Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

Michael Winkler

doi:10.1515/ans-2020-2107

Open Access Published by De Gruyter September 3, 2020

Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

Michael Winkler

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2020-2107

Abstract

The chemotaxis-growth system

($\star$) { u t = D ⁢ Δ ⁢ u - χ ⁢ ∇ ⋅ ( u ⁢ ∇ ⁡ v ) + ρ ⁢ u - μ ⁢ u α , v t = d ⁢ Δ ⁢ v - κ ⁢ v + λ ⁢ u

is considered under homogeneous Neumann boundary conditions in smoothly bounded domains Ω⊂ℝn, n≥1. For any choice of α>1, the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of ($\star$), the present work shows that, whenever α≥2-2n, under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state ((ρμ)1α-1,λκ⁢(ρμ)1α-1) in the large time limit.

Keywords: Chemotaxis; Logistic Source; Stabilization

MSC 2010: 35B40; 92C17; 35Q92; 35K55

1 Introduction

The reduction of regularity belongs to the most intensely studied effects of chemotactic cross-diffusion. In the context of classical Keller–Segel systems, this becomes manifest not only in a comprehensive literature focusing on the detection of blow-up phenomena [13, 12, 24, 4, 41], but also in several findings concerned with more detailed analysis of singularity formation [12, 33], or extension of solutions beyond blow-up [3, 23, 54].

The analysis of corresponding features apparently becomes significantly more challenging in systems which couple chemotactic interaction to further mechanisms. Among accordingly refined variants which attempt to provide more realistic descriptions in situations more complex than those addressed by minimal Keller–Segel systems, of particular importance seem models which account for proliferation and competition-induced death, known as relevant in a noticeably large number of biological contexts such as bacterial pattern formation, self-organization during embryonic development and tumor invasion [48, 29, 6, 34, 37].

Correspondingly, logistic Keller–Segel systems of the form

(1.1) { u t = D ⁢ Δ ⁢ u - χ ⁢ ∇ ⋅ ( u ⁢ ∇ ⁡ v ) + ρ ⁢ u - μ ⁢ u α , v t = d ⁢ Δ ⁢ v - κ ⁢ v + λ ⁢ u

as well as some close relatives have received considerable interest in the past years, and elaborate methods have been developed to identify conditions on the system parameters therein which ensure that the joint dissipative action of diffusion and suitably strong degradation rules out the occurrence of blow-up phenomena. In the best understood case α=2 of quadratic absorption, for instance, associated no-flux initial-boundary value problems in n-dimensional bounded domains Ω are known to admit global bounded classical solutions for all suitably regular initial data if either n≤2 (see [27, 26]), or n≥3 and μ>μ0=μ0⁢(D,d,χ,ρ,κ,λ,Ω) (see [39]); for appropriately large μ, even some results on global asymptotic stability of the corresponding spatially homogeneous equilibria (ρμ,λ⁢ρκ⁢μ) are available [5, 42]. Further findings in these directions, inter alia focusing at refinements with respect to parameter setting, or generalizations to slightly modified systems, or also qualitative facets such as wave-like behavior, can be found in [7, 14, 19, 16, 25, 50, 49, 55, 30, 31, 32], for instance.

In the presence of weaker absorption, however, the knowledge in this regard seems significantly sparser: while in two-dimensional domains already some subquadratic death terms involving certain logarithmic corrections have been shown to rule out blow-up [51], in the case n≥3, it yet appears to be unknown whether or not explosions may occur for small values of μ when α=2; for such parameter choices, only global weak solutions have been shown to exist, and results concerned with their qualitative behavior seem limited to statements on eventual smoothness for small ρ, and on asymptotic decay for ρ≤0, in the special case n=3 (see [18]; cf. also [36]).

Possible dampening effects of yet weaker degradation have been understood to a rudimentary extent only up to now. Indeed, the knowledge in this regard so far reduces to statements on mere global existence in suitably generalized solution frameworks. In [35], certain global solutions have been constructed under the assumptions that n≥2 and α>2-1n, and in [46, 53], a relaxation of these hypotheses could be achieved so as to ensure solvability even for any α>min⁡{2-2n,2⁢n+4n+4} when n≥2. Only recently, in the context of a yet further generalized solution concept it has been found that actually any choice of α>1 is sufficient to ensure global solvability for widely arbitrary initial data [47].

Asymptotics in Weakly Dampened Chemotaxis-Growth Systems. Analysis beyond Blow-up

Beyond quite poor basic regularity features, however, no qualitative information on the behavior of such solutions seems available in such weakly dampened cases. That, in fact, the correspondingly generated dynamics might be considerably complex is indicated by some noticeable caveats contained in the literature. Besides providing numerical evidence that shows remarkably colorful facets in logistic Keller–Segel systems, up to even chaotic behavior [28], previous studies have revealed quite drastic phenomena related to the spontaneous emergence of large population densities, possibly at intermediate timescales, partially even in frameworks of bounded solutions [15, 17, 43, 44, 38]. Yet more drastically, in some parabolic-elliptic simplifications of (1.1), even finite-time blow-up has been detected, e.g. under the hypotheses that n∈{3,4} and α<76 (see [45]), or n≥5 and α<32+12⁢(n-1) (see [40]); recent progress indicates that similar statements are actually available under the mere assumptions that n≥3, α=2 and μ>0 is sufficiently small [10] (cf. also [9, 21] for further blow-up results in this direction).

Despite these complicating circumstances, the present work attempts to develop a basic qualitative theory for generalized solutions to (1.1) within reasonably large parameter ranges. By addressing arbitrarily large initial data, we especially intend to include situations in which the above precedents suggest to expect the occurrence of finite-time explosions, and in which thus a large time analysis of global solutions amounts to describing life beyond blow-up, as having formed the objective, meanwhile quite well-understood, of seminal studies concerned with simpler scalar parabolic problems [2, 11, 22].

To make this more precise, let us henceforth consider the full initial-boundary value problem

(1.2) { u t = D ⁢ Δ ⁢ u - χ ⁢ ∇ ⋅ ( u ⁢ ∇ ⁡ v ) + ρ ⁢ u - μ ⁢ u α , x ∈ Ω , t > 0 , v t = d ⁢ Δ ⁢ v - κ ⁢ v + λ ⁢ u , x ∈ Ω , t > 0 , ∂ ⁡ u ∂ ⁡ ν = ∂ ⁡ v ∂ ⁡ ν = 0 , x ∈ ∂ ⁡ Ω , t > 0 , u ⁢ ( x , 0 ) = u 0 ⁢ ( x ) , v ⁢ ( x , 0 ) = v 0 ⁢ ( x ) , x ∈ Ω ,

in a smoothly bounded domain Ω⊂ℝn, n≥1, with positive parameters D, d, χ, ρ, μ, κ and λ, with α>1, and with initial data complying with the hypotheses that

(1.3) { u 0 ∈ C 0 ⁢ ( Ω ¯ ) such that u 0 > 0 in ⁢ Ω ¯ , v 0 ∈ W 1 , ∞ ⁢ ( Ω ) such that v 0 ≥ 0 in ⁢ Ω .

