Global Renormalised Solutions and Equilibration of Reaction–Diffusion Systems with Nonlinear Diffusion

The global existence of renormalised solutions and convergence to equilibrium for reaction–diffusion systems with nonlinear diffusion are investigated. The system is assumed to have quasi-positive nonlinearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with nonlinear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract renormalised solutions in the same compatibility class. This convergence extends even to a range of nonlinear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.

in Ω. (1.1) Here and below, we set Q T :" Ωˆp0, T q and Γ T :" BΩˆp0, T q for any T ą 0, and we employ ν to denote the unit outward normal vector to BΩ. The diffusion constants d i and the non-linear diffusion exponents m i are assumed to satisfy d i ą 0 and 0 ă m i ă 2. For proving exponential convergence to equilibrium, we further demand m i ą pd´2q`{d with pd´2q`:" maxt0, d´2u. We will comment on the bounds on m i later on. Moreover, we generically assume u i,0 P L 1 pΩq, u i,0 ě 0, and impose the following assumptions on the reaction terms f i puq, i " 1, . . . , I: (F1) (Local Lipschitz continuity) f i : R Ì Ñ R is locally Lipschitz continuous.
(F3) (Entropy inequality) There exist constants pµ i q i"1,...,I P R I satisfying for all u P R Ì , where C ą 0 is independent of u.
The second assumption (F2) guarantees non-negativity of solutions provided that the initial data are non-negative, which is a natural assumption since we consider here u i as concentrations or densities. Assumption (F3), on the other hand, implies existence and control of an entropy of the system and it is only slightly stronger than assuming control of the mass, i.e. In general, the assumptions (F1), (F2), and either (F3) or (1.3) of the non-linearities f i are not enough to obtain suitable a priori estimates which allow to extend local strong solutions globally, see e.g. [25]. On the one hand, for the case of linear diffusion (i.e. m i " 1 for all i " 1, . . . , I), the global existence for (1.1) under extra assumptions on non-linearities has been extensively investigated. For instance, global bounded solutions were shown under (F3) for the case d " 1 with cubic non-linearities and d " 2 with quadratic non-linearities, see [17,28]. Recent works [27,1,12] showed under either (F3) or (1.3) that bounded solutions can be obtained for (slightly super-)quadratic non-linearities in all dimensions. We refer the reader to the extensive review [24] for related results concerning global existence of bounded or weak solutions. It is remarked that in all such results, additional assumptions on non-linearities must be imposed. A breakthrough has been made in [15] where global renormalised solutions were obtained without any extra assumptions on the non-linearities. The notion of renormalised solution was initially introduced for Boltzmann's equation in [7] and it has been applied subsequently to many other problems; see e.g. [4,5,2]. The case of non-linear, possibly degenerate diffusion is, on the other hand, much less understood. Up to our knowledge, there are only two works ( [20] and [11]) which showed global existence of weak or bounded solutions for porous medium type of diffusion, i.e. m i ą 1, under (1.3) and some restricted growth conditions on the non-linearities. A recent preprint, however, provides renormalised solutions to a system as in (1.1) but imposing more regularity on the initial data and the underlying domain [21]. Instead of an entropy condition as in (1.2), the authors of [21] assume an additional bound on cross-absorptive reaction terms. Furthermore, [11] proved the convergence to equilibrium for (1.1) modelling a single reversible chemical reaction.
The present study shows the existence of global renormalised solutions to (1.1) featuring non-linear diffusion of porous medium type or fast diffusion without any extra conditions on the non-linearities (up to assuming (F1)-(F3)). In addition, we show that these solutions, in case (1.1) models general complex balanced chemical reaction networks, converge exponentially to equilibrium with explicitly computable rates. Our paper seems to be the first extensive contribution to the non-linear diffusion type system (1.1) establishing-for specific parameter regimes-the existence of global renormalised (or even weak and bounded) solutions as well as exponential convergence to equilibrium in various L p spaces. A brief summary of our results is given in Table 1 at the end of this section.
The first part of this paper is concerned with the global existence of renormalised solutions to (1.1). We consider the following definition of renormalised solutions.
Definition 1.1 (Renormalised solutions). Let m i ă 2 for all 1 ď i ď I and u 0 " pu i,0 q i P L 1 pΩq I be (componentwise) non-negative. A non-negative function u " pu 1 , . . . , u I q : Ωˆp0, 8q Ñ R Ì is called a renormalised solution to (1.1) with initial data u 0 if u i P L 8 loc p0, 8; L 1 pΩqq and ∇u i m i 2 P L 2 loc p0, 8; L 2 pΩq I q, and if for any smooth function ξ : R Ì Ñ R with compactly supported derivative Dξ and any function ψ P C 8 pΩˆp0, 8qq, we have ż Ω ξpup¨, T qqψp¨, T q dx´ż for a.e. T ą 0.
We notice that all integrals in (1.4) are well-defined due to the compact support of Dξ and as the integrals relating to gradients of solutions can be rewritten as ż  Let Ω Ă R d be a bounded domain with Lipschitz boundary BΩ. Assume m i ă 2 for all i " 1, . . . , I, and that the conditions (F1)-(F3) hold. Then, for any non-negative initial data u 0 " pu i,0 q i P L 1 pΩq I subject to Moreover, any renormalised solution satisfies the following weak entropy-entropy dissipation relation for a.e. 0 ď s ă T : In addition, if there exist some numbers q i P R such that ř I i"1 q i f i puq " 0, then all renormalised solutions satisfy for a.e. t ą 0.
‚ The relation (1.6) usually corresponds to mass conservation laws in chemical reactions, see e.g. [14]. ‚ The uniqueness of renormalised solutions is widely open. For a weaker notion called weakstrong uniqueness, i.e. the renormalised solution is unique as long as a strong solution exists, we refer the interested reader to [16]. ‚ The fact that (1.5) and (1.6) are satisfied by any renormalised solution plays an important role in the next part of this paper where we show that all renormalised solutions, instead of only some of them, converge to equilibrium.
