Large-time asymptotics for degenerate cross-diffusion population models with volume filling

The large-time asymptotics of the solutions to a class of degenerate parabolic cross-diffusion systems is analyzed. The equations model the interaction of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. Compared to previous works, we allow for different diffusivities and degenerate nonlinearities. The proof is based on the relative entropy method, but in contrast to usual arguments, the relative entropy and entropy production are not directly related by a logarithmic Sobolev inequality. The key idea is to apply convex Sobolev inequalities to modified entropy densities including"iterated degenerate"functions.


Introduction
The aim of this note is to extend the large-time asymptotics result of [19] on multispecies cross-diffusion systems with volume-filling effects to the degenerate case. Such systems describe, for instance, the spatial segregation of population species [16], chemotactic cell migration in tissues [15], motility of biological cells [18], or ion transport in fluid mixtures [4]. The main difficulties of the cross-diffusion systems are the lack of positive semidefiniteness of the diffusion matrix and the nonstandard degeneracies. The first issue was overcome by applying the boundedness-by-entropy method [13], which exploits the underlying entropy (or formal gradient-flow) structure. This allows for both a global existence analysis and the proof of lower and upper bounds, without the use of a maximum principle. The second issue was handled by extending the Aubin-Lions compactness lemma [19]. However, the large-time asymptotics in [19] only holds if the problem is not degenerate. In the present note, we remove this restriction.
The evolution of the volume fraction u i (x, t) of the ith species is given by A ij (u)∇u j in Ω, t > 0, i = 1, . . . , n (1) n j=1 A ij (u)∇u j · ν = 0 on ∂Ω, u i (·, 0) = u 0 i in Ω, (2) where u 0 = 1 − n i=1 u i is the solvent volume fraction or the proportion of unoccupied space (depending on the application), Ω ⊂ R d (d ≥ 1) is a bounded domain with Lipschitz boundary, ν is the exterior unit normal vector to ∂Ω, and the diffusion coefficients are given by where i, j = 1, . . . , n, u = (u 1 , . . . , u n ) is the solution vector, D i > 0 are the diffusivities, δ ij denotes the Kronecker symbol, and p i and q are smooth functions. In particular, the bounds 0 ≤ u i ≤ 1 should hold for all i = 0, . . . , n. The boundary condition in (2) means that the physical or biological system is isolated. We note that equations (1) and (3) can be written as In some applications, drift or reaction terms need to be added; see, e.g., [3,9] for systems with drift terms and [6] for reaction rates.
Equations (1) and (3) can be formally derived from a random-walk lattice model in the diffusion limit [19,Appendix A]. The functions p i and q are related to the transition rates of the lattice model with p i measuring the occupancy and q measuring the non-occupancy. This class of systems contains the population model of Shigesada, Kawasaki, and Teramoto [16] (if p i is a linear function and q = 1) and Nernst-Planck-type equations accounting for finite ion sizes (if p i = 1 and q(u 0 ) = u 0 ; see [9]). In this note, we consider the degenerate case q ′ (0) = 0 and assume that there exists a smooth function χ such that p i = exp(∂χ/∂u i ) to guarantee an entropy structure via the entropy density where u ∈ D := {u ∈ (0, 1) n : n i=1 u i < 1}. There exist other approaches to model volume filling. The finite particle size may be taken into account by adding cross-diffusion terms of the type u i ∇ n j=1 b ij u j to the standard Nernst-Planck flux [11] or by using the Bikerman- The global existence of bounded weak solutions to (1)-(3) was shown in [19, Theorem 1] assuming D i = 1 for i = 1, . . . , n and the following conditions: and Ω T = Ω × (0, T ). (H2) Initial datum: u 0 (x) ∈ D for a.e. x ∈ Ω and h(u 0 ) ∈ L 1 (Ω).
q ′ (s) > 0 for all 0 < s ≤ 1. The convexity of Ω in Hypothesis (H1) is used for the convex Sobolev inequality; see Lemma 2 below. For generalized Nernst-Planck systems with p i = const., we may choose χ(u) = n i=1 u i , which satisfies Hypothesis (H3). Moreover, if p i (u) = P i (u i ) for some func- The functions q(s) = s α with α ≥ 1 satisfy Hypothesis (H4). We claim that the existence result also holds for arbitrary D i > 0. Indeed, it is sufficient to define χ(u) = χ(u) + n j=1 u j log D j , since exp(∂ χ/∂u i ) = D i exp(∂χ/∂u i ) = D i p i , and we can apply Theorem 1 in [19] with χ. We observe that the condition q ′ (s)/q(s) ≥ c 1 > 0 in [19] is not needed for the existence analysis.
Our main result is the convergence of the solutions to (1)-(3) towards the constant steady state for large times under the following additional hypothesis: (H5) q is convex, q/q ′ is concave, and there exist β ∈ [0, 1], c 1 > 0 such that Examples of functions satisfying Hypothesis (H5) are q(s) = s α with α ≥ 1. The convergence (with exponential decay rate) was proved in [19] for the nondegenerate case q ′ (0) > 0 only. In the degenerate situation q ′ (0) = 0, the numerical results of [9] indicate that exponential rates cannot be expected. Therefore, we show the convergence without rate.
The idea of the proof is to exploit, as in [19], the relative entropy density (or Bregman distance) where u = (u 1 , . . . , u n ) is the weak solution to (1)-(3). The entropy inequality implies that Unfortunately, the entropy production integral cannot be estimated in terms of the relative entropy directly by applying a logarithmic Sobolev inequality to u i . We overcome this issue by using two ideas. First, we apply the logarithmic Sobolev inequality to q(u 0 )u i , The idea is to relate the integrand of the left-hand side to the relative entropy part For this, we define The key result is the limit (see Lemma 8) lim )ds of the relative entropy density, we analyze the function whereq := |Ω| −1 Ω q(u 0 )dx, which can be seen as an "iterated" version of h * 2 (u|u ∞ ), since it involves q • q instead of q. Then an application of the convex Sobolev inequality yields a bound for the integral over |∇ q(u 0 )| 2 without the need of condition q ′ (0) > 0; see Remark 6 for details. It follows from The convergences f 1 (u(t k )) → 0 and f 2 (u(t k )) → 0 as well as the monotonicity of the entropy imply that h * (u(t k )|u ∞ ) → 0 pointwise. The monotonicity of t → Ω h * (u(t)|u ∞ )dx then implies the convergence for all sequences t → ∞ and finally To conclude the introduction, we mention some results on the large-time asymptotics for diffusion systems. Exponential equilibration rates in L p (Ω) norms were shown for reactiondiffusion systems in [8,7], for electro-reaction-diffusion systems in [10], and for Maxwell-Stefan systems for chemically reacting fluids in [6,14]. The convergence to equilibrium was proved for Shigesada-Kawasaki-Teramoto cross-diffusion systems without rate in [17], for instance. All these results concern nondegenerate diffusion equations. The work [3] is concerned with the large-time asymptotics for systems like (1) with D i = p i = 1 and q(u 0 ) = u 0 without rate. The asymptotics for solutions to Poisson-Nernst-Planck-type equations with quadratic nonlinearity was investigated in [20] using Wasserstein techniques. Decay rates for degenerate diffusion systems without cross-diffusion terms were derived in [5]. An extension of our results to cross-diffusion systems with drift or reactions seems delicate; see Remark 9 for drift terms and [6] for cross-diffusion systems with reversible reactions.

