Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit

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Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit

Elisabetta Brocchieri, Lucilla Corrias, Helge Dietert, Yong-Jung Kim

To cite this version:

Elisabetta Brocchieri, Lucilla Corrias, Helge Dietert, Yong-Jung Kim. Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit. Journal of Mathematical Biology, Springer Verlag (Germany), 2021, �10.1007/s00285-021-01679-y�. �hal-03013229v2�

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Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit

E. Brocchieri¹, L. Corrias¹, H. Dietert², Y.-J. Kim³

11th November 2021

Abstract

We rigorously prove the passage from a Lotka-Volterra reaction- diffusion system towards a cross-diffusion system at the fast reaction limit. The system models a competition of two species, where one species has a more diverse diet than the other. The resulting limit gives a cross-diffusion system of a starvation driven type. We investigate the linear stability of homogeneous equilibria of those systems and rule out the possibility of cross-diffusion induced instability (Turing instability). Numerical simulations are included which are compatible with the theoretical results.

Keywords.Cross-diffusion, starvation-driven diffusion, entropy, Turing instability.

2010 Mathematics Subject Classification. Primary : 35B25, 35B40, 35K57, 35Q92, 92D25. Secondary 35B45, 35K45

1 Introduction

1.1 Problem setting

We consider a semilinear reaction-diffusion system that models a competition dynamics when two species have partially different diets. The population densities of the two species are denoted by u =u(t, x) and v = v(t, x). The species uhas a more diverse diet and is divided into two substatesua=ua(t, x) andub=ub(t, x) so thatu=ua+ub. The system is parametrized by a small parameter ε >0 and written as











∂tu^ε_a=da∆u^ε_a+fa(u^ε_a) +1

εQ(u^ε_a, u^ε_b, v^ε), in (0,+∞)×Ω,

∂tu^ε_b =db∆u^ε_b+fb(u^ε_b, v^ε)−1

εQ(u^ε_a, u^ε_b, v^ε), in (0,+∞)×Ω,

∂tv^ε=dv∆v^ε+fv(u^ε_b, v^ε), in (0,+∞)×Ω,

(1.1)

where Ω⊂R^N, N≥1, is a bounded domain with a smooth boundary, and da, db

and dv are diffusivities for the three populations. The unknown solutions depend

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on the parameter ε and we denote it explicitly if needed. The above system is complemented with nonnegative initial data

u^ε_a(0, x) =uⁱⁿ_a(x), u^ε_b(0, x) =uⁱⁿ_b(x), v^ε(0, x) =vⁱⁿ(x), x∈Ω, (1.2) and zero flux boundary conditions,

da∇u^ε_a·σ=db∇u^ε_b·σ=dv∇v^ε·σ= 0, on (0,+∞)× ∂Ω, (1.3) whereσdenotes the outward unit normal vector on the boundary∂Ω.

In this paper, we explore the effect of diet diversity in a competition context and show the emergence of cross-diffusion triggered by the different substates ua

and ub, as ε→ 0. The competition dynamics is given in the reaction terms. The reaction terms of order one are given by

fa(ua) :=ηaua

1−ua

a , fb(ub, v) :=ηbub

1−ub+v b

, (1.4)

fv(ub, v) :=ηvv

1−ub+v b

,

wherea, b >0 are carrying capacities supported by two different groups of resources andηa, ηb, and ηv >0 are the intrinsic growth rates ofua, ub, andv, respectively.

The competition of the two species, u and v, is for the resource b. However, the species u has a diverse diet and can survive by consuming the other resource a without competition. To model such a competition using a Lotka-Volterra type system, the speciesu is divided into two substatesua and ub depending on their diets. In the above reaction terms,uatakes a logistic equation type reaction, andub

andv take Lotka-Volterra competition equations type reactions as given in (1.4).

Since competition exists only partially for the species u, the competition is weak foru. However, the speciesvcompetes withufor all of its resources and hence the competition is not weak in general and the competition result may depend on the parameterε(see Sections4 and5).

