Boundedness for reaction-diffusion systems with Lyapunov functions and intermediate sum conditions

We study the uniform boundedness of solutions to reaction-diffusion systems possessing a Lyapunov-like function and satisfying an {\it intermediate sum condition}. This significantly generalizes the mass dissipation condition in the literature and thus allows the nonlinearities to have arbitrary polynomial growth. We show that two dimensional reaction-diffusion systems, with quadratic intermediate sum conditions, have global solutions which are bounded uniformly in time. In higher dimension, bounded solutions are obtained under the condition that the diffusion coefficients are {\it quasi-uniform}, i.e. they are close to each other. Applications include boundedness of solutions to chemical reaction networks with diffusion.


Introduction and Main Results
Let Ω ⊂ R n , n ≥ 1 be a bounded domain with smooth boundary ∂Ω. In this paper, we study the global existence of classical solution to the following semilinear reaction-diffusion system where u = (u 1 , . . . , u m ), d i > 0 are diffusion coefficients, ν is the unit outward normal vector on ∂Ω, u i,0 are bounded, non-negative initial data, and the nonlinearities satisfy the following conditions: (A1) f i : R m → R is locally Lipschitz. where |u| = m j=1 |u j |. Let us comment on the assumptions (A1)-(A5). The local Lipschitz continuity in (A1) is common in studying reaction-diffusion systems as it allows us to foremost obtain local existence. Assumption (A2) is relevant for systems arising from biology or chemistry, as it preserves the non-negativity of solutions, meaning that if the initial concentration (or density, population) is non-negative, then the solution is also non-negative as long as it exists. This assumption has a simple physical interpretation: if a concentration is zero, then it cannot be consumed in the reaction. The condition (A3) generalizes several common assumptions in the literature. Namely, • mass conservation: when h i (u i ) = α i u i , α i > 0, K = 0 and equality holds, i.e. m i=1 α i f i (u) = 0; (2) • mass dissipation: when h i (u i ) = α i u i , α i > 0, K = 0, i.e.
• mass control: • entropy dissipation: when h i (u i ) = u i log u i − u i + 1 and K = 0, i.e.
The condition (A5) indicates that the nonlinearities do not grow faster than polynomial of order µ. We remark that our results in this paper will not have any restriction on the growth µ.
The assumption (A4) is called an intermediate sum condition, in the sense that only one of the nonlinearties is assumed to be one-side bounded by a polynomial of order r, while for the others we just need a good "cancellation". In practice, r can be much smaller than µ. To better explain that, we consider the following example which models the reversible reaction pS 1 + qS 2 ⇆ ℓS 3 , for p, q ≥ 1, By choosing the matrix we can see that (A4) satisfies with r = ℓ, while obviously µ = max{p + q; ℓ} which can be much bigger than r when p and q are large.
With (A1), (A2) and non-negative, bounded initial data, the local existence of a non-negative strong solution to (1) on a maximal interval (0, T max ) is classical (see e.g. [Ama85]). The global existence of that local solution, on the other hand, is a challenging issue. Assuming for instance the mass dissipation (3), one can easily show, using the homogeneous Neumann boundary conditions, that d dt Taking into account the non-negativity of the solution, this means that the solution is bounded in L ∞ (0, T ; L 1 (Ω)) uniformly in 0 < T < T max . This is far from enough to obtain global existence. In fact, it was shown by a famous counterexample in [PS00] that there exist systems satisfying (A1)-(A2) and (3) whose solutions blow up in L ∞ -norm in finite time. The global existence of a strong solution to (1) with more structural conditions on the nonlinearities is therefore an interesting problem, and has been studied extensively in the literature. Let us first review some existing results in the literature, and from that highlight the novelty of our paper. Note that in the following, (A1) and (A2) are always assumed.