Proof of Theorem 1
We first recall the convex Sobolev inequality; see [19,Lemma 11].

Lemma 2 (Convex Sobolev inequality).
Let Ω ⊂ R d (d ≥ 1) be a convex domain and let g ∈ C 4 (R) be convex such that 1/g ′′ is concave. Then there exists C S > 0 such that for all v ∈ L 1 (Ω) such that g(v), g ′′ (v)|∇u| 2 ∈ L 1 (Ω), The logarithmic Sobolev inequality is obtained for the choice g(v) = v(log v − 1) + 1: Since h(u ∞ ) is independent of time (because of mass conservation), the entropy inequality (6) implies the relative entropy inequality ≤ Ω h * (u(s)|u ∞ )dx for 0 ≤ s < t and t > 0, where h * (u|u ∞ ) is defined in (7). As mentioned in the introduction, we cannot apply the logarithmic Sobolev inequality (8) with v = u i since q(u 0 ) = 0 for u 0 = 0. Instead we apply this inequality to v = q(u 0 )u i .
We split the relative entropy density h * into three parts, where χ is introduced in Hypothesis (H3).
2.1. Study of some auxiliary functions. The study of the large-time behavior is based on the analysis of the two functions for u ∈ D, where Proof. Set z = q(u 0 )u i /q i and let u ∈ D. Then proving the first claim. To show the nonnegativity of f 2 , we distinguish two cases. If q(u 0 (x, t)) ≥q at some (x, t) ∈ Ω T , then log(q(s)/q(q)) ≥ 0 for anyq ≤ s ≤ q(u 0 (x, t)) and consequently f 2 (u(x, t)) ≥ 0. If q(u 0 (x, t)) <q, we have log(q(s)/q(q)) < 0 for q(u 0 (x, t)) ≤ s ≤q and f 2 (u 0 (x, t)) = q q(u 0 (x,t)) log(q(q)/q(s))ds ≥ 0. It remains to show that f 2 is bounded. Since q is convex, Jensen's inequality shows that q ≥ q(|Ω| −1 Ω u 0 dx) = q(u ∞ 0 ). Then, using integration by parts and arguing as in (10), We already showed above that the last integral is bounded. This finishes the proof.
x ∈ Ω, s ∈ (0, 1], which finishes the proof. Remark 6. In the nondegenerate case q ′ (0) > 0, it was shown in [19, Section 5] that t → h * 2 (u(t)|u ∞ ) converges to zero exponentially fast. Indeed, applying the convex Sobolev inequality similarly as in the previous proof, and we conclude from the entropy inequality (6) and Gronwall's lemma. Since we allow for q ′ (0) = 0, this argument cannot be used here. We solve this issue by considering the "iterated" function f 2 involving q • q and assuming that s → sq ′ (s)/q(s) is bounded; see (14). The iterated use of q gives the term |∇ q(u 0 )| 2 in (14) without requiring the nondegeneracy condition q ′ (0) > 0.
A consequence of the limit for f 2 is the following result.