The individuals of the speciesumay freely change the type of food depending on the availability, which is modelled by the fast reaction term of orderε⁻¹,

1

εQ(ua, ub, v) := 1 ε

φub+v b

ub−ψua

a ua

, ε >0. (1.5) In this reaction term,ǫ⁻¹φub+v

b

is the conversion rate for individuals in the stateub which switch to the other stateua, and ǫ⁻¹ψua

a

is the conversion rate in the other direction. The conversion rateφ ^u^b_b^+v

is assumed as a function of the starvation measure ^u^b_b^+v for the populationsubandv. If the resourcebdwindles or the populationub+vincreases, the resourceb becomes scarce relatively, and more individuals of populationub will convert to ua and consume the other resourcea.

Hence, we assume thatφis an increasing function of the starvation measure (see [22]

for more discussion on the starvation measure). In the same way, the conversion rate ψ is a function of the starvation measure ^u_a^a for the population ua and is assumed to be increasing. For this reason, it makes sense to call the conversion dynamics given by (1.5) a starvation-driven conversion, which eventually results in the starvation-driven cross-diffusion after taking the limitε→0 (see [8,9]). More specifically, we assume the following starvation-driven conversion hypothesis

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(H1) φand ψin (1.5) are increasing functions belonging toC¹([0,+∞)); in addition, there exist strictly positive constantsδψ, δφ, Mφ^′, andMψ^′ such that, for allx≥0,

ψ(x)≥δψ>0, φ(x)≥δφ>0, φ^′(x)≤Mφ^′, and ψ^′(x)≤Mψ^′. The main result of the paper is that, asε→0, the (unique) solution (u^ε_a, u^ε_b, v^ε) of the initial boundary value problem (1.1)–(1.5) converges to a limit (ua, ub, v) and this limit is a weak solution of the reaction cross-diffusion system

(∂tu= ∆(daua+dbub) +fa(ua) +fb(ub, v), in (0,+∞)×Ω,

∂tv=dv∆v+fv(ub, v), in (0,+∞)×Ω, (1.6) whereua andub are (uniquely) determined by the nonlinear system

ua+ub=u and Q(ua, ub, v) = 0, (1.7) complemented by the initial data,

u(0, x) =uⁱⁿ(x) :=uⁱⁿ_a(x) +uⁱⁿ_b(x), v(0, x) =vⁱⁿ(x), x∈Ω, (1.8) and the zero flux boundary condition,

∇(daua+dbub)·σ=dv∇v·σ= 0, in (0,+∞)×∂Ω. (1.9) Note that the zero flux boundary conditions in (1.3) are equivalent to the homogeneous Neumann boundary conditions,

∇u^ε_a·σ=∇u^ε_b·σ=∇v^ε·σ= 0, on (0,+∞)×∂Ω

(see [20] for similar diffusion operator for a single species with two phenotypes).

However, after taking the singular limit, we obtain the zero flux boundary conditions (1.9), but not the homogeneous Neumann boundary conditions.

If da = db, the diffusion for the species ugiven in (1.6) is the homogeneous linear diffusion. However, the diffusivity of a species usually depends on its food (or prey) andda 6=db in general. In that case (da 6=db), the diffusion for the total population in (1.6) contains cross-diffusion dynamics depending on the distribution of the three populations groups,ua, ub andv, through the relations in (1.7). This explains the starvation-driven diffusion for the specific case of the paper, a concept formally introduced by Cho and Kim [7]. Funaki et al.[18] derived a macroscopic cross-diffusion model from a system of two phenotypes and a signaling chemical in the context of chemotaxis.

The proof of the convergence as ε → 0 is rigorously obtained via a priori estimates foru^ε_a, u^ε_b,andv^ε. The main tool is the energy (or entropy) functional

E(ua, ub, v) :=

Z

Ω

h1(ua)dx+ Z

Ω

h2(ub, v)dx , (1.10) where

h1(ua) :=

Z ua

0

ψz a

z dz, and h2(ub, v) :=

Z u_b 0

φz+v b

z dz . (1.11)

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Notice here that the assumption (H1) implies thath1 is positive, increasing, and convex, and thath2is positive, increasing in both variables, and convex with respect to the first variable. Therefore, the name entropy for the function given in (1.10) is justified. We refer to [10] and [15] for the use of such entropies in the context of triangular cross-diffusion systems (that is, systems in which only one of the two equations includes a cross-diffusion term). For more general systems, we refer to [5,6,12,14,21,2] among other works.