Therein, the solution is also proved to be bounded uniformly in time. • In the close-to-equilibrium regime, it was shown in [CC17] that if the initial data is close to equilibrium (in L 2 -norm) then one can obtain global existence with µ = 2 up to dimension n = 4. This was later improved in [Tan18a] with the condition µ = 1 + 4/n. • The case µ = 2 in higher dimensions had remained an open question until recently when it was settled in three different works [CGV19,Sou18,FMT19a]. The first work assumed (2) and (5), while the second relaxed (2) to (3) but still needs (5). The most general result is contained in the last work where the authors only assumed (4). The uniform-in-time bound has been shown recently in [FMT19b]. It's worth to mention the almost unnoticed work [Kan90] where it considered Ω = R n . • Concerning a work assuming the general assumption (A3), we refer to the works [Mor90] and [MW04], in which the later showed the global existence with (4) and quadratic intermediate sum condition but without uniform-in-time bounds. • There are also number of works dealing with weaker notions of solutions. For instance it was shown in [Pie03] under (4) that if the nonlinearities belong to L 1 (Ω × (0, T )) for any T , then one gets global weak solutions. By using a duality method, [DFPV07] showed with (4) global existence of weak solutions in all dimensions assuming µ = 2. An even weaker notion called renormalized solutions was shown global in [Fis15] assuming (5). The interested reader is referred to the extensive survey [Pie10] for more details. It can be seen from the aforementioned works that the global existence of strong solutions with super-quadratic nonlinearities, i.e. µ > 2, is much less studied except in [CDF14] and [FLS16], where the diffusion coefficients are assumed to be quasi-uniform. That is the main motivation of our paper. More precisely, by utilizing the intermediate sum condition (A3), we allow arbitrary polynomial growth of the nonlinearities.
The first main result of this paper is the following.
• The assumption (A4) can be generalized to , provided there exists M > 0 so that g i (z) ≤ M(h i (z) + 1) for z sufficiently large, for all i = 1, . . . , m.
• The condition r = 2 can be slightly improved to r = 2 + ε for sufficiently small ε > 0.
• In fact, to obtain the uniform bound (7) as long as a uniform bound of the functions This bound follows straightforwardly from (A3) with K = 0 (see Lemma 3.2).
In Theorem 1.1, observe that the growth µ in (A5) can be arbitrary. Which means that the nonlinearities can have arbitrarily high polynomial growth, as long as their intermediate sums (in the sense of (A4)) are bounded from the right by a quadratic polynomial. For instance, the system (6) with ℓ = 2 satisfies the assumptions of Theorem 1.1 and it therefore possesses a global strong solution, bounded uniformly in time, in two dimensions.
It's worthwhile to mention that the uniform-in-time bound of the solution is also of importance. This issue was usually left untouched in the literature, except for e.g. [PSY19] or our recent work [CMT19]. In some situations arising from chemical reaction networks with boundary equilibria, the uniform L ∞ -bound plays a very important role. See more details in Examples 4.1 and 4.2.
Let us sketch the proof of Theorem 1.1, which can be roughly divided into several steps.
Step 4: We then construct a sequence {p N } N ≥0 with p N +1 = 2p N 4−p N for p N < 4 and p N +1 = ∞ for p N ≥ 4, such that u i ∈ L q (Ω × (0, T )) for all q < p N .
Step 6: To show the uniform boundedness in time, we repeat the above arguments but now in each cylinder Ω × (τ, τ + 1), τ ∈ N, and eventually show that u i L ∞ (Ω×(τ,τ +1)) ≤ C where the constant C is independent of τ .
When n ≥ 3 and r > 2, the global existence of (1) is largely open, except for the case when h i (u i ) = u i and µ = 2, see e.g. [CGV19,FMT19a,Sou18]. In the next main result, we show that, if the diffusion coefficients are quasi-uniform, meaning that if they are close to each other, then one can obtain global strong solutions to (1). If where p ′ > 1 such that and C A+B 2 ,p ′ is the constant defined in Lemma 2.1 (a), then (1) has a unique global nonnegative, bounded strong solution for any nonnegative, bounded initial data u i0 .
Moreover, if K = 0 in (A3), then the solution is bounded uniformly in time, i.e. It is again remarked that the condition on the closeness of the diffusion coefficients (8) depends only on r in (A4) and independent of the polynomial growth µ in (A5). This greatly improves related results in e.g. [CDF14] or [FLS16].
Remark 1.2. In the recent work [CMT19], global existence and uniform-in-time bounds were also obtained for (1) under mass dissipation condition (3) and quasi-uniform diffusion coefficients. We remark that the latter condition imposed in [CMT19] depends on the growth µ of the nonlinearities, and therefore is much less general than (8).
The rest of this paper is organized as follows: For convenience, we split the proofs of Theorems 1.1 and 1.2 into two parts: global existence and uniform-in-time bounds, which will be proved in Sections 2 and 3 respectively. To highlight the relevance of our results, we give some applications of our results to models arising from chemical reactions in Section 4.
Notation. In this paper, we will use the following notation.
• We denote by Q τ,T = Ω × (τ, T ). When τ = 0, we write simply Q T = Q 0,T . For any 1 ≤ p ≤ ∞, L p (Q τ,T ) stands for L p (τ, T ; L p (Ω)). • For any T > 0, we write C T for a generic constant depending continuously on T , which can be different from line to line or even in the same line. More importantly, Here we write C q,T to indicate that the constant depends on q, and might blow up to infinity when q → p.
