Solution of a class of reaction-diffusion systems via logarithmic Sobolev inequality

We study global existence, uniqueness and positivity of weak solutions of a class of reaction-diffusion systems of chemical kinetics type, under the assumptions of logarithmic Sobolev inequality and appropriate exponential integrability of the initial data.


Introduction
A mixture one gets after esterification of one mole of ethyl alcohol by one mole of ethanoic acid contains products (ethyl acetate and water), but also reactants. This is an example of a double displacement reaction (see [34]). We consider here chemical reactions between q 2 species A i , i = 1, . . . , q, as follows where α i , β i ∈ N. We assume that for any 1 ≤ i ≤ q, α i − β i = 0 which corresponds to the case of a reaction without a catalyst.
If u = (u 1 , · · · , u q ) denotes the concentration of the species A i then the law of action mass proposed by Waage and Guldberg in 1864 (see again [34]) implies that the concentrations are solutions of the system, for all i ∈ {1, · · · , q}, where k, l > 0 are the rate constants of the two reactions.
When considering substances distributed in space, the concentrations change not only under the influence of the chemical reactions but also due to the diffusion of the species over the space, one gets the following kinetic model for a chemical reaction-diffusion equation where for all i = 1, . . . , q, L i is an operator which modelizes how the substance diffuses.
We will assume that L i = C i L for some C i 0 and some reference operator L. Moreover, by a change of variables, one can assume that there exist (a posteriori two) constants λ i > 0 such that the system of reaction-diffusion is given by where u(t, x) = (u 1 (t, x), · · · , u q (t, x)) with t 0 and x belongs to the underlying space.
The two-by-two system, one of the simplest non trivial example, describes the chemical reaction and the system of equations can by formulated as follow where λ,λ > 0 and u i denotes the concentration of the specie A i and v i the concentration of the specie B i for i = 1, 2. To make things even simpler, we will assume later that λ =λ.
More general reaction-diffusion systems, of the following form with prescribed boundary conditions, were intensively studied in the past. Here, Ω is a (possibly unbounded sufficiently smooth) domain of R n , u takes values in R q , C is a usually diagonal q × q matrix which can be degenerate, and F (t, x, ·) is a vector field on R q . Depending on specific choices for C and F (t, x, ·), such systems can present various behaviours with respect to global existence and asymptotic behaviour of the solution. Paragraph 15.4 in [43] is a nice introduction with a lot of classical references.
In the above setting, local existence follows from general textbooks on parabolic type partial differential equations (see [23], [30], or for fully general boundary value problems [1]).
Global existence question (or how to prevent blow up) gave rise to extensive efforts and to different methods adapted to specific cases (see [2], especially remark 5.4. a), [40], [37] and references therein). Most of these methods consist in deducing L ∞ bounds on the maximal solution from bounds in weaker norms.
The survey [37] provides a lot of references, positive and negative results, together with a description of open problems. Its first observation is that, for numerous reaction-diffusion systems of interest in applications, the nonlinearity satisfies two general conditions which ensure respectively positivity and a control of the mass (i.e. the L 1 norm) of a solution. M. Pierre investigates how these L 1 estimates (as well as L 1 bounds on the nonlinearity) help to provide global existence.
Further works provide results on asymptotic behaviour. Spectral gap, logarithmic Sobolev inequality and entropy methods are often used to quantify exponential convergence of the solution of an equation to equilibrium, and in the context of reaction-diffusion equations (mostly of type (2)) they were used to study the convergence (to constant steady states) in [16,17,15,25]. Geometric characteristics and approximations of global and exponential attractors of general reaction-diffusion systems may be found in [49,20,50] (and references therein) in terms of precise estimates of their Kolmogorov ε-entropy. In these papers, C is of positive symmetric part and the nonlinearity must satisfy some moderate growth bound involving the dimension n to ensure global existence. Other cross-diffusion systems are studied by entropy methods in [11].
One way or another, local or global existence results in the above setting rely on regularity theory for the heat semigroup, the maximum principle, and Sobolev inequality through one of its consequences, Gagliardo-Nirenberg inequalities or ultracontractivity of the semigroup (as well as Moser estimates). (Note nevertheless that an approach based on a nonlinear Trotter product formula is proposed in [43], but seems to impose some kind of uniform continuity of the semigroup).
The aim of this article is to prove global existence of a non-negative solution of the reaction-diffusion system (2) with unbounded initial data in a setting where Sobolev inequality possibly does not hold, as e.g. in infinite dimensions or when underlying measure does not satisfy polynomial growth condition. We restrict ourselves to some nonlinearities for which in a finite dimensional setting, L ∞ bounds of the solution (and so global existence) come for free, [37]. Nevertheless, Sobolev inequality has to be replaced by the weaker logarithmic Sobolev inequality (or other coercive inequalities which survive the infinite dimensional limit; see [8], [6], [39], [7]).
The celebrated paper [26] of L.Gross established equivalence of logarithmic Sobolev inequality and hypercontractivity of the semigroup. No compactness embeddings hold in this context.
The paper is organized as follows. In the next section we describe the framework and the main result of the paper: in the two-by-two case, assuming 1) that C 1 = C 3 and C 2 = C 4 , 2) that the linear diffusion term satisfies logarithmic Sobolev inequality and 3) that the initial datum f is nonnegative and satisfies some exponential integrability properties (made more precise later), then there exists a unique weak solution of the system of reaction-diffusion equation (3) which is moreover nonnegative. Section 3 presents the iterative procedure we follow to approximate weak solutions of our reaction-diffusion type problem. This is based on some cornestone linear problem which is stated there. The two following sections are devoted to the details of the proof : section 4 to the convergence of the iterative procedure to the unique nonnegative weak solution of the nonlinear Cauchy problem, whereas section 5 focuses on the cornerstone linear problem.
In Section 6 we extend our result to the general case of system (2), and present how operators C i L can be modified. (To give a comprehensive proof we focus in the rest of the paper on the two-by-two case which already contains non trivial difficulty).
We recall or detail tools used in the proof in three appendices: the entropic inequality, basics on Orlicz spaces, and finally some further topics on Markov semigroups and Orlicz spaces.