Then, by invoking the Aubin-Lions Lemma, we pass to the limit along a subsequence and conclude that the limit is a weak solution of (1.6)–(1.9). To use the energy estimate, we take initial values with bounded energy, which is our second hypothesis

(H2) uⁱⁿ_a ∈L¹₊(Ω), uⁱⁿ_b ∈L¹₊(Ω),vⁱⁿ∈L^∞₊(Ω), andE(uⁱⁿ_a, uⁱⁿ_b , vⁱⁿ)<∞.

Remark 1. Under Hypothesis (H2), the initial datauⁱⁿ_a, uⁱⁿ_b , vⁱⁿfor the reaction diffusion system(1.1)do not satisfy a priori the nonlinear equationQ(uⁱⁿ_a , uⁱⁿ_b , vⁱⁿ) = 0 in (1.7). Thus, the appearance of an initial layer is expected (see also Section5).

We conclude this introduction proposing a formal derivation of (1.1) out of a microscopic system. We shall consider problems left open here (such as regularity, uniqueness, stability and long time asymptotic behaviour of the macroscopic solutions) in a forthcoming paper, where a more general class of cross-diffusion system is analysed.

The rest of the paper is organised as follows.Section2is devoted to the statement of the existence result. InSection 3.1, we prove a priori estimates, which are the preliminary ingredients for the proof of the existence result obtained in Sec- tion 3.2. The paper concludes with the existence and linear stability analysis of trivial and non-trivial spatially homogeneous steady states, inSection 4 and Ap- pendix A, with a particular emphasis put on the coexistence state. Some numerical tests inSection5illustrate the linear stability analysis. The discussion inSection6 completes the article.

1.2 Formal derivation of the reaction-diffusion system with fast switching

We explain here how the mesoscopic scale model (1.1) is obtained at a formal level from a microscopic scale model in which the resources inducing the competition explicitly appear. Consider











∂ts1= 1 δ

hr1s1

1− s1

A1

−p1s1U1

i

∂ts2= 1 δ

hr2s2

1− s2

A2

−p2s2U2−pVs2Vi

∂tU1=D1∆U1+k1p1s1U1+1 ε

hΦp2U2+pVV s2

U2−Ψp1U1

s1

U1

i

∂tU2=D2∆U2+k2p2s2U2−1 ε

hΦp2U2+pVV s2

U2−Ψp1U1

s1

U1

i

∂tV =DV∆V +kVpVs2V,

(1.12)

whereδ >0 is the microscopic reaction time scale andεis the mesoscopic one (hence δ≪ε≪1). These equations describe the time evolution of a small ecosystem with

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two prey population densities (or vegetal resources),s1 and s2, and two predator population densities (or harvesters of the vegetal resources),UandV. Moreover, the populationU is composed of two subpopulationsU1andU2depending on the prey they consume, i.e., s1 ands2, respectively. The prey speciessi follows the logistic dynamics with a carrying capacityAiand an intrinsic growth rateri. The predator species consume a certain amount of preys which is proportional to the prey density with proportionality factorsp1, p2andpV. The harvested prey mass is converted to the predator mass with conversion ratesk1, k2 andkV. The subpopulationsU1and U2convert to each other depending on the availability of the prey. The two functions Φ and Ψ are the conversion rates which are respectively increasing functions of the starvation measures ^p²^U²_s^+p₂ ^V^V and ^p¹_s^U₁¹. The other species V consumes only the second preys2. Hence, the active competition is only betweenV andU2, whileU1

competes with V passively (via conversion). Finally, since the dispersal rate of a predator species usually depends on the nature of its prey,D16=D2in general.

Remark 2. The expression (1.12) has no diffusion terms for the prey species s1

and s2. Since growth is the dominant factor for plant species and their dispersal is negligible, this is especially relevant when the prey species are vegetal resources.

Mathematically, this choice yields an explicit form in the singular limitsδ→0, see (1.13). Adding diffusion terms in the prey species equations could result in a less explicit formulas.

The mescoscopic system with fixedε >0 is obtained in the limitδ→0. This is to say that the time-scale for the reaction of the resourcess1ands2is much faster than all other processes. In simple predator-prey models this corresponds to a fast dynamics of the prey which has been studied more carefully in [30,24,28].