Lemma 7.
If lim N →∞ f 2 (u 0 (x, s + N)) = 0 for some x ∈ Ω, s ∈ (0, 1] then Proof. We write u N i := u i (x, s + N) andq N =q(s + N) to simplify the notation. We recall from Lemma 4 that f 2 is nonnegative and change the variable σ = s/q N in the integral: where we used Jensen's inequality to find thatq N ≥ q(|Ω| −1 This shows that lim N →∞ f 2 (u N 0 ) = 0 if and only if Set A := {(x, s) ∈ Ω × (0, 1] : lim N →∞ f 2 (u 0 (x, s + N)) = 0}. We want to show that lim N →∞ q(u N 0 )/q N = 1 for (x, s) ∈ A. If not, there exist (x 0 , s 0 ) ∈ A and ε 0 > 0 such that either In the former case, we have q(q N σ) ≥ q(q N (1 + ε 0 /2)) for σ ≥ 1 + ε 0 /2, since q is increasing, and therefore, Using the convexity of q, a Taylor expansion shows that q(q N +q N ε 0 /2) ≥ q(q N ) + q ′ (q N )q N ε 0 /2. Then the integrand of the previous integral can be estimated according to where we used Hypothesis (H5) andq N ≥ q(u ∞ 0 ) in the last step, and c 0 > 0 is some constant. As the right-hand side is independent of σ, we infer from (17) that In the latter case q(u N 0 )/q N < 1 − ε 0 , we estimate as We apply again a Taylor expansion, similarly as in the first case, Thus, in both cases, which contradicts (16) and consequently lim N →∞ f 2 (u N 0 ) = 0.

Key lemma.
We show that f 1 (u(·, s + N))/q(s + N) and h * 1 (u(·, s + N)|u ∞ ) are close for sufficiently large N ∈ N. The following lemma is the key of the proof.
Lemma 8. For a.e. x ∈ Ω, s ∈ (0, 1], it holds that Proof. We set u N := u(·, s + N),q N =q(s + N), andq N i = |Ω| −1 Ω q(u N 0 )u N i dx. Inserting definition (11) of f 1 , the lemma is proved if we can show that for any i = 1, . . . , n, Fix i ∈ {1, . . . , n}. We know from Lemmas 5 and 7 that lim N →∞ q(u N 0 )/q N = 1 a.e. Together with the boundedness of u N i , this shows that To show that the limit on the right-hand side equals zero, we observe that, because of mass conservation and dominated convergence, Putting together the previous limits, we have proved (18 in Ω × (0, 1] as N → ∞ for i = 1, . . . , n. We deduce from the continuity of χ that also lim N →∞ h * 3 (u N |u ∞ ) = 0. For the limit of h * 2 , we observe that Since the integrand is a function in L 1 (1/2, 3/2), it follows from the absolute continuity of the integral that lim N →∞ h * 2 (u N |u ∞ ) = 0 a.e. in Ω × (0, 1]. By definition of h * , we have proved that lim N →∞ h * (u N |u ∞ ) = 0.
Using [12,Lemma 16] again, we have The convergence in L p (Ω) for any p < ∞ then follows from the uniform bound for (u i (t)) t>0 , finishing the proof.
Remark 9 (Drift terms). Equations (4) with drift terms read as where Φ i = Φ i (x) are given (electric or environmental) potentials. Adding the associated energy to the entropy density (5), we can compute (formally) the entropy inequality, giving It was shown in [19, Section 3.2] that the entropy production term with Φ i = 0 can be bounded from below by p i (u)(q(u 0 ) n i=1 |∇ √ u i | 2 + |∇ q(u 0 )| 2 ). Such an estimate seems to be impossible in the presence of ∇Φ i . Indeed, the entropy inequality shows that Thus, in the special case q(0) > 0 and if Φ i is bounded from above, we conclude the existence of a subsequence t k → ∞ such that ∇(u i p i (u)e Φ i /q(u 0 )) 1/2 (t k ) → 0 strongly in L 2 (Ω) as k → ∞, and one may proceed similarly as in [2,Section 5]. However, the condition q(0) = 0 is needed to model correctly the transition rate of nonoccupied cells in the lattice model [3,19].