Framework and main result An abstract Reaction-Diffusion equation
In the following we will consider an underlying Polish space M equipped with a probability measure µ. Let L be a (linear) densely defined selfadjoint Markov operator on L 2 (µ) ≡ L 2 (M, µ), that is the infinitesimal generator of a C 0 Markov semigroup (P t ) t 0 symmetric with respect to µ. It is well known that under these assumptions there exist a kernel p t (x, dy) on (M, B M ), that is a measurable family of probability measures such that, for any t 0, any f ∈ L 1 (µ), and for µ almost every x ∈ M, Let us consider the following equation where, in the two-by-two case, • the unknown u(t, x) = (u 1 (t, x), u 2 (t, x), u 3 (t, x), u 4 (t, x)) is a function from [0, ∞) × M to R 4 ; and L u = (Lu 1 , Lu 2 , Lu 3 , Lu 4 ) is defined componentwise.
• the nonlinearity G is quadratic: • C is a diagonal matrix of the following form where we assume that C 1 = C 3 and C 2 = C 4 .
(This condition is weakened in section 6).

Dirichlet form and logarithmic Sobolev inequality
Let (E, D) be the Dirichlet form associated to (L, µ) (see [14], [24], [33], [10]; or [22] for a minimal introduction). For any u ∈ D(L) (the domain of L) and v ∈ D (the domain of the Dirichlet form), one has We will denote E(u) ≡ E(u, u), for any u ∈ D. Recall that D is a real Hilbert space with associated norm u D = (µ(u 2 ) + E(u)) 1/2 .
We will assume that the Dirichlet structure (E, µ) satisfies logarithmic Sobolev inequality with constant C LS ∈ (0, ∞), that is for any u ∈ D.

Classical function spaces
Let I = [0, T ]. For any Banach space (X, · X ), we shall denote by C(I, X) the Banach space of continuous functions from I to X equipped with the supremum norm sup t∈I u(t) X .
Let also L 2 (I, X) be the space of (a.e. classes of) Bochner measurable functions from I to X such that T 0 u(t) 2 X ds < ∞. As for vector valued functions, let L 2 (I, All these are Banach spaces. We'll furthermore consider the space L ∞ (I, X) of Bochner measurable X-valued functions on I such that ess sup 0≤t≤T u(t) X < +∞.
The reader may refer to [41] for Bochner measurability, Bochner integration and other Banach space integration topics.