In this formal limit δ→0, we find s1

r1−r1s1

A1 −p1U1

= 0 =⇒ s1= 0 or s1=A1

1−p1U1

r1

, and

s2

hr2 1− s2

A2

−p2U2−pVVi

= 0 =⇒ s2= 0 or s2=A2

1−p2U2+pVV r2

. Only the nontrivial case, s1 6= 0 6= s2, is meaningful (since s1 = 0 and s2 = 0 correspond to unstable equilibria), and we obtain two relations

p1U1

s1

=r1

s1 − r1

A1

and p2U2+pVV s2

=r2

s2 − r2

A2

. Therefore, the last three equations in (1.12) turn into











∂tU1=D1∆U1+A1k1p1U1 1−^p¹_r^U1¹

+1 ε

ΦU2−ΨU1

∂tU2=D2∆U2+A2k2p2U2 1−^p²^U²r^+p2 ^V^V

−1 ε

ΦU2−ΨU1

∂tV =DV∆V +A2kVpVV 1−^p²^U²r^+p2 ^V^V

,

(1.13)

where the conversion rates Φ and Ψ read as Φ = Φr2

s2 − r2

A2

and Ψ = Ψr1

s1− r1

A1

,

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and the Lotka-Volterra reaction dynamics of competition type naturally appears.

Now we consider the relationship between the variables in (1.1) and in (1.13).

First, we define

u^ε_a :=U1, u^ε_b:=U2, v^ε:= pV

p2

V , and keep the same diffusivity coefficients

da :=D1, db:=D2, dv:=DV .

Then, the coefficients in the Lotka-Volterra type competition dynamics,fa, fb and fv, are given as

ηa:=p1A1k1, ηb:=p2A2k2, ηv :=pVA2kV, a:= r1

p1

, b:= r2

p2

. (1.14) Finally, the mesoscopic conversion rates are as follows

φ(x) := Φr2

A2

x 1−x

, ψ(x) := Ψr1

A1

x 1−x

. (1.15)

After replacing variables, coefficients and functions with the above new ones, system (1.13) becomes our system (1.1).

Remark 3. (i)The conversion rates of the microscopic model, ΦandΨ, are functions of the starvation measures ^p²Û²_s^+p₂ ^V^V and ^p¹_sÛ₁¹, instead of simply Û²_s^+V₂ and

U1

s1, in order to take into account the difference in the harvesting rates p2 andpV. (ii) The mesoscopic conversion rates φ and ψ in (1.15) are increasing functions, sinceΦandΨare chosen to be increasing functions.(iii)It is worth noticing that the carrying capacitiesaandbfor the predator species are proportional to the growth ratesri’s of the prey species and that the prey carrying capacitiesAi’s are also in- volved in deciding φ and ψ (see (1.14) and (1.15)). (iv) The macroscopic system reduces to the classical Lotka-Volterra system of competition type with linear diffusion, whenever the conversion ratesφandψare both constant, (see the discussion section6).

2 Statement of the main result

Before stating our main result in Theorem2.2below, we introduce some notations that will be used in the sequel, and the definition of the very weak solutions of (1.6)–(1.9), with the reaction terms in (1.4).

We denote

C_c^k:=C_c^k([0,+∞)×Ω)¯ :=n

u=u(t, x) :∃T >0 s.t. u∈C^k [0, T)×Ω¯

and suppu⋐[0, T)×Ω¯o , and, for allp∈[1,+∞),

L^p_loc:=L^p_loc((0,+∞)×Ω) :=n

u=u(t, x) :∀T >0, u∈L^p(ΩT)o , with ΩT := (0, T)×Ω. Similarly, forp= +∞,

L^∞_loc:=L^∞_loc((0,+∞)×Ω) :=n

u=u(t, x) :∀T >0, ess sup

(t,x)∈ΩT

|u(t, x)|<+∞o .

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It is worth noticing here that, due to hypothesis (H1), the function q(ub, u, v) :=Q(u−ub, ub, v) =φub+v

b

ub−ψu−ub

a

(u−ub), (2.1) defined for (ub, u, v)∈[0, u]×(0,+∞)×(0,+∞), satisfies (for givenu >0, v >0)

∂u_bq(ub, u, v) =φub+v b

+ub

b φ^′ub+v b

+ψu−ub

a

+u−ub

a ψ^′u−ub

a

>0 and

q(0, u, v)<0, q(u, u, v)>0.

Hence, for any given (u, v)∈R²₊, there exists a uniqueu^∗_b(u, v)∈(0, u) zero ofq, and thus a unique solution of the nonlinear system (1.7) is well-defined. Furthermore, the implicit function theorem guarantees the continuity (and even theC¹character) ofu^∗_b with respect to (u, v).