Theorem 2.1 (Local existence of strong solutions). Assuming (A1)-(A2). For any bounded, nonnegative initial data, (1) possesses a local nonnegative strong solution on a maximal x ∈ Ω. (10) We have the following: (a) There exists a constant C m,p depending on m and p, but independent of τ, T such that Proof. At first glance, (10) looks like a backward heat equation. However, with the change of variable T + τ − t, we get the usual forward heat equation Therefore, part (a) can be found in [Lam87] . Also from [Lam87] we have Part (b) then follows from this and the embeddings of space-time spaces, see e.g. [LSU88].
Lemma 2.2. Suppose the conditions of Theorem 1.2 are satisfied. If for some p ′ > 1. Then we have where p is the Hölder conjugate exponent of p ′ , i.e. 1 p + 1 p ′ = 1. Consequently, Proof. We compute Then we have and ∇Z · ν = 0 on ∂Ω × (0, T ). By integrating (15) on Ω we get thus, thanks to Gronwall's lemma and the nonnegativity of Z, Define m = A+B 2 and let φ be the nonnegative solution to for all s < (n+2)p ′ n+2−2p ′ . By using integration by parts and (15), we compute for some constant s < (n+2)p ′ n+2−2p ′ and s ′ = s s−1 . We will estimate the last three terms on the right-hand side of (19) separately. First, using (18), Therefore, by interpolation inequality for some α ∈ (0, 1), where we have used (16) at the last step. Using this, we can estimate the second term on the right-hand side of (19) as Inserting (20), (21) and (22) into (19), we obtain for all 0 ≤ θ ∈ L p ′ (Q T ). By duality, , and thanks to the assumption (12), Since α ∈ (0, 1), we can use Young's inequality to finally obtain (14), we use the convexity of h i and the fact that for some constant C > 0. It then follows from (13) that Lemma 2.3. Suppose the conditions of Theorem 1.2 are satisfied. There exists p > 2 such that Proof. We only need to show that (12) Proof. To prove Proposition 2.1, we construct a sequence {p N } N ≥0 with for all N ≥ 0. We will do that in three steps.
Step 1. We claim that Let q < p N +1 arbitrary. We choose q such that q < q < p N +1 , and define s := (n + 2)r q n + 2 + 2 q , which is equivalent to q = (n + 2) s r n + 2 − 2 s r .
From q < p N +1 and (25), it follows that s < p N . Therefore, From the intermediate sum condition (A4), we have From (28), the right hand side of (29) satisfies Using comparison principle and maximal regularity in Lemma 2.1 we have Recalling that q < q, it follows Since q < p N +1 arbitrary, we obtain the claim (26).
Step 2. We claim that, for any k = 2, . . . , m if sup Fix q < p N +1 . Let 0 ≤ θ ∈ L q ′ (Q T ) with q ′ = q q−1 , and let φ be the nonnegative solution to From Lemma 2.1, and From the intermediate sums condition (A4) we have Using integration by parts we have We will estimate these four terms separately. Firstly, by Hölder's inequality Thanks to (32) it follows that and therefore For (II), we use (31), (32) and Hölder's inequality to estimate The term (III) is easily estimated that Since q < p N +1 and q ′ = q q−1 , (n + 2)q ′ n + 2 − 2q ′ = (n + 2)q nq − (n + 2) > (n + 2)p N +1 np N +1 − (n + 2) = p N p N − r where we used (25) at the last step. From this we can choose s such that and it follows from (33) that We now estimate the (IV ) as, with s ′ = s s−1 , where at the last step we used rs ′ = rs s−1 < p N , thanks to (39), and thus u i L rs ′ (Q T ) ≤ C T for all i = 1, . . . , m.

From this and
Step 1. and Step 2. we get for all i = 1, . . . , m. To finally obtain the L ∞ -bound of the solution, we use (41) and the assumption (A5) that all f i (u) are one-side bounded by a polynomial. Indeed, from (A5) and the comparison principle we have x ∈ Ω.
Thanks to (41), we can apply Lemma 2.1 to obtain v i L ∞ (Q T ) ≤ C T and consequently which finishes the proof of Lemma 2.1.
We are now ready to prove the global existence parts in Theorems 1.2 and 1.1.