Within this general setting, it follows from the results in [47] that (1.2) indeed admits globally defined solutions in an appropriately generalized sense, and that these can be approximated by solutions to suitably regularized variants of (1.2).

Proposition 1.1.

Let α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0, and assume (1.3). Then there exist nonnegative functions

{ u ∈ L loc α ⁢ ( Ω ¯ × [ 0 , ∞ ) ) , v ∈ L loc 1 ⁢ ( [ 0 , ∞ ) ; W 1 , 1 ⁢ ( Ω ) )

such that (u,v) forms a global generalized solution of (1.2) in the sense of Definition A.1, and that (u,v) can be approximated by solutions to the regularized problems (2.1) below in the following sense: for each ε∈(0,1), (2.1) admits a global classical solution (uε,vε)∈(C0⁢(Ω¯×[0,∞))∩C2,1⁢(Ω¯×(0,∞)))2, and there exist (εj)j∈N⊂(0,1) and a null set N⊂(0,∞) such that εj↘0 as j→∞, and that

(1.4) u ε → u ⁢ 𝑖𝑛 ⁢ L loc 1 ⁢ ( Ω ¯ × [ 0 , ∞ ) ) ⁢ and a.e. in ⁢ Ω × ( 0 , ∞ ) ,

(1.5) v ε → v ⁢ 𝑖𝑛 ⁢ L loc 1 ⁢ ( Ω ¯ × [ 0 , ∞ ) ) ⁢ and a.e. in ⁢ Ω × ( 0 , ∞ ) ,

(1.6) v ε ⁢ ( ⋅ , t ) → v ⁢ ( ⋅ , t ) ⁢ 𝑖𝑛 ⁢ L 1 ⁢ ( Ω ) ⁢ and a.e. in ⁢ Ω for all ⁢ t ∈ ( 0 , ∞ ) ∖ N

as ε=εj↘0.

In order to describe the large time behavior of these solutions, we shall examine how far expressions of the form

(1.7) ℱ ( t ) : = ∫ Ω ( u ( ⋅ , t ) - u ⋆ - u ⋆ ln u ⁢ ( ⋅ , t ) u ⋆ ) + b 2 ∫ Ω ( v ( ⋅ , t ) - v ⁢ ( ⋅ , t ) ¯ ) 2 , t > 0 ,

enjoy certain Lyapunov-type properties for (1.2) if the free parameter b>0 therein is chosen appropriately, and if the number u⋆ denotes the first component of the associated nontrivial spatially homogeneous equilibrium (u⋆,v⋆) of (1.2) given by

(1.8) u ⋆ : = ( ρ μ ) 1 α - 1 and v ⋆ : = λ ⁢ u ⋆ κ .

Here and throughout the sequel, we adapt standard notation by abbreviating

(1.9) ψ ¯ : = 1 | Ω | ∫ Ω ψ for ψ ∈ L 1 ( Ω ) .

In spite of evident challenges linked to the poor information on regularity and the topological setting in the approximation statements in (1.4)–(1.6), a suitably designed analysis of ℱ, as well as of a natural counterpart ℱε thereof at the level of approximate solutions, will reveal that, whenever α≥2-2n and χ is appropriately small, ℱ plays the role of an eventual energy functional in the sense that, for each individual trajectory, ℱ becomes nonincreasing after an adequate waiting time. On the basis of this observation, we shall see that, within this framework, any such solution approaches (u⋆,v⋆) in the sense substantiated in the following main result of this work.

Theorem 1.2.

Let n≥1, D>0, d>0, ρ>0, μ>0 and λ>0, and suppose that α>1 is such that

α ≥ 2 - 2 n .

Then, given any bounded domain Ω⊂Rn with smooth boundary, one can find C⁢(Ω)>0 with the property that, whenever χ>0 satisfies

(1.10) χ 2 ≤ C ⁢ ( Ω ) ⋅ d 2 ⁢ D λ 2 ⋅ ρ - 3 - α α - 1 ⁢ μ 2 α - 1 ,

for arbitrary initial data fulfilling (1.3), the problem (1.2) possesses a global generalized solution (u,v), in the sense of Definition A.1 below, such that, with some T>0 and some null set N⊂(T,∞), we have

(1.11) u ⁢ ( ⋅ , t ) → u ⋆ ⁢ 𝑖𝑛 ⁢ L 1 ⁢ ( Ω ) ⁢ ⁢ 𝑎𝑠 ⁢ ( T , ∞ ) ∖ N ∋ t → ∞ ,

(1.12) v ⁢ ( ⋅ , t ) → v ⋆ ⁢ 𝑖𝑛 ⁢ L 2 ⁢ ( Ω ) ⁢ ⁢ 𝑎𝑠 ⁢ ( T , ∞ ) ∖ N ∋ t → ∞ ,

where u⋆>0 and v⋆>0 are given by (1.8).

Remark.

(i) We underline that the key condition α≥2-2n in Theorem 1.2 firstly allows for actually any α>1 when n=2, but moreover includes some choices of α for which the literature indicates the possibility of finite-time blow-up: in fact, when n=5, the assumption α≥2-2n=85 can simultaneously be fulfilled with the assumption α<32+12⁢(n-1)=138 from [40], and corresponding consistency with the hypotheses from the yet unpublished work [10] can be achieved even for any n≥3.

(ii) By essentially asserting relaxation into constant equilibria, Theorem 1.2 reveals that, with respect to solution behavior after a possible singularity formation, (1.1) considerably differs from corresponding proliferation-free Keller–Segel systems in which, at least in two-dimensional parabolic-elliptic cases, extensions beyond blow-up seem to reflect eternal persistence of Dirac-type singularities, as typically emerging during explosion processes at some finite time [3, 23].