Definition 2.1. Let Ωbe a smooth bounded domain of R^N,N≥1. Assumeuⁱⁿ∈ L¹₊(Ω), and vⁱⁿ∈L^∞₊(Ω) be nonnegative initial densities. We say that the pair of nonnegative functions(u, v)is avery weak solutionof (1.6)–(1.9)over(0,+∞)× Ω, with reaction terms (1.4), if the following conditions are satisfied

• (u, v) belongs toL²_loc×L^∞_loc,

• for all test functionsξ1, ξ2∈C_c²,with∇ξ1·σ=∇ξ2·σ= 0on[0,+∞)×∂Ω, and forua, ub defined as the unique solution of (1.7), a.e. on (0,+∞)×Ω, it holds

− Z +∞

0

Z

Ω

(∂tξ1)u dxdt − Z

Ω

ξ1(0,·)uⁱⁿdx− Z +∞

0

Z

Ω

∆ξ1 daua+dbub dxdt

= Z +∞

0

Z

Ω

ξ1 fa(ua) +fb(ub, v)

dx dt , (2.2) and

− Z +∞

0

Z

Ω

(∂tξ2)v dxdt− Z

Ω

ξ2(0,·)vⁱⁿdx−dv

Z +∞

0

Z

Ω

∆ξ2v dxdt

= Z +∞

0

Z

Ω

ξ2fv(ub, v)dxdt . (2.3)

We observe that all terms in (2.2)–(2.3) are well-defined thanks to the as- sumptions (H2) on the initial densitiesuⁱⁿ, vⁱⁿ, to the L² integrability of the subpopulation densitiesua, uband to theL^∞ bound forv. Remember that the logistic structure of the reaction functionsfa, fb, fv involves at most quadratic nonlinear- ities.

Theorem 2.2. LetΩbe a smooth bounded domain of R^N,N≥1. Assume (H1) and (H2) on parameters and initial datauⁱⁿ_a , uⁱⁿ_b ,vⁱⁿ, respectively. We denote(u^ε_a, u^ε_b, v^ε) the unique global strong (for t >0) solution of system (1.1)–(1.3) with those initial data. Then, the triplet (u^ε_a, u^ε_b, v^ε) converges a.e. (t, x)∈(0,+∞)×Ω (up to extraction of a subsequence) towards a nonnegative triplet (ua, ub, v), as ε → 0.

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Moreover, the functionsu:=ua+ub,v satisfy the nonlinear system (1.7), for a.e.

(t, x) ∈ (0,+∞)×Ω, and the following bounds: u ∈ L^q(ΩT) for q = 2 + 2/N if N ≥ 3, q < 3 if N = 2 and q = 3 if N = 1; v ∈ L^∞(ΩT); |∇u| ∈ L²(ΩT); and for the same previous q, |∇v| ∈ L^2q(ΩT); ∂xi,xjv , ∂tv ∈ L^q(ΩT), i, j = 1, . . . ,N.

Finally, (u, v) is a very weak solution of the macroscopic system (1.6)–(1.9) with the reaction terms (1.4), in the sense of Definition 2.1.

3 Proof of the main Theorem

We first recall that for any ε >0, there exists a unique global strong (for t >0) solution (u^ε_a, u^ε_b, v^ε) solution to system (1.1)–(1.3), under the assumption on the initial data of Theorem 2.2. We refer for example to [13, 29] for obtaining such a result.

3.1 A priori estimates

In this section we shall obtain a priori estimates on the subpopulation densities u^ε_a, u^ε_b, on the total population densitiesu^ε:=u^ε_a+u^ε_bandv^ε, and onQ(u^ε_a, u^ε_b, v^ε).

More specifically, we take advantage of the triangular structure of the system that give usa prioriestimates on the densityv^εand its derivatives (see Lemma3.1). The reaction functionsfa andfbof competition type allow us to control the total mass R

Ωu^ε(t)dx, and to get anL²(ΩT) estimate onu^ε(see Lemma3.2). The latter will be employed in Lemma3.3to obtain estimates on∇u^ε_a,∇u^ε_bandQ(u^ε_a, u^ε_b, v^ε), through the use of the energy functional (1.10)–(1.11). In addition, the triplet (u^ε_a, u^ε_b, v^ε) will be shown to have finite energyE(T) as well, for allT >0.