The strategy for constructing a global renormalised solution follows the ideas in [15]. Given a sequence of approximate solutions u ε , one proves compactness of the family u ε by deriving bounds on truncations ϕ E i pu ε q, E P N, i " 1, . . . , I, and subsequently employing an Aubin-Lions lemma. The smooth functions ϕ E i defined in (2.6) essentially truncate the mappings u Þ Ñ u i if u becomes too large (measured in terms of E) while leaving u Þ Ñ u i unchanged for sufficiently small u. The main difficulty in our current situation is caused by the different diffusion exponents m i for each species u i , which makes the truncations ϕ E i provided in [15] not applicable. We overcome this issue by modifying the functions ϕ E i in such a way that the truncation is not decided by ř j u j but by a weighted sum ř j E´α i j u j with suitable constants α i j ě 1 chosen depending on the diffusion exponents m i . We then pass to the limit ε Ñ 0 in the equation for ϕ E i pu ε q at the cost of an additional defect measure which, nevertheless, vanishes in the subsequent limit E Ñ 8. Finally, an equation for ξpϕ E i puqq is derived, where ξ is subject to the same assumptions as in Definition 1.1, and the limit E Ñ 8 is performed resulting in the desired equation (1.4) for ξpuq.
The restriction m i ă 2 for i " 1, . . . , I appears already in the definition of renormalised solutions in Definition 1.1. It is also crucial in showing the vanishing of the defect measure when establishing the renormalised solution. We do not know whether this restriction is purely technical or due to a deeper reason. It is remarked that the same condition was crucially imposed in [20], where global weak solutions to (1.1) were investigated.
In the second part of this paper, we study the large-time behaviour of reaction-diffusion systems of type (1.1) modelling chemical reaction networks subject to the complex balance condition. The main vocabulary is introduced below but we refer to [6] for a more detailed discussion of the involved concepts. We assume that there are I chemicals S 1 , . . . , S I reacting via the following R reactions y r,1 S 1`¨¨¨`yr,I S I kr Ý Ñ y 1 r,1 S 1`¨¨¨`y 1 r,I S I for all r " 1, . . . , R (1 .7) where y r,i , y 1 r,i P t0u Y r1, 8q are stoichiometric coefficients, and k r ą 0 are reaction rate constants. Utilizing the notation y r " py r,i q i"1,...,I and y 1 r " py 1 r,i q i"1,...,I , we can rewrite the reactions in (1.7) as y r kr Ý Ñ y 1 r for all r " 1, . . . , R. (1.8) Denote by u i px, tq the concentration of S i at position x P Ω and time t ě 0. The reaction network (1.7) results in the following reaction-diffusion system with non-linear diffusion for u " pu 1 , . . . , u I q: (1.9) in which the reaction term f i puq is determined using the mass action law, It is obvious that f i puq is locally Lipschitz continuous, hence (F1) is satisfied. For (F2), it is easy to check that f i puq ě 0 holds true for all u P R Ì satisfying u i " 0 since we assume k r ą 0 and y P pt0u Y r1, 8qq I for all y P ty r , y 1 r u r"1,...,R . Before verifying (F3), we recall the definition of a complex balanced equilibrium. A constant state u 8 " pu i,8 q i P r0, 8q I is called a complex balanced equilibrium for (1.9)-(1.10) if ÿ tr:yr"yu k r u yr 8 " ÿ ts:y 1 s "yu k s u ys 8 for all y P ty r , y 1 r u r"1,...,R . (1.11) Intuitively, this condition means that for any complex the total outflow from the complex and the total inflow into the complex are balanced at such an equilibrium. Using complex balanced equilibria, one can show that the non-linearities in (1.10) satisfy (F3) by choosing µ i "´log u i,8 with a strictly positive complex balanced equilibrium u 8 (see [6,Proposition 2.1]). Therefore, one can apply Theorem 1.2 to obtain global renormalised solutions to (1.9)-(1.10).
In general, there might exist infinitely many solutions to (1.11). To uniquely identify a positive equilibrium, we need a set of conservation laws. More precisely, we consider the Wegscheider matrix (or stoichiometric coefficient matrix) W " " py 1 r´y r q r"1,...,R ‰ T P R RˆI , and define m " dimpkerpW qq. If m ą 0, then we can choose a matrix Q P R mˆI , where the rows form a basis of kerpW q. Note that this implies Q py 1 r´y r q " 0 for all r " 1, . . . , R. Therefore, by recalling f puq " pf i puqq i"1,...,I " ř R r"1 k r u yr py 1 r´y r q we obtain Q f puq " R ÿ r"1 k r u yr Qpy 1 r´y r q " 0 for any u P R I .
By using the homogeneous Neumann boundary condition, we can then (formally) compute that d dt ż Ω Q f puq dx " 0 and, consequently, for a.e. t ą 0, In other words, system (1.9)-(1.10) possesses m linearly independent mass conservation laws. By the vocabulary used in the literature on chemical reaction network theory, for any fixed nonnegative mass vector M P R m , the set tu P R Ì : Q u " M u is called the compatibility class corresponding to M . In case that for each strictly positive vector M P R m , there exists a strictly positive complex balanced equilibrium, one says that the chemical reaction network is complex balanced. This terminology is well-defined since it was proved in [18] that if one equilibrium is complex balanced, then all equilibria are complex balanced. It was further shown that for each non-negative initial mass M P R m with M ‰ 0, there exists a unique strictly positive complex balanced equilibrium u 8 " pu i,8 q i P R Ì in the compatibility class corresponding to M (cf. [19,10]). We stress that additionally there might exist many so-called boundary equilibria, which are complex balanced equilibria lying on BR Ì . It is remarked that there exists a large class of complex balanced chemical reaction networks, called concordant networks, which do not have boundary equilibria, see [26]. For the sake of brevity, from now on we call the unique strictly positive complex balanced equilibrium simply the complex balanced equilibrium. Note that all the previous considerationsalthough established for ODE models for chemical reaction networks-also apply to our PDE setting thanks to the homogeneous Neumann boundary conditions, which allow for spatially homogeneous equilibria. We emphasise that the convergence to equilibrium in Theorem 1.4 is proved for all renormalised solutions, not only the ones obtained via an approximation procedure. This is possible because the proof of the convergence uses only the weak entropy-entropy dissipation law (1.5) and the conservation laws (1.6) which are satisfied by all renormalised solutions. It is noted that, when a system possesses boundary equilibria, the convergence to the positive equilibrium (more precisely, the instability of boundary equilibria) is very subtle and strongly connected to the famous Global Attractor Conjecture, see for instance [6,14,3].