Bochner measurability in an Orlicz space
Let Φ : R → R + given by Φ(x) = exp(|x|) − 1 and Φ α (x) = Φ(|x| α ), α 1. These are Young functions and the Orlicz space associated to Φ α is denoted by L Φα (µ). This is the space of measurable functions f such that for some γ > 0 (or functions whose α power is exponentially integrable). An important closed subspace E Φα (µ) of L Φα (µ) consists of those functions such that (7) holds for any γ > 0. This is the closure of the space of simple functions (finitely valued measurable functions) in L Φα (µ).
A stricking property of Markov semigroups is that C 0 property in L 2 (µ) implies C 0 property in any L p (µ); 1 ≤ p < +∞ (see [14]). We will need the following weakened result in the context of Orlicz spaces.

First regularity result and weak solutions
The following lemma exhibits the main role the entropic inequality (see appendix A) and the logarithmic Sobolev inequality play to deal with the nonlinearity we consider. In short, the multiplication operator by a function in L Φ 2 (µ) is a bounded operator, mapping the domain of the Dirichlet form D to L 2 (µ).
The reader may note that we will use this lemma to define properly a weak solution of the nonlinear problem below.
As for continuity of (8), what precedes shows that, for any t a.e., Integrating w.r.t. t on [0, T ], one gets the result. Finally, continuity of the trilinear mapping follows by Cauchy-Schwarz inequality in L 2 . ⊲ Weak solutions. Let T > 0. We say that a function is a weak solution of (RDP) on [0, T ] provided, for any φ ∈ C ∞ ([0, T ], D 4 ) and any t ∈ [0, T ), When this is satisfied for any T > 0, we'll say that u is a weak solution on [0, ∞).
In section 6, we will state the extension of this theorem to the general problem (2).
In short, to prove this theorem, we linearize the system of equations by means of an approximation sequence ( u (n) ) n . We show recursively that u (n) (t) is nonnegative, belongs to L ∞ ([0, T ], L Φ 2 (µ)) so that lemma 2.1 guarantees u (n+1) is well defined. This propagation is made precise in a lemma studying the linear cornerstone problem which underlies the recursive approach.
We will first focus our efforts to prove convergence of the approximation sequence in the space 4 . Afterwards, we detail a way to study the cornerstone existence lemma.
Remark 2 We will exhibit in appendix B a sufficient condition to ensure that f ∈ E Φα (µ), namely, that there exist β > α and γ > 0 such that µ(e γ|f | β ) < +∞. In particular, it implies that, provided f 0 belongs to (E Φ 2 (µ)) 4 , one may choosẽ γ > 0 large enough such that 4 min(C 1 , C 2 ) λC LS <γ, and which will be useful in the proof of existence and uniqueness.

Iterative procedure
Let us define the approximation sequence ( u (n) ) n∈N in the following way. (First of all, note the parenthesis in u (n) has nothing to do with differentiation, and has been introduced to distinguish the index from powers).
Lemma 3.1 (Cornerstone existence lemma) Let L be a Markov generator satisfying logarithmic Sobolev inequality with constant C LS ∈ (0, ∞). Let T > 0 and has a unique weak solution on [0, T ]. Futhermore, provided f , A and B are assumed nonnegative, then the solution u is nonnegative.
Recursive equivalence of both systems (RDP n ) and (14) may be seen as follows. Starting from (RDP n ), one easily gets and writting u 1 (t) (and similarly for the other coordinates) gives the announced decoupled system. Conversely, deducing from the decoupled system that u

and similarly) follows by induction and uniqueness in lemma 3.1.
To be able to define u (n+1) , and hence prove that the iterative sequence is well defined, it remains to check that u , for all i = 1, . . . , 4. This is based on results stated in appendix C and can be shown as follows.
We may focus on u , the contraction property of the semigroup stated in lemma C.1 implies that, for any γ > 0, for any t a.e., So that, in particular, for any t ∈ [0, T ], u (n) . Following lemma C.2, what remains to be checked is Bochner measurability of the mapping t → u From the corresponding weak formulation (weak-CS) applied to a constant (in