2 Preliminaries

2.1 Approximation of Generalized Solutions

As already announced, given parameters α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0, as well as initial data fulfilling (1.3), we follow the regularization procedure in [47] and hence consider the approximate problems

(2.1) { u ε ⁢ t = D ⁢ Δ ⁢ u ε - χ ⁢ ∇ ⋅ ( u ε ⁢ ∇ ⁡ v ε ) + ρ ⁢ u ε - μ ⁢ u ε α , x ∈ Ω , t > 0 , v ε ⁢ t = d ⁢ Δ ⁢ v ε - κ ⁢ v ε + λ ⁢ u ε 1 + ε ⁢ u ε , x ∈ Ω , t > 0 , ∂ ⁡ u ε ∂ ⁡ ν = ∂ ⁡ v ε ∂ ⁡ ν = 0 , x ∈ ∂ ⁡ Ω , t > 0 , u ε ⁢ ( x , 0 ) = u 0 ⁢ ( x ) , v ε ⁢ ( x , 0 ) = v 0 ⁢ ( x ) , x ∈ Ω ,

for ε∈(0,1). Each of these indeed admits a globally defined classical solution

( u ε , v ε ) ∈ ( C 0 ⁢ ( Ω ¯ × [ 0 , ∞ ) ) ∩ C 2 , 1 ⁢ ( Ω ¯ × ( 0 , ∞ ) ) ) 2

such that, according to (1.3) and the strong maximum principle, uε>0 and vε>0 in Ω¯×(0,∞) (cf. also [47, Lemma 2.1]), and in fact, the arguments detailed in [47, Lemma 7.1, Lemma 8.2] for the prototypical choices D=d=χ=κ=λ=1 show that, in the limit of vanishing ε, these solutions approach a solution of (1.2) in the sense documented in Proposition 1.1.

2.2 Basic Bounds for uε. Absorbing Sets in L1

Let us first apply an essentially straightforward argument to the first equation in (2.1) to achieve the following basic regularity information.

Lemma 2.1.

Let α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0, and assume (1.3). Then, for all t≥0 and ε∈(0,1),

(2.2) ∫ Ω u ε ⁢ ( ⋅ , t ) ≤ { 2 α - 1 ⁢ μ ( 3 ⁢ | Ω | ) α - 1 ⁢ ρ + ( { ∫ Ω u 0 } 1 - α - 2 α - 1 ⁢ μ ( 3 ⁢ | Ω | ) α - 1 ⁢ ρ ) ⋅ e - ( α - 1 ) ⁢ ρ ⁢ t } - 1 α - 1 ,

(2.3) ∫ Ω u ε ( ⋅ , t ) ≤ m : = max { ∫ Ω u 0 , 3 2 ⋅ ( ρ μ ) 1 α - 1 | Ω | }

as well as

(2.4) ∫ t t + 1 ∫ Ω u ε α ≤ ( ρ + 1 ) ⁢ m ( 1 - ( 2 3 ) α - 1 ) ⋅ μ .

Proof.

We abbreviate y0:=∫Ωu0 and θ:=(23)α-1μ∈(0,μ), and let y∈C1⁢([0,∞)) denote the solution of

{ y ′ ⁢ ( t ) = ρ ⁢ y ⁢ ( t ) - θ ⁢ | Ω | 1 - α ⁢ y α ⁢ ( t ) , t > 0 , y ⁢ ( 0 ) = y 0 ,

that is, we let

y ( t ) : = { θ ⁢ | Ω | 1 - α ρ + ( y 0 1 - α - θ ⁢ | Ω | 1 - α ρ ) ⋅ e - ( α - 1 ) ⁢ ρ ⁢ t } - 1 α - 1 , t ≥ 0 .

Since, in view of the Hölder inequality, the first equation in (2.1) ensures that, for each ε∈(0,1),

d d ⁢ t ⁢ ∫ Ω u ε = ρ ⁢ ∫ Ω u ε - μ ⁢ ∫ Ω u ε α ≤ ρ ⁢ ∫ Ω u ε - θ ⁢ | Ω | 1 - α ⋅ { ∫ Ω u ε } α - ( μ - θ ) ⁢ ∫ Ω u ε α for all ⁢ t > 0 ,

and that thus yε(t):=∫Ωuε(⋅,t), t≥0, and hε(t):=(μ-θ)∫Ωuεα(⋅,t), t>0, satisfy

(2.5) y ε ′ ⁢ ( t ) + h ε ⁢ ( t ) ≤ ρ ⁢ y ε ⁢ ( t ) - θ ⁢ | Ω | 1 - α ⁢ y ε α ⁢ ( t ) for all ⁢ t > 0 ,

by nonnegativity of hε, a comparison argument shows that

y ε ⁢ ( t ) ≤ y ⁢ ( t ) ≤ max ⁡ { y 0 , ( ρ θ ⁢ | Ω | 1 - α ) 1 α - 1 } = m for all ⁢ t > 0

and thereby establishes both (2.2) and (2.3). An integration in (2.5) thereupon warrants that

∫ t t + 1 h ε ⁢ ( s ) ⁢ d ⁢ s ≤ y ε ⁢ ( t ) + ρ ⁢ ∫ t t + 1 y ε ⁢ ( s ) ⁢ d ⁢ s ≤ m + ρ ⁢ m for all ⁢ t > 0 ,

which by definition of hε entails (2.4). ∎

In particular, this entails an absorption feature of suitably large balls in L1, in the following flavor.

Lemma 2.2.

Let α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0, and let u⋆>0 be as in (1.8). Then, given any (u0,v0) fulfilling (1.3), one can find T⋆=T⋆⁢(u0,v0)>0 with the property that

∫ Ω u ε ⁢ ( ⋅ , t ) ≤ 2 ⁢ u ⋆ ⁢ | Ω | for all ⁢ t > T ⋆ ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) .

Proof.

We only need to employ (2.2) and observe that therein

{ 2 α - 1 ⁢ μ ( 3 ⁢ | Ω | ) α - 1 ⁢ ρ + ( { ∫ Ω u 0 } 1 - α - 2 α - 1 ⁢ μ ( 3 ⁢ | Ω | ) α - 1 ⁢ ρ ) ⋅ e - ( α - 1 ) ⁢ ρ ⁢ t } - 1 α - 1 → { 2 α - 1 ⁢ μ ( 3 ⁢ | Ω | ) α - 1 ⁢ ρ } - 1 α - 1 = 3 2 ⁢ u ⋆ ⁢ | Ω |

as t→∞. ∎

3 Construction of Eventual Energy Functionals at the Approximate Level

The purpose of this section is to make sure that, for appropriately small χ>0, at the stage of approximate solutions, the functional in (1.7) enjoys a genuine Lyapunov property after some relaxation time possibly depending on the initial data. Here, in comparison to the case α=2 addressed in several related precedents in the literature [1, 8, 52], in the presence especially of subquadratic degradation, some further technical efforts seem necessary so as to facilitate an efficient quantitative analysis of how far the steady state u⋆ from (1.8) inherits attractiveness features from corresponding taxis-free frameworks.

3.1 The Time Evolution of ∫Ω(uε-u⋆-u⋆⁢ln⁡uεu⋆)

Our first objective in this regard is the function appearing in the first integral from (1.7). Some useful of its properties are summarized in the following lemma.

Lemma 3.1.