Theorem 1.4, up to our knowledge, is the first result of trend to equilibrium for general complex balanced reaction networks with non-linear diffusion. A special case of a single reversible reaction was considered recently in [11] in which the authors utilised a so-called indirect diffusion effect (see e.g. [9]), which seems difficult to be generalised to general systems. The proof of Theorem 1.4 uses the relative entropy and its corresponding entropy dissipation Drus " where Ψpx; yq " x logpx{yq´x`y. It holds, at least formally, that d dt Eru|u 8 s " Drus along any trajectory of (1.9)-(1.10). The degeneracy of the non-linear diffusion makes the classical logarithmic Sobolev inequality not applicable. Our main idea here is to utilise some generalised logarithmic Sobolev inequalities (see Lemmas 3.3 and 3.4), which are suited for non-linear diffusion, and the established results in [14] for the case of linear diffusion, to firstly derive an entropy-entropy dissipation inequality of the form Drus ě CpEru|u 8 sq α (1.14) where α " max i"1,...,I t1, m i u. Note that this functional inequality is proved for all non-negative functions u : Ω Ñ R Ì satisfying the conservation laws Q ş Ω u dx " |Ω|Qu 8 , and therefore is suitable for renormalised solutions, which have very low regularity. If α " 1 in (1.14), one immediately gets exponential convergence of the relative entropy to zero and, consequently, exponential convergence of the solutions to equilibrium in L 1 thanks to a Csiszár-Kullback-Pinsker type inequality. If α ą 1, we first obtain an algebraic decay of the relative entropy to zero. Thanks to this, we can explicitly compute a finite time T 0 ą 0 from which onwards (in time) the averages of concentrations are strictly bounded below by a positive constant. This helps to compensate the degeneracy of the diffusion and, therefore, to show that solutions with such lower bounds satisfy the linear entropyentropy dissipation inequality, i.e.
Druptqs ě CEruptq|u 8 s for all t ě T 0 , which recovers exponential convergence to equilibrium.
The use of renormalised solutions allows to deal with a large class of non-linearities, but on the other hand it restricts Theorem 1.4 to the case pd´2qd ă m i ă 2. When the non-linearities are of polynomial type, it was shown in [20,11], under the assumption of the mass dissipation (1.3) instead of the entropy inequality (1.2), that one can get global weak or even bounded solutions if the porous medium exponents m i are large enough. In the following theorem, we show an analog result under the entropy inequality condition (1.2). Moreover, the solution is shown to converge exponentially to the positive complex balanced equilibrium. the weak solution to (1.1) subject to bounded non-negative initial data u 0 P L 8 pΩq I is locally bounded in time, i.e.
where C T is a constant growing at most polynomially in T ą 0.
for some C, λ ą 0. ‚ If (1.16) holds and 0 ď u 0 P L 8 pΩq I , weak solutions to (1.9)-(1.10) are bounded locally in time and converge exponentially to equilibrium in any L p norm with p P r1, 8q, i.e.
with positive constants C p , λ p ą 0.
As one can see from Theorems 1.4 and 1.5, the global existence and trend to equilibrium for (1.9)-(1.10) are well-established for either m i ă 2 for all i " 1, . . . , I, or m i large enough in the sense of (1.15) or (1.16) for all i " 1, . . . , I. There exists, therefore, a gap in which the global existence of any kind of solution remains open. Remarkably, the proof of the convergence to equilibrium in this paper does not rely on the restriction m i ă 2, and it therefore is applicable to any global solution to (1.9)-(1.10) as long as it satisfies the entropy law (1.5) and the mass conservation laws (1.6). We, thus, arrive at the following result on the convergence to equilibrium for approximating systems of (1.9)-(1.10) uniformly in the approximation parameter.
ş Ω u i,0 log u i,0 dx ă 8, and m i ą pd´2qd for all i " 1, . . . , I admits a complex balanced equilibrium but no boundary equilibria. For any ε ą 0, let u ε " pu ε i q i"1,...,I be the solution to the approximate system Moreover, the approximated initial data u ε 0 are chosen such that, for all ε ą 0, (This implies that the system (1.19) admits a unique positive complex balanced equilibrium u 8 P R Ì , which is independent of ε ą 0.) Then, u ε converges exponentially to u 8 with a rate which is uniform in ε, i.e.
To prove Theorem 1.6, we make use of the same relative entropy as in (1.12), where u ε is the solution to (1.19). The corresponding non-negative entropy dissipation can be calculated as x.
Since there is no uniform-in-ε L 8 bound for u ε available, the factor 1 1`ε|f pu ε q| is not bounded below uniformly in ε ą 0. Therefore, it seems to be impossible to use the entropy dissipation (for limit solutions) in (1.13) as a lower bound for Dru ε s. We overcome this issue by using the ideas in [14]. Roughly speaking, we deal with the second sum in Dru ε s by estimating it below by a term involving only spatial averages of u ε (rather than u ε pointwise as above), and then exploiting the fact that these averages are bounded uniformly in ε ą 0.