Proof of Theorem 2.2
Convergence of the approximation procedure (RDP n ) From now on, we'll use the notation The main idea is to show that, with for some κ > 0 (specified later), the supremum sup t∈[0,T ] Σ n (t) goes to 0 exponentially fast as n goes to ∞ provided T > 0 is small enough. From lemma 3.1, u (n) is defined recursively as a weak solution of the cornerstone linear problem. To make things simpler at this stage, we here perform formal computations to get a priori estimates. Getting the estimates rigorously makes use of Steklov regularisation, which we will illustrate in the proof of the next proposition. * see [46], [19] or [41] for a proof, [21], appendix E.5, theorem 7, for a statement.
Estimate of the L 2 -norm derivative We will focus on the L 2 -norm of u and after natural multilinear handlings, Since u (n−1) is nonnegative, using the quadratic inequality ab ≤ a 2 /2 + b 2 /2, one gets All the similar terms are then estimated thanks to the relative entropy inequality (36). For instance, The logarithmic Sobolev inequality (6) and bound (15) give Using the same arguments for all the terms leads to Completely similar terms are obtained when dealing with the L 2 -norms of the other components. After summation in all the components, one gets which is positive thanks to the assumed constraint (13).
Use the absolute continuity and the positivity of Reminding the definition (16) of Σ n and that u (n) (0) = u (n−1) (0), after integration over [0, t], t ∈ [0, T ], we obtain the following main estimate

Gronwall argument and convergence
Gronwall type arguments applied to the estimate (18) give for any t ∈ [0, T ], It follows that sup Performing a similar estimate for 1 It follows that Hence, ( u (n) ) n∈N is a Cauchy sequence: it converges to some function