Let α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0, and with u⋆ taken from (1.8), let

(3.1) H ( ξ ) : = ξ - u ⋆ - u ⋆ ln ξ u ⋆ , ξ > 0 .

Then H⁢(ξ)≥0 for all ξ>0, and there exists C>0 such that, with q:=min{α,2}, we have

(3.2) ∫ Ω H q ⁢ ( ψ ) ≤ ∫ Ω ψ α + C ⁢ ∫ Ω | ∇ ⁡ ψ | 2 ψ 2 + C ⋅ { ∫ Ω | ln ⁡ ψ | } 2 + C for all ⁢ ψ ∈ L α ⁢ ( Ω ; ( 0 , ∞ ) ) such that ⁢ ln ⁡ ψ ∈ W 1 , 2 ⁢ ( Ω ) .

Proof.

Noting that nonnegativity of H can be seen by elementary analysis, to verify (3.2), we use that, according to a Poincaré inequality, we can fix c1>0 such that

∫ Ω ζ 2 ≤ c 1 ⁢ ∫ Ω | ∇ ⁡ ζ | 2 + c 1 ⋅ { ∫ Ω | ζ | } 2 for all ⁢ ζ ∈ W 1 , 2 ⁢ ( Ω ) .

Therefore, given a positive ψ∈Lα⁢(Ω) such that ln⁡ψ∈W1,2⁢(Ω), observing that

H ⁢ ( ξ ) ≤ - u ⋆ ⁢ ln ⁡ ξ + c 2 for all ⁢ ξ ∈ ( 0 , 2 ⁢ u ⋆ )

with c2:=u⋆+u⋆|lnu⋆|>0, we can set ζ:=lnψ to see that, thanks to Young’s inequality and the fact that q≤2,

∫ { ψ < 2 u ⋆ } H q ⁢ ( ψ ) ≤ ∫ { ψ < 2 u ⋆ } H 2 ⁢ ( ψ ) + | Ω | ≤ 2 ⁢ u ⋆ 2 ⁢ ∫ { ψ < 2 u ⋆ } | ln ⁡ ψ | 2 + 2 ⁢ c 2 2 ⁢ | Ω | + | Ω | ≤ 2 ⁢ c 1 ⁢ u ⋆ 2 ⁢ ∫ Ω | ∇ ⁡ ln ⁡ ψ | 2 + 2 ⁢ c 1 ⁢ u ⋆ 2 ⋅ { ∫ Ω | ln ⁡ ψ | } 2 + 2 ⁢ c 2 2 ⁢ | Ω | + | Ω | .

In the corresponding complementary region, however, we may simply use that ln⁡ξu⋆≥0 for ξ≥2⁢u⋆, and that hence

H ⁢ ( ξ ) ≤ ξ for all ⁢ ξ ≥ 2 ⁢ u ⋆ ,

to find that, for any such ψ, again by Young’s inequality, and by the restriction q≤α,

∫ { ψ ≥ 2 u ⋆ } H q ⁢ ( ψ ) ≤ ∫ { ψ ≥ 2 u ⋆ } H α ⁢ ( ψ ) + | Ω | ≤ ∫ { ψ ≥ 2 u ⋆ } ψ α + | Ω | .

Therefore, (3.2) results upon choosing C:=max{2c1u⋆2,2c22|Ω|+2|Ω|}, for instance. ∎

Let us already here add a second preparation in this regard, albeit only used in our final proof of Theorem 1.2 in Section 5.3, which asserts that ∫ΩH⁢(ψ) controls differences to u⋆ actually with respect to the norm in L1⁢(Ω).

Lemma 3.2.

Let α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0, let u⋆ and H be as in (1.8) and (3.1), and let ψ:Ω→(0,∞) be measurable. Then

(3.3) ∫ Ω | ψ - u ⋆ | ≤ 1 1 - ln ⁡ 2 ⁢ ∫ Ω H ⁢ ( ψ ) + 8 ⁢ u ⋆ ⁢ | Ω | ⋅ { ∫ Ω H ⁢ ( ψ ) } 1 2 .

Proof.

Using that H⁢(u⋆)=H′⁢(u⋆)=0 and that H′′⁢(ξ)=u⋆ξ2≥14⁢u⋆ for all ξ∈(0,2⁢u⋆), we first observe that

H ⁢ ( ξ ) ≥ 1 2 ⋅ { inf σ ∈ ( 0 , 2 ⁢ u ⋆ ) ⁡ H ′′ ⁢ ( σ ) } ⋅ ( ξ - u ⋆ ) 2 ≥ 1 8 ⁢ u ⋆ ⋅ ( ξ - u ⋆ ) 2 for all ⁢ ξ ∈ ( 0 , 2 ⁢ u ⋆ )

so that, given any measurable ψ:Ω→(0,∞), we can estimate

(3.4) ∫ { ψ < 2 u ⋆ } | ψ - u ⋆ | ≤ ∫ { ψ < 2 u ⋆ } 8 ⁢ u ⋆ ⁢ H ⁢ ( ψ ) ≤ 8 ⁢ u ⋆ ⁢ | Ω | ⋅ { ∫ { ψ < 2 u ⋆ } H ⁢ ( ψ ) } 1 2

by means of the Cauchy–Schwarz inequality. Apart from that, since writing c1:=1-ln2, we see that, for

φ ( ξ ) : = H ( ξ ) - c 1 ⋅ ( ξ - u ⋆ ) , ξ ≥ 2 u ⋆ ,

we have

φ ⁢ ( 2 ⁢ u ⋆ ) = H ⁢ ( 2 ⁢ u ⋆ ) - c 1 ⁢ u ⋆ = ( 1 - c 1 ) ⁢ u ⋆ - u ⋆ ⁢ ln ⁡ 2 = 0

as well as

φ ′ ⁢ ( ξ ) = H ′ ⁢ ( ξ ) - c 1 = 1 - u ⋆ ξ - c 1 ≥ 1 - 1 2 - c 1 = ln ⁡ 2 - 1 2 > 0 for all ⁢ ξ ≥ 2 ⁢ u ⋆

so that φ⁢(ξ)≥0 for all ξ≥2⁢u⋆. Accordingly, for any ψ as given above, we have

( 1 - ln ⁡ 2 ) ⁢ ∫ { ψ ≥ 2 u ⋆ } ( ψ - u ⋆ ) ≤ ∫ { ψ ≥ 2 u ⋆ } H ⁢ ( ψ ) ,

which together with (3.4) readily yields (3.3). ∎

The role of H in our asymptotic analysis of (1.2) is foreshadowed by the following straightforward observation.

Lemma 3.3.