Proof of global existence in Theorem 1.1. The global existence of strong solutions follows directly from Lemma 2.3 and Proposition 2.1, by checking the blow-up criterion (9). To show uniqueness, we assume that u (1) = (u i ) are two strong solutions on (0, T ) for arbitrary T > 0, with the same initial data. By multiplying (1) (2) i L ∞ (Q T ) ≤ C T , and f i are locally Lipschitz continuous (see (A1)), Inserting this into (42) then summing up with respect to i = 1, . . . , m, we obtain in particular Using Gronwall's inequality and the fact that u (1) (0) = u (2) (0) we finally have i (t) L 2 (Ω) = 0 for all t ∈ (0, T ) and all i = 1, . . . , m, which proves the uniqueness of strong solutions.

Proof of global existence in Theorem 1.2. From the assumption (8) and Lemma 2.2 we have
2 (r − 1). Thanks to Proposition 2.1 we get the bound (24), and therefore the global existence of strong solution. The uniqueness is proved similarly to Theorem 1.1 and therefore omitted.

Uniform-in-time bounds
From Section 2, we know that the strong solution exists globally. This section aims to show that if K = 0 in (A3), then the solution is bounded uniformly in time. The idea is to study (1) on each cylinder Ω × (τ, τ + 1), τ ≥ 0, and show that the solution is bounded on this cylinder independently of τ .
In this section, we always denote by C a generic constant which is independent of τ , which can be different from line to line, or even in the same line.
We will need the following elementary lemma, whose proof is straightforward.
From Lemma 2.1, and From (34), Therefore, by using integration by parts, we have τ Ω φu j dxdt τ Ω For (I) we use Hölder's inequality and (59) to estimate where we used u j L q (Q τ,τ +2 ) ≤ C for all j = 1, . . . , k − 1. For (II) we first choose γ = where we used u j L q (Q τ,τ +2 ) ≤ C for all j = 1, . . . , k − 1. Note that Thus, we can use the interpolation inequality where we used u k (t) L 1 (Ω) ≤ C, which is implied from (45) and (23). We can then estimate (II) further as The term (III) can be easily estimated as Since q < p N +1 , we choose s such that (39) holds, and thus, thanks to (60), We now can estimate the last term (IV ) as where we used at the last step τ Ω for all 0 ≤ θ ∈ L q ′ (Q τ,τ +2 ). Therefore, by duality, Then, using ϕ τ ≡ 1 on [τ + 1, τ + 2] we get , which implies, thanks to Young's inequality and α ∈ (0, 1), for all τ such that (66) holds. Thanks to Lemma 3.1, this inequality in fact holds for all τ ∈ N, which proves the claim (58) since q < p N +1 arbitrary.
Thanks to the claims (56) and (58), we can proceed exactly as the last part of the proof of Proposition 2.1 to see that there exists N 0 such that Thus, for all τ ≥ 0, Using this and the fact that with, thanks to (A5), Due to (67), we have Therefore, the comparison principle and Lemma 2.1 gives ϕ τ u i L ∞ (Q τ,τ +2 ) ≤ C and consequently u i L ∞ (Q τ,τ +1 ) ≤ C, which finishes the proof of Proposition 3.1.
Proof of uniform bounds in Theorem 1.1. From [CDF14, Lemma 3.2], there exists p ′ < 2 such that B − A 2 C A+B 2 ,p ′ < 1. Therefore, it follows from Lemma 3.2 that u i L p (Q τ,τ +1 ) ≤ C for all i = 1, . . . , m and all τ ≥ 0. The uniform boundedness in time the follows directly from Proposition 3.1.
Proof of uniform bounds in Theorem 1.2. From the assumption (8) and Lemma 3.2 we have u i L p (Q τ,τ +1 ) ≤ C for all i = 1, . . . , m, τ ≥ 0, for p > n+2 2 (r − 1). Thanks to Proposition 3.1 we get the bound (55), and therefore the uniform-in-time bound of the solution.

Applications
In this section, we provide five examples to demonstrate applications of our main results to reaction-diffusion systems arising from chemical reaction networks. We emphasize that the uniform bounds in time that we get play an important role in studying the large time behavior of these systems.
In [CJPT18], the author considered the reversible reaction A + kB The idea of [CJPT18] is to first show that the solution to (68) is bounded uniformly in time, then an entropy-entropy dissipation technique leads to the exponential decay to equilibrium. While the latter part is dimensional independent, they explicitly need n = 1 to get the uniform-in-time bound of the solution. Therefore, they can only obtain the decay to equilibrium in one dimension. Looking at system (68), we can easily see that it satisfies our assumptions (A1)-(A5) with h 1 (a) = a, h 2 (b) = b and h 3 (c) = 2c, K = 0, r = 2, µ = k + 1 and the matrix It is remarked also that in this case, the assumption (A3) becomes the usual mass dissipation condition (2). Our Theorem 1.1 is therefore applicable and consequently, we obtain the convergence result in [CJPT18] also in two dimensions.