Global existence of the weak solution
Let T > 0 fixed as in the previous computation. We will first prove that the limit We now show we can pass to the limit n → ∞ in all the terms. (Dealing with other coordinates u (n) i is similar by symmetry). Thanks to the continuity of the scalar Moreover, as the convergence also holds in C([0, T ], L 2 (µ)), then lim n→∞ µ(u Dealing with the convergence of the term t 0 µ(φu belongs to L ∞ ([0, T ], E Φ 2 ) which will follow indirectly. The details are as follows.
By lemma 2.1, τ n ≡ τ . Let us show that this sequence is Cauchy, and so converges to, say, τ (12) But by (15), and again entropic and log-Sobolev inequalities, . This goes to 0 as n, m → +∞.
From lemma C.2, what remains to do is to prove E Φ 2 Bochner measurability. Let us summarize what we obtained. One has after taking limit n → +∞, In particular, choosing φ(t) = ϕ ∈ D, the mapping t ∈ [0, T ] → µ(ϕu (∞) 1 (t)) ∈ R is continuous. Then, arguments detailed on page 9 ensure that u Letting separately n (resp. m) to +∞ in (20) shows that All this implies that  Assume the diffusion coefficients C 1 and C 2 , the logarithmic Sobolev constant C LS of L, the reaction rate λ and the exponential integrability parameter γ are linked by the constraint Then a weak solution of the Reaction-Diffusion problem (RDP) with initial datum f is unique.
We recall basics on Steklov calculus (see [30] for instance), i.e. appropriate time regularization to deal with weak solutions. For any Banach space X, and any v ∈ L 2 ([0, T ], X), the Steklov average, defined by ) in X, and a h (v)(t) converges to v(t) in X, for every t ∈ [0, T ]. The space X will be here L 2 (µ) or D depending on the context. Proof of Proposition 4.1 ⊳ Let u and v be two weak solutions of (RDP) with the same initial datum f 0. Let M ∈ (0, ∞) such that, ∀i = 1, . . . , 4, µ(e γ|u i (t)| ) ≤ M , t a.e., (and similarly for v). Let w ≡ u − v and a h (w i )(t) the Steklov average of the i component of w as defined before. Integrating 1 We then use the definition of a weak solution with the constant test function And the other term is bounded from above by We can deal with the four similar terms by the same way: let us focus on the first one. One first uses Once gain, entropic inequality followed by logarithmic Sobolev inequality give Note that, up to a constant, the first term of the RHS is the Steklov average of the , so that, as h → 0, it converges in L 1 ([0, T ]) to that function. Going back to (21) and performing all the explained bounds before passing to the limit h → 0, one gets the estimate (note that w i (0) = 0) Summing over all i's, one gets provided the announced constraint 4 λC LS γ ≤ min(C 1 , C 2 ) is satisfied. Uniqueness follows by Gronwall arguments. ⊲ 5 Proof of Lemma 3.1 Our approach to study the cornerstone linear problem introduced in lemma 3.1 will be as follows. We first complete regularity lemma 2.1 by another preliminary lemma (relative to differentiability) which allow us to perform a recursive approximation of the solution of a mollified problem (with a small action of the semigroup on the extra affine term). On the way, we show a priori estimates which will be useful later to remove the mollification and get a solution of our initial problem. Uniqueness and preservation of positivity are tackled in specific sections.
Such an approach was already proposed in [22], and computations look quite similar. The main difference consists in the fact that, as A(t) ∈ L Φ 2 (µ), then one has µ(e γ|A(t)| ) < ∞ for any γ (see appendix B), so that, using of the entropic inequality, contribution of the affine extra term may be made small enough to be dominated by the log-Sobolev constant without further constraint.
We now turn our attention to the second term, by another use of (24). (Strong) Absolute continuity follows.
Indeed, we deal with the first term as for absolute continuity of Ψ ε (z) above. One has which goes to 0 as h → 0.
Convergence of (II) to 0 in L 2 (µ), and this for any t a.e., follows from the easy part of the fundamental theorem of calculus for Bochner integrable functions with values in L 2 (µ) (proved via comparison with strongly Henstock-Kurzweil integrable functions and Vitali covering arguments in [ Finally, we focus on (III). For any s a.e., as 0 < h goes to 0, in L 2 (µ) as P ε z(s) ∈ D(L). And we can use dominated convergence theorem as, for g ε (τ, s) ≡ P τ (P ε z(s)), still using (24).
At the end of the day, u is a solution a.e. of (22). Deducing that u is a weak solution is easy. If φ ∈ C ∞ ([0, T ], L 2 (µ)), by bilinearity, uφ is absolutely continuous in L 1 (µ) on [a, T ], 0 < a < T , and so is the real valued function t → µ(u(t)φ(t)). The weak formulation follows when a → 0 in the integration by parts formula The proof is complete. ⊲