We summarise the global existence and convergence to equilibrium for the mass action system (1.9)-(1.10) in Table 1 (assuming the existence of a unique strictly positive complex balanced equilibrium and the absence of boundary equilibria). Note that the global existence of any kind of solution remains open for diffusion exponents satisfying 2 ď m i ă µ´1, where µ is defined in (1.17). The rest of this paper is organised as follows. In Section 2, we show the global existence of renormalised solutions for the general system (1.1). The proofs of Theorems 1.4, 1.5, and 1.6 on the convergence to equilibrium of chemical reaction networks are presented in Section 3. Some technical proofs of auxiliary results are postponed to the Appendix.

Global existence of renormalised solutions
2.1. Existence of approximate solutions. Let ε ą 0 and consider the following approximating system for u ε " pu ε 1 , . . . , u ε I q,

1)
Diffusion exponents Global existence Convergence to equilibrium Exponential in L 1 for approximate solutions uniformly in the approximation parameter Exponential in any L p , p ă 8 Table 1. Summarisation of global existence and convergence to equilibrium for mass action reaction-diffusion systems (1.9)-(1.10) with non-linear diffusion in various parameter regimes.
where u ε i,0 P L 8 pΩq is non-negative and u ε With this approximation, it is easy to check that the approximated non-linearities f ε i still satisfy the assumptions (F1)-(F3).
Proof. For each ε ą 0, we see by recalling |f pu ε q| " The existence of bounded weak solutions to (2.1) is therefore standard. However, since we are not able to find a precise reference, a proof is given in Appendix A. By multiplying (2.1) by logpu ε i q`µ i (or more rigorously by logpu ε i`δ q`µ i for some δ ą 0, then let δ Ñ 0), summing the resultants over i " 1, . . . , I, and then integrating over Ω, we obtain d dt where we used (F3) and x ď δx log x`C δ for all x ě 0 and any δ ą 0 at the last step. Hence, by integrating (2.5) over pt, T q and using Gronwall's inequality, we obtain the desired estimate 2.2. Existence of renormalised solutions. As it can be seen from Lemma 2.1, the a priori estimates of u ε i are not enough to extract a convergent subsequence. Following the idea from [15], we consider another approximation of u ε i by defining for E P N, and a smooth function ξ : R Ñ r0, 1s satisfying ξ " 1 on p´8, 0q and ξ " 0 on p1, 8q.
The constants α i j are given by Remark 2.2. The α i j are chosen in (2.7) for the sake of simplicity. In fact, we can choose any α i j such that See the proof of (E2) in Lemma 2.3.
Lemma 2.3. The smooth truncations ϕ E i defined in (2.6) have the following properties: Proof. Properties (E1), (E3), (E4), and (E6) are immediate. To prove (E2), we first compute From that, one further gets the second derivatives To show (2.8), we will consider the following cases: Employing the identities ξ 1 p¨¨¨q " ξ 2 p¨¨¨q " 0 for v i ě 2E α i i as well as the bound |ξ 1 p¨¨¨q|| Therefore, by choosing α i i " 1, we have the desired estimate (2.8) for i " j " k. ‚ When k " i and j ‰ i, we estimate using ‚ When j " i and k ‰ i, we estimate similarly to the previous cases If we choose then obviously (2.8) holds true for j " i ‰ k.
It is easy to see that from (2.9) and (2.10) it follows and hence (2.8) is proved in this case.
This entails y ď 0 and, thus, completes the proof of the lemma.
Lemma 2.4. Consider non-negative functions u 0 " pu i,0 q i P L 1 pΩq I which satisfy Let u ε " pu ε 1 , . . . , u ε I q for ε Ñ 0 be the sequence of solutions to the regularised problems as stated in Lemma 2.1. Then, there exists a subsequence u ε converging a.e. on Ωˆr0, 8q to a limit u P L 8 loc p0, 8; Proof. Due to the lack of uniform-in-ε estimates of the non-linearities, it is difficult to show directly, for instance by means of an Aubin-Lions lemma, that u ε i has a convergent subsequence. Following the ideas from [15], we first prove that ϕ E i pu ε q converges (up to a subsequence) to z E i as ε Ñ 0, and then that z E i converges to u i (up to a subsequence) as E Ñ 8. In combination with the convergence of ϕ E i pu ε q to u ε i for E Ñ 8, this leads to the desired result.
Due to the bound ϕ E i ď 3E, it follows at once that tϕ E i pu ε qu is bounded in L 2 p0, T ; L 2 pΩqq uniformly in ε ą 0 for each E P N. Next, by the chain rule we have thanks to the compact support of Dϕ E i , m j ă 2, and the L 2 p0, T ; L 2 pΩqq bound on ∇pu ε j q m j 2 from Lemma 2.1. As a consequence, we know that tϕ E i pu ε qu is bounded (uniformly in ε) in L 2 p0, T ; H 1 pΩqq.
To apply the Aubin-Lions Lemma, we need an estimate concerning the time derivative of ϕ E i pu ε q. Using B j ϕ E i pu ε qψ as a test function in (2.3) and summing over j P t1, . . . , Iu, we obtain for almost all t 2 ą t 1 ě 0, The third term on the right hand side is clearly bounded uniformly in ε for each fixed E P N since Dϕ E i has a compact support. The first and second terms are bounded uniformly in ε thanks to the boundedness of ∇pu ε k q m k 2 in L 2 p0, T ; L 2 pΩqq for all k " 1, . . . , I, and properties (E2), (E3) in Lemma 2.3. It follows then that B t ϕ E i pu ε q is bounded uniformly (w.r.t. ε ą 0) in L 1 p0, T ; pW 1,8 pΩqq 1 q for each fixed E P N. Therefore, by applying an Aubin-Lions lemma to the sequence tϕ E i pu ε qu εą0 , for fixed E P N, there exists a subsequence (not relabeled) of ϕ E i pu ε q converging strongly in L 2 p0, T ; L 2 pΩqq and, thus, almost everywhere as ε Ñ 0. Using a diagonal sequence argument, one can extract a further subsequence such that ϕ E i pu ε q converges a.e. in Ω to a measurable function z E i for all E P N and i " 1, . . . , I.
