 y r, j >1 = 0 otherwise. Hence Inserting this into (4.3) we obtain desired estimate (4.2).
The local existence of classical solution to (4.1) follows from Theorem 4.1. Moreover, thanks to Corollary 4.3, there exists t 0 > 0 such that where β > 0 is a constant independent of t.
Proof. Multiplying (4.1) with v i /u i,∞ and summing over i = 1, . . . , N , it yields with the help of (4.2), Lemma 3.3 and Hölder's inequality where δ is defined in Lemma 4.4. Therefore for some β > 0 (4.5) Since d ≤ 4 and δ ∈ (0, 1], the Sobolev embedding yields H 1 ( ) → L 4 ( ) → L 2+2δ ( ). By using the interpolation inequality we get where we used v i 1+δ ≤ C v i 2 in the second inequality, since δ ∈ (0, 1]. For the term v i μ+1 μ+1 we apply the Gagliardo-Nirenberg inequality and recall that μ = d+4 Therefore, it follows from (4.5) that and consequently, by a Gronwall's lemma, Vol. 18 (2018) Close-to-equilibrium regularity for reaction-diffusion systems 859 Important notation: From now on we always denote by C T a positive constant which depends at most polynomially on T , that is there exists a polynomial P(x) such that C T ≤ P(T ) for all T > 0.
The following two lemmas are important in getting global classical solutions to (4.1). LEMMA 4.6. [2, Lemma 3.3] Let f ∈ L p (Q T ) with p ≥ 1 satisfy f L p (Q T ) ≤ C T and u be the solution to the heat equation with y 0 ∈ L ∞ ( ).
Proof. The proof of (i) and (ii) is given in [2,Lemma 3.3]. We give here the proof for (4.9). Denote by S(t) := e t (d ) the semigroup generated by the operator d with Neumann boundary condition. Then the solution to (4.6) is represented as By the classical estimate S(t)y 0 r ≤ C y 0 q 1 + t With p > (d + 2)/2 with have d/(2( p − 1)) < 1, and thus the integral on the righthand side converges, and has polynomial behaviour in t, and consequently in T . This completes the proof. We are now ready to give Proof of Theorem 1.1. Multiplying (4.1) with p|v i | p−2 v i (or more precisely a smoothed approximation of p|v i | p−2 v i and letting the smoothing go to zero) and then integrating over , we have (4.11) For the second term we have For the right-hand side of (4.11), since L i v is linear (see (3.2)) we estimate by Hölder's inequality For the term concerning the nonlinearities we use (4.2) and Hölder's inequality to obtain Therefore we obtain from summing (4.11) over i = 1, . . . , N that (4.12) Vol. 18 (2018) Close-to-equilibrium regularity for reaction-diffusion systems 861 Since is bounded and μ ≥ 2 we have

Then by adding
to both sides of (4.12) we obtain We now consider four cases d = 1, d = 2, d = 3 and d = 4 separately due to the different Sobolev embeddings in each case. The case d = 1, which implies μ = 5. Choosing p = 4 in (4.13) we have Applying the Gagliardo-Nirenberg inequality in one dimension (4.14) Thanks to Lemma 4.5 2 ≤ 2ε for all t 0 ≤ t < T max , and hence it follows from (4.14) and the one-dimensional embedding That means for all i = 1, . . . , N it holds v i ∈ L ∞ (t 0 , T ; L 4 ( )) ∩ L 4 (t 0 , T ; L ∞ ( )). It then follows from interpolation that v i ∈ L 8 (Q t 0 ,T ) for all i = 1, . . . , N . This implies that the right-hand side of the equation in (4.1) belongs to L 8/5 (Q t 0 ,T ), which with the help of Lemma 4.6(iii) leads to v i L ∞ (Q t 0 ,T ) ≤ C T . Since C T grows at most polynomially in T , this implies hence lim t→T max − u i (t) ∞ < +∞ and therefore T max = +∞.
To prove the convergence to equilibrium of u i to u i,∞ in L ∞ -norm, we first show Therefore, we can apply regularising effect of linear parabolic equation Therefore, by interpolation 15) for some 0 < β < β (1 − θ). This in combination with v i ∈ L ∞ ((0, t 0 ) × ) (implied from Theorem 4.1) completes the proof of Theorem 1.1 in one dimension.

By the interpolation inequality
which means v i ∈ L ∞ (t 0 , T ; L p ( )) for any p ∈ [1, ∞), and hence the right-hand side of (4.13) belongs to L ∞ (t 0 , T ; L p ( )) for any p ∈ [1, ∞). This is enough to apply Lemma 4.7 to obtain v i ∈ L ∞ (Q t 0 ,T ) with v i L ∞ (Q t 0 ,T ) ≤ C T and thus concludes Theorem 1.1 in three dimensions thanks to an argument similarly to (4.15). The case d = 4 and μ = 2. In this case we also choose p > 2 arbitrary. Using the Sobolev embedding H 1 ( ) → L 4 ( ) in four dimensions to (4.13) we have