A mollified problem
Remark 3 In sections 5.2 to 5.4 below, we use notation introduced in the statement of lemma 3.1. So T > 0 is fixed, Let us fix ε > 0 and let us consider the following mollified problem We will prove that, for any ε > 0 (and with some more work still at the limit ε → 0), the problem (CS ε ) has a weak solution in [0, T ] that is u (ε) ∈ L 2 ([0, T ], D) ∩ C([0, T ], L 2 (µ)) and, for any φ ∈ C ∞ ([0, T ], D), and any 0 ≤ t ≤ T , To handle this problem, let us consider the following iteration scheme which, as we will prove later, converge to the unique weak solution u (ε) of our problem (CS ε ). Initially, 0 |t=0 = f and then define It follows from Lemmas 2.1 and 5.1 that, for any f ∈ L 2 (µ), u The convergence scheme we detail below is adapted from the one presented in [22] in another context. Proposition 4 (Uniform bound) Fix ε > 0 and f ∈ L 2 (µ). Let u (ε) n be the recursive solution of the mollified problem introduced above.
There exists β ∈ (0, +∞) and 0 < T 0 ≤ T both independent of ε and of the initial condition f such that for any n ∈ N, n . For any t a.e., Note that 1 ≤ M γ < ∞ for any γ > 0 since A ∈ L ∞ ([0, T ], L Φ 2 (µ)). By a similar argument, the entropic and the logarithmic Sobolev inequalities give and similarly for the other term. So that .
) 2 ) and integrating with respect to t, Choosing γ > C LS 2 , κ γ ≡ 1 − C LS 2γ > 0 and setting the above inequality implies Hence, by Gronwall type arguments, one gets Let us denote Z n = sup t∈[0,T 0 ] θ n (t). Now, provided we choose γ > C LS , C LS 2γ−C LS < 1, so that, for T 0 > 0 small enough, we end up with Hence, by induction, Note that since the map s → µ(P t (f ) 2 ) + 2 t 0 E(P s (f ))ds is decreasing. It follows that, for any n 0, which is the expected bound. ⊲ Proposition 5 (Existence for mollified problem; ε > 0) For any ε > 0 and any initial datum f ∈ L 2 (µ), there exists a weak solution u (ε) on [0, T ] of the mollified problem (CS ε ) as defined in (weak-CS ε ).
n ). For any t 0 a.e., Again thanks to the entropic and the logarithmic Sobolev inequalities, where M γ were defined in the proof of Proposition 4. By the same arguments as before, where T 0 has been defined in the previous proposition, and mimicking what we have done to prove that proposition, this leads to n+1 )(s)ds and wherẽ If we choose γ > C LS , we may take 0 <T 0 ≤ T 0 small enough (and independent of the initial condition f ) so thatηT 0 < 1. Iterating and using uniform bound (27) for n = 1 (and n = 0), one gets . It converges to some u (ε) which is a weak solution in [0,T 0 ] of (CS ε ) (see page 12, but note that things are much simpler here). AsT 0 does not depend on f , one easily extends the solution to the entire interval [0, T ]. ⊲

Uniqueness
We now state uniqueness of a weak solution for both cases : with or without a mollification.
We omit the proof which is quite similar to the one of proposition 4.1.
Proof ⊳ Let ε 1 > ε 0 > 0 and let u 0 = u (ε 0 ) and u 1 = u (ε 1 ) be the associated solutions of the mollified problem (weak-CS ε ). Using Steklov calculus as in the previous proof, we get the same estimate as if we were dealing with strong solutions. Here we avoid such technicalities to focus on the main arguments. Let us denote w = u 1 − u 0 and w = P ε 1 w. One has .

Term (I) is bounded by
µ(w 2 (t)) as in the previous proof. After integration, using symmetry of the semigroup, one gets , (which is the estimate we would get rigorously after letting h → 0 in the Steklov regularisation). After using Gronwall type arguments and taking the supremum over t ∈ [0, T 0 ], 0 < T 0 ≤ T , we note that, if we prove term (II) goes to 0 as ε 1 > ε 0 > 0 both go to 0, then (u (ε) ) ε>0 is Cauchy (as ε goes to 0) in the Banach space L 2 ([0, T 0 ], D) ∩ C([0, T 0 ], L 2 (µ)). Now, by Cauchy-Schwarz inequality, Following lemma 2.1, Choosing T 0 as in Proposition 4, one may pass to the limit n → ∞ in the uniform bound (27) to get that, for any ε > 0, So the second factor of (29) is bounded uniformly in ε 0 . In order to prove convergence to 0 of the other factor t 0 dsµ [(P ε 1 − P ε 0 ) (w(s))] 2 when ε 1 > ε 0 > 0 both go to 0, one makes use of spectral theory and the above uniform bound (30). Details are given in [22,Theorem 4.10].

Non-negativity
We prove here that, provided A and B are nonnegative, the weak solution u of problem (CS), with a nonnegative initial datum f , is nonnegative.
Rigorous arguments to get this are as follows. We consider the Steklov average a h (u)(t) and its negative part a − h (u)(t) ≡ max(0, −a h (u)(t)). Recall that, as h goes to 0, for any t . Namely, from any sequence going to 0, extract a subsequence (h n ) such that, for any t a.e. in [0, T ], a hn (u)(t) → u(t) in D. By continuity of contractions [3], it follows a − hn (u)(t) → u − (t) , in D, t a.e. and one may check easily that the sequence ( a − hn (u)(t) − u − (t) 2 D ) n is uniformly integrable in L 1 ([0, T ]).
Moreover, in W 1,2 ((0, T ), L 2 (µ)), where χ denotes the indicator function. Hence, using the definition of a weak solution (with the constant test function a − h (u)(s) ∈ D), we get We can pass to the limit with h → 0 which yields (as µ (f − ) 2 = 0) for the same reason as above.
The proof of Lemma 3.1 is complete.