We next prove that z E i converges a.e. to some measurable function u i as E Ñ 8. First, since ř I j"1 u ε j log u ε j is uniformly bounded w.r.t. to ε ą 0 in L 8 p0, T ; L 1 pΩqq, it follows that ϕ E i pu ε q log ϕ E i pu ε q is uniformly bounded w.r.t. to ε ą 0 and E P N in L 8 p0, T ; L 1 pΩqq. This is trivial in case E ď 1, ϕ E i pu ε q ď 1, or ř I j"1 u ε j E´α i j ď 1. Otherwise, there exists some j such that u ε j E´α i j ą 1{I holds true. And as α i j ą 1, we derive The bound ϕ E i pu ε q ą 1 ensures ϕ E i pu ε q log ϕ E i pu ε q ď 3Iu ε j logp3Iu ε j q and the claim follows. Thus, z E i log z E i is uniformly bounded in L 8 p0, T ; L 1 pΩqq w.r.t. E P N by Fatou's Lemma. Secondly, guarantees that ř I j"1 ϕ E j pu ε px, tqqE´α i j ă 1 holds true for sufficiently small ε ą 0. Thanks to the property (E8) in Lemma 2.3, it follows that ϕ E i pu ε px, tqq " u ε i px, tq " ϕ r E i pu ε px, tqq for small enough ε and all r E ą E. Therefore, due to z E i px, tq " lim εÑ0 ϕ E i pu ε px, tqq, we obtain z E i px, tq " z r E i px, tq for all r E ą E as desired. As a result, if ř I j"1 z E j px, tqE´α i j ă 1 for some px, tq, then lim EÑ8 z E i px, tq exists and is finite for all i " 1, . . . , I. Using the fact that where L n`1 is the Lebesgue measure in R n`1 . Hence, the limit u i px, tq " lim EÑ8 z E i px, tq exists for a.e. px, tq P Q T . Moreover, u i log u i P L 8 p0, T ; L 1 pΩqq due to Fatou's Lemma and z E i log z E i P L 8 p0, T ; L 1 pΩqq. Since ř I j"1 u ε j is uniformly bounded in L 1 pQ T q, we find 0 " lim EÑ8 L n`1˜# px, tq P Q T : As we have proved that ϕ E i pu ε q Ñ z E i a.e. in Q T for ε Ñ 0 and fixed E P N, and that z E i Ñ u i a.e. in Q T as E Ñ 8, we infer the convergence in measure of u ε i to u i for ε Ñ 0, and convergence a.e. of another subsequence. The uniform bound on u ε i log u ε i in L 8 p0, T ; L 1 pΩqq and the convergence u ε i Ñ u i a.e. ensure that u ε i Ñ u i strongly in L p p0, T ; L 1 pΩqq for all p ě 1. One can easily prove this by truncating u ε i at a sufficiently large threshold. By the strong convergence of u ε i to u i in L 1 p0, T ; L 1 pΩqq, we are able to prove the distributional convergence of pu ε i q At this point, we expect the function u " pu 1 , . . . , u I q in Lemma 2.4 to be a renormalised solution to (1.1). We will first derive an equation admitting ϕ E i puq as a solution. This equation is already "almost" matching the formulation for a renormalised solution (1.4) except for a "defect measure". Lemma 2.5. Let u " pu 1 , . . . , u I q be the functions constructed in Lemma 2.4. Then, for any ψ P C 8 pr0, T s; for all T ą 0 and i P t1, . . . , Iu.
Proof. Choosing t 1 " 0, t 2 " T , and using the convergence u ε Ñ u a.e. in Q T , the weak convergence ∇pu ε i q m i 2 á ∇pu i q m i 2 in L 2 p0, T ; L 2 pΩqq, and the fact that Dϕ E i pu ε q vanishes when u ε is too large, we straightforwardly obtain the convergence of the left hand side and the last two terms on the right hand side of (2.11). It remains to establish with a signed Radon measure µ E i satisfying (2.13). By denoting we can use property (E2) of the truncation function ϕ E i and the ε-uniform bound of ∇pu ε i q m i 2 in L 2 p0, T ; L 2 pΩqq, to obtain |µ E i,ε |pQ T q ď C for all ε ą 0. Therefore, by passing to a subsequence, we know that µ E i,ε weak-˚converges on Q T to a signed Radon measure µ E i as ε Ñ 0. It remains to prove (2.13). Due to Young's inequality, we have We stress that ν ε l,K is uniformly bounded w.r.t. l, K, and ε. Consequently, we may pass to a subsequence ν ε l,K which converges weak-˚on Q T to a Radon measure ν l,K as ε Ñ 0. Together with the weak-˚convergence of µ E i,ε to µ E i on Q T , we derive which is bounded uniformly in ε. Fatou's Lemma applied to the counting measure on N now entails Therefore, employing the dominated convergence theorem, we can finally estimate thanks to the properties (E2) and (E7) of the truncation function ϕ E i .
To prove Theorem 1.2, we use the following technical lemma whose proof can be found in [15].

Lemma 2.6 (A weak chain rule for the time derivative). [15, Lemma 4]
Let Ω be a bounded domain with Lipschitz boundary. Assume that T ą 0, v P L 2 p0, T ; H 1 pΩq I q, and v 0 P L 1 pΩq I . Let ν i be a Radon measure on Q T , w i P L 1 pQ T q, and z i P L 2 p0, T ; L 2 pΩq I q for 1 ď i ď I. Assume moreover that for any ψ P C 8 pQ T q with compact support we have Let ξ : R I Ñ R be a smooth function with compactly supported first derivatives. Then, we have for all ψ P C 8 pQ T q with compact supporťˇˇˇż Ω ξpvpT qqψpT q dx´ż We are now ready to prove the existence of global renormalised solutions.