Let α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0, and let u⋆ and H be as in (1.8) and (3.1). Then, for all t>0 and ε∈(0,1),

(3.5) d d ⁢ t ⁢ ∫ Ω H ⁢ ( u ε ) + D ⁢ u ⋆ ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + μ ⁢ ∫ Ω ( u ε α - 1 - u ⋆ α - 1 ) ⋅ ( u ε - u ⋆ ) = χ ⁢ u ⋆ ⁢ ∫ Ω ∇ ⁡ u ε u ε ⋅ ∇ ⁡ v ε .

Proof.

Using that

H ′ ⁢ ( ξ ) = 1 - u ⋆ ξ and H ′′ ⁢ ( ξ ) = u ⋆ ξ 2 for all ⁢ ξ > 0 ,

thanks to the positivity of uε, we may use the first equation in (2.1) to compute

(3.6) d d ⁢ t ⁢ ∫ Ω H ⁢ ( u ε ) = - ∫ Ω H ′′ ⁢ ( u ε ) ⁢ ∇ ⁡ u ε ⋅ ( D ⁢ ∇ ⁡ u ε - χ ⁢ u ε ⁢ ∇ ⁡ v ε ) + ∫ Ω H ′ ⁢ ( u ε ) ⋅ ( ρ ⁢ u ε - μ ⁢ u ε α ) = - D ⁢ u ⋆ ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + χ ⁢ u ⋆ ⁢ ∫ Ω ∇ ⁡ u ε u ε ⋅ ∇ ⁡ v ε + ∫ Ω ( 1 - u ⋆ u ε ) ⋅ ( ρ ⁢ u ε - μ ⁢ u ε α )

for t>0 and ε∈(0,1). Here, by (1.8), we may replace ρμ=u⋆α-1 in verifying that

∫ Ω ( 1 - u ⋆ u ε ) ⋅ ( ρ ⁢ u ε - μ ⁢ u ε α ) = μ ⁢ ∫ Ω ( u ε - u ⋆ ) ⋅ ( ρ μ - u ε α - 1 ) = - μ ⁢ ∫ Ω ( u ε - u ⋆ ) ⋅ ( u ε α - 1 - u ⋆ α - 1 ) for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 )

so that (3.6) indeed is equivalent to (3.5). ∎

3.2 Estimating ∫Ω(uεα-1-u⋆α-1)⋅(uε-u⋆) from below

Now, in order to appropriately estimate the last summand on the left of (3.5) from below, we first rely on elementary calculus to verify the following.

Lemma 3.4.

Let α>1. Then

(3.7) ( ξ α - 1 - 1 ) ⋅ ( ξ - 1 ) ≥ K ⋅ ( ξ - 1 ) 2 for all ⁢ ξ ∈ [ 0 , 2 ] ,

where

K : = { ( α - 1 ) ⋅ 2 α - 2 𝑖𝑓 ⁢ α ∈ ( 1 , 2 ) , 1 𝑖𝑓 ⁢ α ≥ 2 .

Proof.

If α<2, then φ1(ξ):=1-ξα-1-(α-1)⋅(1-ξ), ξ∈[0,1], satisfies

φ 1 ⁢ ( 1 ) = 0 and φ 1 ′ ⁢ ( ξ ) = - ( α - 1 ) ⁢ ξ α - 2 + α - 1 ≤ 0 for all ⁢ ξ ∈ ( 0 , 1 )

so that φ1≥0 on [0,1], and hence

(3.8) ( ξ α - 1 - 1 ) ⋅ ( ξ - 1 ) = ( φ 1 ⁢ ( ξ ) + ( α - 1 ) ⋅ ( 1 - ξ ) ) ⋅ ( 1 - ξ ) ≥ ( α - 1 ) ⋅ ( 1 - ξ ) 2 for all ⁢ ξ ∈ [ 0 , 1 ] .

Moreover, for such α, we see that φ2(ξ):=ξα-1-1-(α-1)⋅2α-2(ξ-1), ξ∈[1,2], has the properties that φ2⁢(1)=0 and φ2′⁢(ξ)=(α-1)⋅ξα-2-(α-1)⋅2α-2≥0 for all ξ∈(1,2); whence φ2 is nonnegative on [1,2]. Therefore,

( ξ α - 1 - 1 ) ⋅ ( ξ - 1 ) = ( φ 2 ⁢ ( ξ ) + ( α - 1 ) ⋅ 2 α - 2 ⁢ ( ξ - 1 ) ) ⋅ ( ξ - 1 ) ≥ ( α - 1 ) ⋅ 2 α - 2 ⁢ ( ξ - 1 ) 2 for all ⁢ ξ ∈ [ 1 , 2 ] ,

which together with (3.8) proves (3.7) in this case because 2α-2≤1.

If, conversely, α≥2, then ξα-1≤ξ for ξ∈[0,1], and thus

(3.9) ( ξ α - 1 - 1 ) ⋅ ( ξ - 1 ) = ( 1 - ξ α - 1 ) ⋅ ( 1 - ξ ) ≥ ( 1 - ξ ) 2 for all ⁢ ξ ∈ [ 0 , 1 ] ,

whereas letting φ3(ξ):=ξα-1-1-(α-1)⋅(ξ-1), ξ∈[1,2], we obtain

φ 3 ⁢ ( 1 ) = 0 and φ 3 ′ ⁢ ( ξ ) = ( α - 1 ) ⋅ ξ α - 2 - ( α - 1 ) ≥ 0 for all ⁢ ξ ∈ ( 1 , 2 )

so that also φ3≥0 on [1,2], and hence

( ξ α - 1 - 1 ) ⋅ ( ξ - 1 ) = ( φ 3 ⁢ ( ξ ) + ( α - 1 ) ⋅ ( ξ - 1 ) ) ⋅ ( ξ - 1 ) ≥ ( α - 1 ) ⋅ ( ξ - 1 ) 2 for all ⁢ ξ ∈ [ 1 , 2 ] .

Along with (3.9), this establishes (3.7) for any such α because then α-1≥1. ∎

Combining the latter with the eventual L1 bound from Lemma 2.2, by means of a suitable splitting of the integration domain, we obtain the following important lower bound for the degradation-induced contribution to (3.5), underlining its independence of the initial data.

Lemma 3.5.

Let α>1, and let

(3.10) p : = { 2 3 - α 𝑖𝑓 ⁢ α ≤ 2 , 2 𝑖𝑓 ⁢ α > 2 .

Then there exists K1>0 with the property that, for any choice of D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0, given any (u0,v0) fulfilling (1.3), one can find T⋆>0 such that

(3.11) ∥ u ε ⁢ ( ⋅ , t ) - u ⋆ ∥ L p ⁢ ( Ω ) 2 ≤ K 1 ⁢ u ⋆ 2 - α ⁢ ∫ Ω ( u ε α - 1 - u ⋆ α - 1 ) ⋅ ( u ε - u ⋆ ) for all ⁢ t > T ⋆ ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) ,

where u⋆>0 is taken from (1.8).