Extension to the general case
The chemical reactions we consider here are of the following form for some given integers α i = β i , for any i ∈ F . F = {1, . . . , q} is a finite set. The associated reaction-diffusion equation is (after appropriate change of variables) This equation is a particular form of the abstract equation (RDP) on page 4 with constant vector λ i (β i − α i ), i = 1, . . . , q and nonlinearity G( u) = q j=1 u α j j − q j=1 u β j j . The method we detailed for the two-by-two case may be adapted to this general situation provided the following assumptions hold.

Linearity assumptions
We assume that i. F may be partitioned as F = ⊔ k∈K F k so that, L i only depends on which F k , k ∈ K, the index i belongs to. We denote byL k the common operator for any i ∈ F k . For any k ∈ K, one has the following: ii.L k is a Markov generator with (selfadjoint in the L 2 space associated with the) invariant probability measure µ k on (M, B M ) (with the same assumptions as in page 4).
iii. (L k , µ k ) satisfies logarithmic Sobolev inequality with constant C k .
iv. The measures (µ k ) k∈K are mutually equivalent in the strong sense that there exists a measure µ on (M, B M ) and C ∈ (1, +∞) such that

Nonlinearity assumptions
We assume that, for any k ∈ K, F − k and F + k are not empty. (Note that this replaces, in the present context, the hypothesis we made in the two-by-two case that C 1 = C 3 and C 2 = C 4 .)

Initial data assumptions
We assume the following common exponential integrability on the initial data.
Common integrability assumption. We assume that, for any

Iterative sequence
We now define an approximation sequence ( u (n) (t)) n∈N which converges to the solution of problem (31). It is obtained recursively as solutions of the following linear problems.
Let us fix a nonnegative initial datum f satisfying the integrability assumptions introduced before.
For any n 0, we will impose u (n) (0) = f and, for n = 0, ∂ t u the other case is similar by symmetry). Let us label elements of F ± k in the following way We consider an onto mapping ν k : Define furthermore, for any i, j ∈ F , α and similarly for β's. Let us note here that, for any i ∈ F + k and j ∈ F − k , β i > 0 and α j > 0. Finally, The iterated sequence is then defined as follows † . In the case i ∈ F − k , And, in the case i ∈ F + k , where Z k,i = r∈ν −1 k (i) δ r .
Why the sequence is well defined.
Recall Young inequality: for any a 1 , . . . , a q 0, Hence, using also Hölder inequality, ⊲ To prove recursively that the sequence ( u (n) ) n is well defined, we have to split the cornerstone existence lemma into the following two lemmas. Lemma 6.1 (Matrix cornerstone existence lemma) Let (L, µ) be a Markov generator satisfying logarithmic Sobolev inequality with constant C LS ∈ (0, ∞). Let T > 0 and A = A(t) be an N × N matrix with coefficients in L ∞ ([0, T ], L Φ 2 (µ)) and B ∈ (L 2 ([0, T ], L 2 (µ))) N . Then the Cauchy problems and with u + = ((u 1 ) + , . . . (u q ) + ), both have a unique weak solution on [0, ∞) Note that we use that u → u + is a contraction so that it contracts both the L 2 (µ) norm and the Dirichlet form E.
In the system defined by (32) and (33) only blocks made of some i ∈ F + k and j's in ν −1 k (i) interact. We now focus on these coordinates. The following lemma ensures that positivity and Bochner measurability (34) propagate along the approximation sequence. Lemma 6.2 (Positivity and propagation of measurability.) Let N 2 and let δ 1 , . . . , δ N −1 0 such that Z ≡ N −1 i=1 δ i > 0. Assume furthermore B(t) = 0 and A(t) is of the following form where a i ∈ L ∞ ([0, T ], E Φ 2 (µ)), i = 1, . . . , N , are all nonnegative. Assume the initial datum f ∈ (L 2 (µ)) N is nonnegative. Then the solution u of (MCS) is nonnegative. Moreover, one has We detail a bit positivity argument (the remaining is similar to the two-by-two case).
Let v be the unique weak solution of problem (MCS + ) with initial condition f . We now show v is nonnegative and so it coincides to the unique solution of (MCS) with initial condition f . Thanks to Steklov calculus, the following computation is made rigorous. We focus on the last component (which is the most complicated one).
And the third term is trivially nonpositive as the a i 's are assumed nonnegative. Hence, We can state the following theorem. Theorem 6.3 Let L i , i = 1, . . . q, be Markov generators satisfying the linearity assumptions described before. Assume the nonlinearity assumptions are satisfied as well and that f 0 belongs to E Φ 2θ (µ), with θ as in the initial data assumption.
Then, for any reaction rates λ i > 0, there exists a unique nonnegative weak solution u of problem (31) on [0, ∞).
Lemma A.1 (Entropic inequality) Let µ be a probability measure and let f and g be two measurable functions. Assume f 0 (excluding f = 0 µ-a.e.) such that f log + f ∈ L 1 (µ) and µ(e γg ) < +∞ for some γ > 0. Then f g ∈ L 1,− ext (µ) and The proof is based on the following inequality ∀x ∈ R + , ∀y ∈ R, x y ≤ x log x − x + e y .