Proof of Theorem 1.2. By applying Lemma 2.6 to (2.12), we obtain an approximate relation for the weak time derivative of ξpϕ E puqq, which leads to the desired equation for ξpuq when passing to the limit E Ñ 8. In detail, we set v i :" ϕ E i puq, pv 0 q i :" ϕ E i pu 0 q, ν i :"´µ E i , and obtain for any smooth function ξ with compactly supported first derivativešˇˇˇż (2.14) We now want to pass to the limit E Ñ 8 in (2.14) to obtain (1.4). Note first that due to (2.13), we obtain in the limit an equality instead of just an estimate. Since ξ has compactly supported first derivatives, B j ϕ E i puq is bounded (see (E5)), and ϕ E i puq Ñ u i a.e. as E Ñ 8, we directly obtain the following limits for the first line in (2.14): ż It remains to ensure the convergence of the third and fourth line of (2.14). To this end, we recall pu j q m j 2 P L 2 p0, T ; H 1 pΩqq, and we utilise the following observation: there exists a constant E 0 ą 0 such that for all E ą E 0 the inequality ř I i"1 u i ě E 0 implies B i ξpϕ E puqq " B i ξpuq " 0 and B i B k ξpϕ E puqq " B i B k ξpuq " 0 for all i, k P t1, . . . , Iu. The convergence of the third and fourth line above is now a consequence of this auxiliary result, as the derivatives of ξ are zero provided max i u i is larger than E 0 :

The previous observation readily follows by choosing
which proves the claim.

2.3.
Entropy-entropy dissipation law and conservation laws. We prove in this subsection that any global renormalised solution to (1.1) satisfies the entropy law (1.5) and (1.6). These relations become helpful in the next section to investigate the convergence to equilibrium of renormalised solutions.
Such a function θ M exists for C ą 0 large enough as in this situation In the following, we first choose the renormalising function ξ from Definition 1.1 to be then let ε Ñ 0 and M Ñ 8 consecutively. Using this ξ in (1.4) with ψ " 1, we derive ż Here we write p¨¨¨q :" ř I i"1 pu i`ε qplogpu i`ε q`µ i´1 q for simplicity. In the limit ε Ñ 0, the convergence of the left hand side of (2.15) is immediate by the Lebesgue dominated convergence theorem. By rewriting we can use the fact that ∇u m i 2 i is bounded in L 2 pQ T q and θ 1 M psq " θ 2 M psq " 0 for s ě M C to pass to the limit ε Ñ 0 for the first and second terms on the right hand side of (2.15). Concerning the last term, we observe that θ 1 M p¨¨¨qf i puq logpu i`ε q ď CpM qu i | logpu i`ε q| thanks to the Lipschitz continuity of f i and the fact that θ 1 M psq " 0 for s ě M C . Note that tu i | logpu i`ε q|u εą0 is bounded uniformly (in ε) in L 1 pQ T q due to Lemma 2.4. We now use the assumption (1.2) and Fatou's Lemma to get lim sup Therefore, by letting ε Ñ 0 in (2.15), we obtain ż (2.16) Note that here p¨¨¨q stands for ř I i"1 u i plog u i`µi´1 q. We now pass to the limit M Ñ 8. The convergence of the left hand side of (2.16) follows from Fatou's Lemma and the dominated convergence theorem thanks to the fact that u i log u i P L 8 p0, T ; L 1 pΩqq. Using ∇pu i q m i 2 P L 2 pQ T q and u m i´2 2ˇ2 , the convergence of the first term on the right hand side of (2. 16) follows via dominated convergence. The reaction term can also be dealt by Fatou's Lemma, the entropy dissipation assumption (1.2) and the fact that θ 1 M ě 0. We are left to consider the second term on the right hand side of (2.16). It is straightforward that the integrand converges pointwise to zero as M Ñ 8, since θ M is an approximation of the identity. By the fact that ∇pu i q m i 2 P L 2 pQ T q and rewriting is bounded uniformly in M . Indeed, using the assumption on θ M as well as 0 ă m i 2 and 1´m j 2 ă 1, we deducěˇˇˇθ In conclusion, by letting M Ñ 8 in (2.16), we obtain the desired inequality (1.5).
Proof of the conservation law (1.6). Let ̺ P R be arbitrary. Using the renormalisation nd ψ " 1 in (1.4), we can proceed similarly to the previous proof of the entropy law (1.5), now with the additional information that ř I i"1 ̺θ 1 M p¨¨¨qq i f i puq " 0, to obtain For ̺ ą 0, we divide both sides by ̺ and let ̺ Ñ`8 to get Repeating the argument for ̺ ă 0, we obtain the reverse inequality and, therefore, the desired conservation law (1.6).

Exponential convergence to equilibrium
3.1. Entropy-entropy dissipation inequality. Without loss of generality, we assume that the domain Ω has unit volume, i.e. |Ω| " 1. We employ the following notation: ż Ω u i dx and u :" pu i q i"1,...,I .
To show the convergence to equilibrium for (1.9), we exploit the so-called entropy method. Assuming the complex balanced condition, system (1.9) possesses the relative entropy functional and the corresponding entropy dissipation function Drus " with Ψpx; yq " x logpx{yq´x`y (see [6, Proposition 2.1] for a derivation of Drus). Formally we have Drus "´d dt Eru|u 8 s along the trajectory of (1.9). The following lemma shows that the L 1 norm can be bounded by the relative entropy.
Proof. The elementary inequalities x logpx{yq´x`y ě p ? x´?yq 2 ě 1 2 x´y ensure that Taking the non-negativity of the solution u into account allows to conclude.
in which C CKP ą 0 is a constant depending only on Ω and u 8 .
The following two lemmas provide generalisations of the classical logarithmic Sobolev inequality, which are suited for non-linear diffusion.  The cornerstone of the entropy method is the entropy-entropy dissipation inequality which is proved in the following lemma.