Proof.

We let K>0 be as introduced in Lemma 3.4 and claim that then, for any u0 and v0 fulfilling (1.3), we can find T⋆>0 such that (3.11) holds with

(3.12) K 1 : = c 1 + c 2 , where c 1 : = { 2 2 p 1 - 2 1 - α ⋅ ( 2 ⁢ | Ω | ) 2 - α if ⁢ α ≤ 2 , 2 2 p 1 - 2 1 - α ⋅ 2 2 - α if ⁢ α > 2 and c 2 : = ( 2 ⁢ | Ω | ) 2 - p p K .

To this end, given any such (u0,v0), we first invoke Lemma 2.2 to pick T⋆=T⋆⁢(u0,v0)>0 fulfilling

(3.13) ∥ u ε ⁢ ( ⋅ , t ) ∥ L 1 ⁢ ( Ω ) ≤ 2 ⁢ u ⋆ ⁢ | Ω | for all ⁢ t > T ⋆ ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

and use the fact that, since p≥2, we have (a+b)2p≤22p-1⁢(a2p+b2p) for a≥0 and b≥0,

(3.14) ∥ u ε - u ⋆ ∥ L p ⁢ ( Ω ) 2 = { ∫ { u ε < 2 u ⋆ } | u ε - u ⋆ | p + ∫ { u ε ≥ 2 u ⋆ } ( u ε - u ⋆ ) p } 2 p ≤ 2 2 p - 1 ⋅ { ∫ { u ε < 2 u ⋆ } | u ε - u ⋆ | p } 2 p + 2 2 p - 1 ⋅ { ∫ { u ε ≥ 2 u ⋆ } u ε p } 2 p for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Here, in the latter integral, we can make use of (3.13) to see that if α≤2, then due to the Hölder inequality and thanks to our choice of p,

(3.15) { ∫ { u ε ≥ 2 u ⋆ } u ε p } 2 p = ∥ u ε ∥ L 2 3 - α ( { u ε ≥ 2 u ⋆ } ) 2 ≤ ∥ u ε ∥ L α ( { u ε ≥ 2 u ⋆ } ) α ⁢ ∥ u ε ∥ L 1 ( { u ε ≥ 2 u ⋆ } ) 2 - α ≤ ( 2 ⁢ u ⋆ ⁢ | Ω | ) 2 - α ⁢ ∫ { u ε ≥ 2 u ⋆ } u ε α for all ⁢ t > T ⋆ ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

while in the case when α>2 and thus p=2, we can trivially estimate

(3.16) { ∫ { u ε ≥ 2 u ⋆ } u ε p } 2 p = ∫ { u ε ≥ 2 u ⋆ } u ε 2 ≤ ( 2 ⁢ u ⋆ ) 2 - α ⁢ ∫ { u ε ≥ 2 u ⋆ } u ε α for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

As, on the other hand, regardless of the sign of α-2, we have

∫ { u ε ≥ 2 u ⋆ } ( u ε α - 1 - u ⋆ α - 1 ) ⋅ ( u ε - u ⋆ ) ≥ ∫ { u ε ≥ 2 u ⋆ } ( u ε α - 1 - ( u ε 2 ) α - 1 ) ⋅ ( u ε - u ε 2 ) = 1 - 2 1 - α 2 ⁢ ∫ { u ε ≥ 2 u ⋆ } u ε α for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

from (3.15) and (3.16), we infer that

(3.17) 2 2 p - 1 ⋅ { ∫ { u ε ≥ 2 u ⋆ } u ε p } 2 p ≤ c 1 ⁢ u ⋆ 2 - α ⁢ ∫ { u ε ≥ 2 u ⋆ } ( u ε α - 1 - u ⋆ α - 1 ) ⋅ ( u ε - u ⋆ ) for all ⁢ t > T ⋆ ⁢ and ⁢ ε ∈ ( 0 , 1 )

with c1>0 as defined in (3.12).

Next, in order to control the first summand on the right of (3.14), we note that, again by applying the Hölder inequality and relying on the fact that p≤2, we may utilize Lemma 3.4 to find that, for all t>0 and ε∈(0,1),

2 2 p - 1 ⋅ { ∫ { u ε < 2 u ⋆ } | u ε - u ⋆ | p } 2 p ≤ 2 2 p - 1 ⁢ | Ω | 2 - p p ⁢ ∫ { u ε < 2 u ⋆ } ( u ε - u ⋆ ) 2 = ( 2 ⁢ | Ω | ) 2 - p p ⁢ u ⋆ 2 ⁢ ∫ { u ε < 2 u ⋆ } ( u ε u ⋆ - 1 ) 2 ≤ ( 2 ⁢ | Ω | ) 2 - p p ⁢ u ⋆ 2 K ⁢ ∫ { u ε < 2 u ⋆ } ( ( u ε u ⋆ ) α - 1 - 1 ) ⋅ ( u ε u ⋆ - 1 ) = ( 2 ⁢ | Ω | ) 2 - p p ⁢ u ⋆ 2 - α K ⁢ ∫ { u ε < 2 u ⋆ } ( u ε α - 1 - u ⋆ α - 1 ) ⋅ ( u ε - u ⋆ ) .

Together with (3.17), this shows that indeed (3.11) is valid if we take K1>0 as in (3.12). ∎

3.3 The Evolution of ∫Ω(vε-vε¯)2

Next, turning our attention to the second summand making up (1.7), we note the following basic description of its evolution at the level of approximate solutions.

Lemma 3.6.

Let α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0. Then, for any choice of a∈R,

(3.18) 1 2 ⁢ d d ⁢ t ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) 2 + d ⁢ ∫ Ω | ∇ ⁡ v ε ⁢ ( ⋅ , t ) | 2 + κ ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) 2 = λ ⁢ ∫ Ω ( u ε ⁢ ( ⋅ , t ) 1 + ε ⁢ u ε ⁢ ( ⋅ , t ) - a ) ⋅ ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) for all ⁢ t > 0 ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) .

Proof.