B Basics on Orlicz spaces
Classical properties of Orlicz spaces can be found in [38].

Young functions
Let Φ be a Young function, that is Φ : R → R convex, even such that Φ(0) = 0 and Φ is not constant. Note that from this, it follows that Φ(x) 0, that Φ(x) → +∞ when x → ∞ and that Φ is an increasing function on [0, +∞).

Gauge norm
From these properties, it follows that the gauge norm associated to B Φ is indeed a norm. One has The space (L Φ (µ), · Φ ) is a Banach space.

Comparison of norms
We often have to compare Orlicz norms associated to different Young functions. We already have seen in a footnote that any Young function Φ satisfies |x| Φ(x). It leads to the following lemma.
Lemma B.5 (Separability) Assume M is a separable metric space. Then, for any Young function Φ, E Φ (µ) is separable.
(Use that B M is countably generated, monotone class theorem and density of simple functions).

Duality
What follows may be found in [13].
In the case of Young functions with rapid growth (as the Φ α 's introduced before), ∆ 2 condition fails. Consequently E Φ (µ) is a proper Banach subspace of L Φ (µ) (assuming the support of µ is infinite) and L Φ (µ) is not separable.
Recall that the conjugate function Ψ * of a Young function Ψ is the Young function defined by Ψ * (y) ≡ sup x 0 (x|y| − Ψ(x)).
C Markov Semigroups and Orlicz spaces C.1 Contraction property Lemma C.1 Let Φ : R → R + be a nonnegative convex function. Let (P t ) t 0 be a Markov semigroup on L 2 (µ), for a probability measure µ, as introduced in section 2.
In particular, in the case when Φ is a Young function (with domain R), provided f ∈ L Φ (µ), then P t f ∈ L Φ (µ) and (P t ) t 0 is a contraction semigroup on L Φ (µ).
Then (43) follows by integration w.r.t. µ and invariance property of P t .

C.3 Bochner measurability
Let X be a Banach space. Recall that an X-valued function u : I → X defined on a compact interval I is Bochner measurable provided it is an a.e. limit of a sequence of X-valued simple functions on I (see [41] for instance).

Proof of proposition 1
By density of L 2 (µ) in L Φ * α and contraction of P t in L Φ * α , C 0 property of P t in L Φ * α follows from C 0 property in L 2 (µ). Indeed, let f ∈ L Φ * α . ε > 0 being fixed, let g ∈ L 2 (µ) such that f − g Φ * α < ε 3 . Then allows to conclude. As a consequence, provided f ∈ E Φα , t → P t f ∈ E Φα is weakly continuous, and so Bochner measurable as E Φα is separable, following Pettis measurability theorem (see page 9 for references).