Hereafter, all constantsC andCT are strictly positive and may depend on Ω, the initial datauⁱⁿ_a, uⁱⁿ_b, vⁱⁿ, the coefficients in system (1.1), the transition functions φ, ψ and on T, but never on ε. They may change also from line to line in the computations.

Lemma 3.1. Under the hypothesis of Theorem 2.2, the following statements hold:

(i) there exists a constantC >0 such that for all ε >0

kv^εkL^∞((0,+∞)×Ω)6C; (3.1) (ii) for all q∈(1,+∞)there exists a constantC(q)>0 such that, for all ε >0,

T >0 and all i, j= 1, ..,N,

k∂tv^εkL^q(ΩT)+k∂xi,xjv^εkL^q(ΩT)6C(q)(1 +ku^ε_bkL^q(ΩT)) ; (3.2) (iii) for all q ∈(1,+∞)there exist C(q,N)>0 andC(q)>0 such that, for all

ε >0and all T >0,

k∇v^εk^2q_L²^q_(Ω_T₎≤C(q,N)(1 +ku^ε_bk^q_L^q_(Ω_T₎) +C(q)T. (3.3) Remark 4.

In the sequel, the value of q in (3.2)–(3.3) will be first chosen equal to 2 (see Lemma3.2), and then to a different number after Corollary 3.4.

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Proof. It is easily seen that 06v^ε(t, x)6 K:= max

kvⁱⁿkL^∞(Ω); b , for a.e. (t, x) ∈ (0,+∞)×Ω. (3.4) Indeed, by the existence result of strong solution for (1.1), we know that the nonneg- ativity ofv^εis preserved in time. Concerning the upper bound in (3.4), it is obtained by multiplying the equation forv^ε in (1.1) by (v^ε−K)⁺ := max{0, v^ε−K} and integrating over Ω, to obtain for allt >0,

Z

Ω

(v^ε(t)−K)²₊dx≤ Z

Ω

(v^in,ε−K)²₊dx= 0.

Next, by the maximal regularity property of the heat equation (see [25] and the references therein), for allq∈(1,+∞) there exists a strictly positive constantC, which only depends on Ω andq, such that for alli, j= 1, ..,N,

k∂tv^εkL^q(ΩT)+k∂xi,xjv^εkL^q(Ω_T)≤C(kfv(u^ε_b, v^ε)kL^q(Ω_T)+kvⁱⁿkL^q(Ω)

≤C 1 +ku^ε_bkL^q(Ω_T)

, (3.5)

so that estimate (3.2) holds. Then, thanks to the Gagliardo-Nirenberg inequality [27], for all q ∈ (1,+∞), there exists C(q) > 0, such that, for all t > 0 and i= 1, . . .N, we have

k∂x_iv^ε(t)kL²^q(Ω)≤C(q)

N

X

j=1

k∂x_i,x_jv^ε(t)k^1/2L^q(Ω)kv^ε(t)k^1/2L^∞(Ω)+C(q)kv^ε(t)kL^∞(Ω). Integrating the above inequality over (0, T) and using (3.1) and (3.5), we get estimate (3.3).

Lemma 3.2. Under the hypothesis of Theorem 2.2, for all T > 0 there exists CT >0such that for all ε >0 the following estimates hold:

sup

t∈[0,T]

Z

Ω

(u^ε_a+u^ε_b)(t)dx≤CT and ku^ε_a+u^ε_bkL²(ΩT)≤CT. (3.6) Proof. Adding the first two equations in (1.1) and using the positivity ofu^ε_a, u^ε_b, v^ε, we get

∂t(u^ε_a+u^ε_b)≤ da∆u^ε_a+db∆u^ε_b+ηau^ε_a

1−u^ε_a a

+ηbu^ε_b

1−u^ε_b

b

(3.7)

≤ da∆u^ε_a+db∆u^ε_b+1

4(aηa+bηb). (3.8)

Then, integrating (3.8) over Ω, the inequality becomes d

dt Z

Ω

u^ε_a+u^ε_b

(t)dx≤C , implying, for allt in [0, T], that

ku^ε_a(t) +u^ε_b(t)kL¹(Ω)≤ kuⁱⁿ_a +uⁱⁿ_bkL¹(Ω)+C T . (3.9)