(3.19) ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) = 0 for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 )

by (1.9), on the basis of the second equation in (2.1), we see that

1 2 ⁢ d d ⁢ t ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) 2 = ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) ⋅ ( v ε ⁢ t ⁢ ( ⋅ , t ) - ∂ t ⁡ v ε ⁢ ( ⋅ , t ) ¯ ) = ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) ⋅ ( d ⁢ Δ ⁢ v ε ⁢ ( ⋅ , t ) - κ ⁢ v ε ⁢ ( ⋅ , t ) + λ ⁢ u ε ⁢ ( ⋅ , t ) 1 + ε ⁢ u ε ⁢ ( ⋅ , t ) ) = - d ⁢ ∫ Ω | ∇ ⁡ v ε ⁢ ( ⋅ , t ) | 2 - κ ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) ⋅ v ε ⁢ ( ⋅ , t ) + λ ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) ⋅ u ε ⁢ ( ⋅ , t ) 1 + ε ⁢ u ε ⁢ ( ⋅ , t ) for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Since, again due to (3.19),

- κ ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) ⋅ v ε ⁢ ( ⋅ , t ) = - κ ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) 2

as well as

λ ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) ⋅ u ε ⁢ ( ⋅ , t ) 1 + ε ⁢ u ε ⁢ ( ⋅ , t ) = λ ⁢ ∫ Ω ( v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) ⋅ ( u ε ⁢ ( ⋅ , t ) 1 + ε ⁢ u ε ⁢ ( ⋅ , t ) - a )

for t>0 and ε∈(0,1) whenever a∈ℝ, this already establishes (3.18). ∎

3.4 An Eventual Lyapunov Property for Small χ

Now it turns out that, for adequately small χ, the right-hand side contributions to both (3.5) and (3.18) can simultaneously be absorbed by suitable of the respectively dissipated quantities in such a way that, for some choice of b, a linear combination of the form in (1.7) indeed eventually plays the role of an energy functional for (2.1).

Lemma 3.7.

Let α>1 be such that α≥2-2n, let D>0, d>0, ρ>0, μ>0, κ>0 and λ>0, and suppose that χ>0 is such that

(3.20) χ 2 ≤ d 2 ⁢ D K 1 ⁢ K 2 ⁢ λ 2 ⁢ ρ - 3 - α α - 1 ⁢ μ 2 α - 1 ,

where K1 is taken from Lemma 3.5, and where K2>0 is such that, with p>1 taken from (3.10), we have

(3.21) ∥ φ - φ ¯ ∥ L p p - 1 ⁢ ( Ω ) 2 ≤ K 2 ⁢ ∥ ∇ ⁡ φ ∥ L 2 ⁢ ( Ω ) 2 for all ⁢ φ ∈ W 1 , 2 ⁢ ( Ω ) .

Then there exists b>0 such that, whenever (u0,v0) satisfies (1.3), one can find

C = C ⁢ ( u 0 , v 0 ) > 0 𝑎𝑛𝑑 T ⋆ = T ⋆ ⁢ ( u 0 , v 0 ) > 0

such that, writing

(3.22) ℱ ε ( t ) : = ∫ Ω ( u ε ( ⋅ , t ) - u ⋆ - u ⋆ ln u ε ⁢ ( ⋅ , t ) u ⋆ ) + b 2 ∫ Ω ( v ε ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) 2 ,

(3.23) 𝒟 ε ( t ) : = C ∫ Ω | ∇ ⁡ u ε ⁢ ( ⋅ , t ) | 2 u ε 2 ⁢ ( ⋅ , t ) + C ∥ u ε ( ⋅ , t ) - u ⋆ ∥ L p ⁢ ( Ω ) 2 + C ∫ Ω ( v ε ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) 2 , t > 0 , ε ∈ ( 0 , 1 ) ,

we have

(3.24) ℱ ε ′ ⁢ ( t ) + 𝒟 ε ⁢ ( t ) ≤ 0 for all ⁢ t > T ⋆ ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) .

Proof.

We let

(3.25) b : = χ 2 ⁢ u ⋆ d ⁢ D

and recall that u⋆=(ρμ)1α-1 to see that, according to (3.20), we have

(3.26) b ⁢ K 1 ⁢ K 2 ⁢ λ 2 ⁢ u ⋆ 2 - α d = K 1 ⁢ K 2 ⁢ λ 2 ⁢ u ⋆ 3 - α ⁢ χ 2 d 2 ⁢ D = K 1 ⁢ K 2 ⁢ λ 2 ⁢ ρ 3 - α α - 1 ⁢ μ - 2 α - 1 ⁢ χ 2 d 2 ⁢ D ⋅ μ ≤ μ .

Assuming (1.3) and taking ℱε as accordingly defined by (3.22), on the basis of Lemma 3.3 and Lemma 3.6, the latter applied to a:=u⋆1+ε⁢u⋆, we then obtain that

(3.27) ℱ ε ′ ⁢ ( t ) + D ⁢ u ⋆ ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + b ⁢ d ⁢ ∫ Ω | ∇ ⁡ v ε | 2 + μ ⁢ ∫ Ω ( u ε α - 1 - u ⋆ α - 1 ) ⋅ ( u ε - u ⋆ ) + b ⁢ κ ⁢ ∫ Ω ( v ε - v ε ⁢ ( ⋅ , t ) ¯ ) 2 = χ ⁢ u ⋆ ⁢ ∫ Ω ∇ ⁡ u ε u ε ⋅ ∇ ⁡ v ε + b ⁢ λ ⁢ ∫ Ω ( u ε 1 + ε ⁢ u ε - u ⋆ 1 + ε ⁢ u ⋆ ) ⋅ ( v ε - v ε ⁢ ( ⋅ , t ) ¯ ) for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Here, by Young’s inequality and (3.25),

(3.28) χ ⁢ u ⋆ ⁢ ∫ Ω ∇ ⁡ u ε u ε ⋅ ∇ ⁡ v ε ≤ D ⁢ u ⋆ 2 ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + χ 2 ⁢ u ⋆ 2 ⁢ D ⁢ ∫ Ω | ∇ ⁡ v ε | 2 ≤ D ⁢ u ⋆ 2 ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + b ⁢ d 2 ⁢ ∫ Ω | ∇ ⁡ v ε | 2 for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

whereas due to the Hölder inequality, (3.21) and Young’s inequality,

(3.29) b ⁢ λ ⁢ ∫ Ω ( u ε 1 + ε ⁢ u ε - u ⋆ 1 + ε ⁢ u ⋆ ) ⋅ ( v ε - v ε ⁢ ( ⋅ , t ) ¯ ) ≤ b ⁢ λ ⁢ ∥ u ε 1 + ε ⁢ u ε - u ⋆ 1 + ε ⁢ u ⋆ ∥ L p ⁢ ( Ω ) ⁢ ∥ v ε - v ε ⁢ ( ⋅ , t ) ¯ ∥ L p p - 1 ⁢ ( Ω ) ≤ b ⁢ λ ⁢ K 2 ⁢ ∥ u ε 1 + ε ⁢ u ε - u ⋆ 1 + ε ⁢ u ⋆ ∥ L p ⁢ ( Ω ) ⁢ ∥ ∇ ⁡ v ε ∥ L 2 ⁢ ( Ω ) ≤ b ⁢ d 2 ⁢ ∥ ∇ ⁡ v ε ∥ L 2 ⁢ ( Ω ) 2 + b ⁢ K 2 ⁢ λ 2 2 ⁢ d ⁢ ∥ u ε 1 + ε ⁢ u ε - u ⋆ 1 + ε ⁢ u ⋆ ∥ L p ⁢ ( Ω ) 2 for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Since