Theorem 4.1 (Exponential convergence to equilibrium for (68) in two dimensions). Let n = 2 and assume that a 0 , b 0 , c 0 ∈ L ∞ (Ω) are nonnegative with for some δ > 0. Then (68) has a unique global strong solution, which is bounded uniformly in time, i.e. sup t≥0 Moreover, as t → ∞, the solution converges exponentially fast to the unique positive equilibrium in L p -norm for any 1 ≤ p < ∞, i.e.
Proof. The global existence and uniform boundedness follow directly from Theorem 1.1. The exponential equilibration in L 1 -norm was shown in [CJPT18], i.e.
for some C, λ 1 > 0. The convergence in L p -norm (70) then follows from the uniform L ∞bound (69) and the interpolation inequality Example 4.2. We consider in this example the following reactions where k, ℓ ≥ 2. For simplicity, we assume that all the reaction rate constants are equal to one. Using the law of mass action, we obtain the following system with homogeneous Neumann boundary conditions ∇u i · ν = ∇v j · ν = 0, for (x, t) ∈ ∂Ω × (0, ∞) and initial data for all i = 1, . . . , m and j = 1, . . . , h. Here we denote by u i and v j the concentration densities of A i and B j , respectively. System (72) was left in [CJPT18] as an open problem, both for the global existence as well as the large time behavior.
For simplicity, we write u = (u 1 , . . . , u m , u m+1 , v 1 , . . . , v h ) and f i (u), i = 1, . . . , m + 1, the nonlinearity for the equation of u i , and f m+1+j (u), j = 1, . . . , h, the nonlinearity for the equation of v j . By straightforward calculations, one can show that there do not exist positive constants α 1 , . . . , α m+1+h such that Therefore, the existing works using the mass control condition (4) are not applicable. Luckily, due to the reversibility nature of (71), we have a dissipative structure of the entropy. More precisely, denote by h i (z) = z log z − z + 1, we can easily see that and thus (A3) is satisfied. With direct computations, we see that (A1), (A2), (A4) and (A5) are also satisfied with r = 2, µ = max{m + k, ℓ + h} and the matrix where − → 1 ⊤ is a column of size h with all elements are one, and I m−1 and I h are identities matrices of size (m − 1) and h, respectively. Therefore, we can apply Theorem 1.1 to get global, uniform-in-time bounded solutions to (72) in two dimensions.
Theorem 4.2. Let n = 2. Then for any bounded, nonnegative initial data u i0 , v j0 ∈ L ∞ (Ω), (72) has a unique, bounded strong solution with We believe this uniform bound will help to obtain convergence to equilibrium for (72). We leave this for future investigation. In the last example, we consider the following reaction network 2S 1 + S 3 2S 2 whose ODE setting was studied in [And11]. Denote u i (x, t) as the concentration densities of S i , we obtain the following system thanks to the law of mass action with homogeneous Neumann boundary conditions ∇u i ·ν = 0 on ∂Ω×(0, T ) and nonnegative initial data u i (x, 0) = u i,0 (x) on Ω. We can again easily check that the assumptions (A1), Similarly to Example 4.2, there do not exist positive constants α i , i = 1, . . . , 3 such that (4) holds. To justify (A3), we use again an entropic dissipative structure of (73). More precisely, by using h i (u i ) = u i log u i − u i + 1 we can show with direct computations that where Φ(x, y) = x log(x/y) − x + y. Therefore, we have the following Example 4.4.
The last example deals with the system (6) in the Introduction. This type of system was considered in [Laa11] in which the following cases were treated: (1) p + q < ℓ; (2) (d 1 = d 3 or d 2 = d 3 ) and for any (p, q, ℓ); (3) d 1 = d 2 and for any (p, q, ℓ) such that p + q = ℓ; (4) 1 < ℓ < n+6 n+2 and for any (p, q). The results of our paper allow us to deal with the case ℓ = 2, n = 2 and arbitrary (p, q) and arbitrary diffusion coefficients d i > 0, which is not included in [Laa11].

(76)
Proof. The global existence and uniform-in-time bound of strong solutions follow immediately from Theorem 1.2 since r = 1. The exponential convergence to equilibrium in L p -norm (76) can be obtained similarly to example 4.1 thanks to the following L 1 -convergence, which was shown in [FT18],