Lemma 3.5. Let K ą 0 and m i ą pd´2q`{d for all 1 ď i ď I. There exist constants C ą 0 and α ě 1 depending on K such that for any measurable function u : Ω Ñ R Ì subject to Qu " Qu 8 and Eru|u 8 s ď K, we have Proof. First, similarly to Lemma 3.1, we have the following L 1 bounds: For all i with pd´2q`{d ă m i ă 1, we apply Lemma 3.4 to have ż since m i ă 1 and u i ď M 1 . Therefore, by applying Lemma 3.3, we can estimate the entropy dissipation from below as follows: For simplicity, we denote α " max i"1,...,I t1, m i u, x.
If D i ě 1 for some i P t1, . . . , Iu or F ě 1, then we have using Eru|u 8 s ď K. It remains to consider the case D i ď 1 for all i " 1, . . . , I and F ď 1. Here, we find By applying now the entropy-entropy production estimate for the case of linear diffusion (cf. [14, Theorem 1.1]), we have and, thus, From (3.4) and (3.6), we obtain the desired estimate. If α " 1, we immediately get the exponential convergence It follows from (3.7) that ψptq`Cϕpsq ď ψpsq.
Applying a non-linear Gronwall inequality yields thanks to ψptq`Cϕp0q ď ψp0q and α ą 1. Setting s " t and using ϕptq " 0 leads us to the desired estimate (3.8). Owing to the Csiszár-Kullback-Pinsker inequality, it follows Therefore, there exists an explicit T 0 ą 0 such that 8 for all t ě T 0 and, thus, Using this property, we estimate Druptqs for t ě T 0 as follows. If m i ă 1, then it is similar to and for m i ě 1, ż since u i ptq ě 1 2 u i,8 for all t ě T 0 . Therefore, for all t ě T 0 , we can estimate Druptqs as Since T 0 can be explicitly computed, we in fact get the exponential convergence for all t ě 0, i.e.
Eruptq|u 8 s ď Ce´λ t Eru 0 |u 8 s for all t ě 0, for some constants C ą 0 and λ ą 0. Thanks to the Csiszár-Kullback-Pinsker inequality, we finally obtain the desired exponential convergence to equilibrium.

3.3.
Convergence of weak and bounded solutions. We consider the approximate u ε to the regularised system (2.1). From Lemma 2.4, we know that, up to a subsequence, The next lemma establishes uniform-in-ε a priori bounds. Proof of Theorem 1.5. By testing (2.1) with smooth test functions ψ P C 8 pr0, T s; C 8 0 pΩqq satisfying ψp¨, T q " 0, we havé ż It follows from (3.11) that pu ε i q m i Ñ u m i i a.e. in Q T . Due to the bound (3.12), it holds that the set tpu ε i q m i u εą0 is uniformly integrable in Q T . Thus, thanks to Vitali's lemma, we deduce that pu ε i q m i converges strongly to u m i i in L 1 pQ T q, which implies ż From the definition of f ε i in (2.1) as well as (3.11), we have f ε i pu ε q Ñ f i puq a.e. in Q T . (3.14) It remains to show that the set tf ε i pu ε qu εą0 is uniformly integrable. Indeed, let K Ă Q T be a measurable and compact set with its measure being denoted by |K|. From (1.17) and (1.15), we have ż It is easy to see that for any δ ą 0, there exists some C δ ą 0 such that for all x ě 0, It follows from (3.15) and Lemma 3.6 that ż Therefore, for any τ ą 0, we first choose δ ą 0 fixed such that Cδ ă τ {2 and then K Ă Q T arbitrarily according to C δ |K| ă τ {2 to ultimately obtain ż K |f ε i pu ε q| dx dt ď τ, which is exactly the uniform integrability of tf ε i pu ε qu εą0 . From this and (3.14), we can apply Vitali's lemma again to get f ε i pu ε q Ñ f i puq strongly in L 1 pQ T q and, therefore, ż This gives us the existence of a global weak solution. The exponential convergence to equilibrium in L 1 pΩq follows from Theorem 1.4.
Concerning bounded solutions, we first obtain from Lemma 3.6 that }u i } L m i`1pQ T q ď C T for all i " 1, . . . , I.
Under the assumptions (1.16) on the porous medium exponent, we can apply [11, Lemma 2.2] to get }u i } L 8 pQ T q ď C T for all i " 1, . . . , I.
Note that the constant C T grows at most polynomially in T . This implies the exponential convergence to equilibrium in L p pΩq for any 1 ď p ă 8 in (1.18). Indeed, from the exponential convergence in L 1 pΩq, we can use the interpolation inequality to have T`C e´λ T˘1 {p ď C p e´λ pT for some 0 ă λ p ă λ, since C T grows at most polynomially in T .
3.4. Uniform convergence of approximate solutions. We remark that all the involved constants in the following results do not depend on ε.
Note that on Ω 1 we have Therefore, there exists a constant K 8 ą 0 such that By using Taylor's expansion, we have for all x P Ω 1 , for some K 9 ą 0. From (3.20) and (3.21), we can estimate the second sum on the left hand side of (3.18) as On the other hand, the first sum on the left hand side of (3.18) is estimated as (3.23) By combining (3.22) and (3.23), we have for any θ P p0, 1s, LHS of (3.18) ěˆθ (3.24) Thus, by choosing θ " min , we finally obtain the desired inequality (3.18) with L 2 " .
Lemma 3.9. There exists a constant L 3 ą 0 such that Proof. The proof of this lemma follows by the same arguments as in [14, Lemma 2.7], so we omit it here.
We are now ready to prove Theorem 1.6.
Proof of Theorem 1.6. We proceed similarly to the proof of Lemma 3.5. If pd´2q`{d ă m i ă 1, we have, thanks to Lemma 3.4 and the bound (3.16), ż with a constant C " CpΩ, m i q ą 0. If m i ě 1, we apply Lemma 3.3 to get ż Therefore, by using Ψpx, yq " x logpx{yq´x`y ě`?x´?y˘2 and noting that ż there exists a constant K 10 ą 0 such that Dru ε s ě K 10 (3.25) We define for some K 11 " K 11 pK 10 , αq. Thanks to Lemmas 3.8 and 3.9, we have Eru ε |u 8 s. The rest of this proof follows exactly from that of Theorem 1.4, which helps us to eventually obtain the exponential convergence to equilibrium where C, λ ą 0 are independent of ε.
Appendix A. Proof of the global existence of approximate solutions In this appendix, we provide a proof for the existence of a global solution to the approximate system (2.1). While a similar proof, with Dirichlet instead of Neumann boundary conditions, can be found in [20, Proof of Lemma 2.3], we mainly follow a standard Galerkin approach as presented in many textbooks.
For 0 ă δ ď 1 and 1 ď i ď I, we consider the regularised system with initial data u ε,δ i px, 0q " u ε i,0 pxq`δ, where we extend f ε i to a locally Lipschitz continuous function on R I by setting f ε i pvq :" f ε i pmaxtv 1 , 0u, . . . , maxtv I , 0uq. For the sake of readability, we subsequently write w i :" u ε,δ i and H δ pw i q :" m i |w i | m i´1 1`δm i |w i | m i´1`δ . We employ the ansatz w k i px, tq " ř k j"1 ξ j,k i ptqe j pxq with k P N, scalar coefficient functions ξ j,k i ptq, and the Schauder basis e j P H 1 pΩq of L 2 pΩq given by the orthonormal eigenfunctions of the Laplace operator satisfying ν¨∇e j " 0 on BΩ. Along with the initial condition ξ j,k i p0q " ş Ω pw 0 q i e j dx, the functions ξ j,k i , 1 ď j ď k, are solutions to the set of equations ż for 1 ď j ď k, which can be recast into an ODE system for the time-dependent coefficient vector ξ k i ptq :" pξ 1,k i ptq, . . . , ξ k,k i ptqq. As the right hand side continuously depends onξ k i ptq via the bounded and continuous functions H δ and f ε i , we know that a solutionξ k i ptq P C 1 pr0, τ s, R I q exists for some τ ą 0. The energy estimate below, which is essentially obtained by multiplying (A.2) with ξ j,k i ptq and summing over j, will indeed show that the solution exists for all times t ě 0. Integrating (A.2) over r0, ts Ă r0, τ q, we have where C ą 0 is independent of k by the uniform bounds H δ pw k i q ě δ and f ε i pw k i q ď ε´1. In order to control the H 1 pΩq norm of w k i , we first observe that e 1 is constant in space. Hence, (A.2) leads for all t P p0, τ q tǒˇˇˇż Ω w k i px, tq dxˇˇˇˇ"ˇˇˇˇż Poincaré's inequality now yields }w k i ptq} 2 H 1 pΩq ď Cˆ}∇w k i ptq} 2 L 2 pΩq`} w k i p0q} 2 L 2 pΩq`| Ω| 2 τ 2 ε 2˙.
As a result, we arrive at }w k i ptq} 2 where C ą 0 is independent of k, and which shows that the solution is bounded on r0, τ q and, thus, existing for all t ě 0. Note that we applied the elementary estimate }w k i p0q} L 2 pΩq ď }w i p0q} L 2 pΩq in the last step.
Let P k : L 2 pΩq Ñ L 2 pΩq be the orthogonal projection in L 2 pΩq onto spante 1 , . . . , e k u, and let φ P L 2 p0, T ; H 1 pΩqq for some arbitrary but fixed T ą 0. We recall the uniform bounds H δ pw k i q ď δ´1`δ and f ε i pw k i q ď ε´1 leading to }∇w k i ptq} L 2 pΩq }∇pP k φqptq} L 2 pΩq dt`C ż T 0 }pP k φqptq} L 2 pΩq dt ď Cp1`}w k i } L 2 p0,T ;H 1 pΩqq q}φ} L 2 p0,T ;H 1 pΩqq with a constant C ą 0 independent of k. We stress that the last estimate also employs the bound }pP k φqptq} H 1 pΩq ď }φptq} H 1 pΩq for a.e. t P r0, T s, which provides a uniform-in-k bound for B t w k i P L 2 p0, T ; pH 1 pΩqq 1 q together with (A.3). We may choose a (not relabelled) subsequence w k i being weakly converging for k Ñ 8 to some w i in L 2 p0, T ; H 1 pΩqq and in H 1 p0, T ; pH 1 pΩqq 1 q. Strong convergence w k i Ñ w i in L 2 p0, T ; L 2 pΩqq -L 2 pQ T q now follows by means of the Aubin-Lions Lemma. Choosing another subsequence, we obtain for all i " 1, . . . I, w k i Ñ w i and, hence, H δ pw k i q Ñ H δ pw i q and f ε i pw k i q Ñ f ε i pw i q a.e. in Ω due to the continuity of H δ and f ε i . The boundedness of H δ and f ε i further guarantees H δ pw k i q Ñ H δ pw i q and f ε i pw k i q Ñ f ε i pw i q strongly in L 2 pQ T q. This allows us to pass to the limit k Ñ 8 in an integrated version of (A.2) resulting in ż for all φ P C 8 pΩˆr0, T sq. By a density argument, (A.1) holds as an identity in L 2 p0, T ; pH 1 pΩqq 1 q.
Thanks to the quasi-positivity assumption (F2), the solution w i is also non-negative. For p ą 1, we have Applying the Gronwall lemma y 1 ď αptqy 1´1{p ñ yptq ď´yp0q 1{p`1 p ş t 0 αpsqds¯p for yptq " }w i ptq} p L p pΩq entails }w i ptq} p L p pΩq ďˆ}w i p0q} L p pΩq`ż t 0 }f ε i pwpsqq} L p pΩq ds˙p ď´}w i p0q} L p pΩq`} f ε i pwq} L p pQ T q T p´1 p¯p .
Using (B.2) and the above bound on y, it follows that ż Q T˜I From the definition of w ε i and z ε i , it is easy to see that for any δ ą 0, there exists some C δ ą 0 such that We then derive from (B.4) that ż The desired estimate (3.12) is now an immediate consequence.