∥ u ε 1 + ε ⁢ u ε - u ⋆ 1 + ε ⁢ u ⋆ ∥ L p ⁢ ( Ω ) ≤ ∥ u ε - u ⋆ ∥ L p ⁢ ( Ω ) for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 )

by the mean value theorem, and since thus Lemma 3.5 in conjunction with (3.26) say that, with T⋆>0 as provided there, we have

b ⁢ K 2 ⁢ λ 2 2 ⁢ d ⁢ ∥ u ε 1 + ε ⁢ u ε - u ⋆ 1 + ε ⁢ u ⋆ ∥ L p ⁢ ( Ω ) 2 + b ⁢ K 2 ⁢ λ 2 2 ⁢ d ⁢ ∥ u ε - u ⋆ ∥ L p ⁢ ( Ω ) 2 ≤ b ⁢ K 2 ⁢ λ 2 d ⁢ ∥ u ε - u ⋆ ∥ L p ⁢ ( Ω ) 2 ≤ b ⁢ K 1 ⁢ K 2 ⁢ λ 2 ⁢ u ⋆ 2 - α d ⁢ ∫ Ω ( u ε α - 1 - u ⋆ α - 1 ) ⋅ ( u ε - u ⋆ ) ≤ μ ⁢ ∫ Ω ( u ε α - 1 - u ⋆ α - 1 ) ⋅ ( u ε - u ⋆ ) for all ⁢ t > T ⋆ ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

from (3.27), (3.28) and (3.29), we thus infer that

ℱ ε ′ ⁢ ( t ) + D ⁢ u ⋆ 2 ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + b ⁢ K 2 ⁢ λ 2 d ⁢ ∥ u ε - u ⋆ ∥ L p ⁢ ( Ω ) p + b ⁢ κ ⁢ ∫ Ω ( v ε - v ε ⁢ ( ⋅ , t ) ¯ ) 2 ≤ 0

for all t>T⋆ and ε∈(0,1). We therefore readily arrive at (3.24) upon taking C>0 suitably small and then defining 𝒟ε through (3.23). ∎

Intending to make use of (3.24) for suitably large times only, we next aim at providing bounds for ℱε from above, possibly depending on time but not on ε. This will be based on the following elementary evolution property.

Lemma 3.8.

Let α>1, D>0, d>0, χ>0, ρ>0, μ>0, κ>0 and λ>0. Then

- d d ⁢ t ⁢ ∫ Ω ln ⁡ u ε + D ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 = χ ⁢ ∫ Ω ∇ ⁡ u ε u ε ⋅ ∇ ⁡ v ε - ρ ⁢ | Ω | + μ ⁢ ∫ Ω u ε α - 1 for all ⁢ t > 0 ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) .

Proof.

This can be seen by straightforward computation using the first equation in (2.1). ∎

In fact, in conjunction with another standard testing procedure, the latter enables us to derive the following.

Lemma 3.9.

Let α>1 be such that α≥2-2n, and let D>0, d>0, ρ>0, μ>0, κ>0 and λ>0. Then, for all T>0, there exists C⁢(T)>0 such that

- ∫ Ω ln ⁡ u ε ⁢ ( ⋅ , t ) ≤ C ⁢ ( T ) for all ⁢ t ∈ ( 0 , T ) ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) ,

∫ Ω v ε 2 ⁢ ( ⋅ , t ) ≤ C ⁢ ( T ) for all ⁢ t ∈ ( 0 , T ) ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) .

Proof.

We test the second equation in (2.1) against vε to see that, taking p as in (3.10), due to the Hölder inequality, we have

(3.30) 1 2 ⁢ d d ⁢ t ⁢ ∫ Ω v ε 2 + d ⁢ ∫ Ω | ∇ ⁡ v ε | 2 + κ ⁢ ∫ Ω v ε 2 = λ ⁢ ∫ Ω u ε 1 + ε ⁢ u ε ⁢ v ε ≤ λ ⁢ ∥ u ε 1 + ε ⁢ u ε ∥ L p ⁢ ( Ω ) ⁢ ∥ v ε ∥ L p p - 1 ⁢ ( Ω ) ≤ λ ⁢ ∥ u ε ∥ L p ⁢ ( Ω ) ⁢ ∥ v ε ∥ L p p - 1 ⁢ ( Ω ) for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Since pp-1 is finite and satisfies pp-1≤2⁢n(n-2)+ thanks to the assumption that α≥2-2n, and since thus W1,2⁢(Ω) is continuously embedded into Lpp-1⁢(Ω), we can find c1>0 such that ∥φ∥Lpp-1⁢(Ω)2≤c1⋅(∥∇⁡φ∥L2⁢(Ω)2+∥φ∥L2⁢(Ω)2) for all φ∈W1,2⁢(Ω) so that, letting c2:=min{d2,κ}>0, from (3.30), we infer that, by Young’s inequality,

(3.31) d d ⁢ t ⁢ ∫ Ω v ε 2 + d ⁢ ∫ Ω | ∇ ⁡ v ε | 2 ≤ - 2 ⁢ c 2 ⋅ { ∫ Ω | ∇ ⁡ v ε | 2 + ∫ Ω v ε 2 } + 2 ⁢ c 1 ⁢ λ ⁢ ∥ u ε ∥ L p ⁢ ( Ω ) ⋅ { ∫ Ω | ∇ ⁡ v ε | 2 + ∫ Ω v ε 2 } 1 2 ≤ c 1 ⁢ λ 2 2 ⁢ c 2 ⁢ ∥ u ε ∥ L p ⁢ ( Ω ) 2 for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Here, in the case α≤2, we can use the Hölder inequality along with (2.3) to find c3>0 such that

c 1 ⁢ λ 2 2 ⁢ c 2 ⁢ ∥ u ε ∥ L p ⁢ ( Ω ) 2 ≤ c 1 ⁢ λ 2 2 ⁢ c 2 ⁢ ∥ u ε ∥ L α ⁢ ( Ω ) α ⁢ ∥ u ε ∥ L 1 ⁢ ( Ω ) 2 - α ≤ c 3 ⁢ ∫ Ω u ε α for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

while if α>2, then again by Young’s inequality,

c 1 ⁢ λ 2 2 ⁢ c 2 ⁢ ∥ u ε ∥ L p ⁢ ( Ω ) 2 = c 1 ⁢ λ 2 2 ⁢ c 2 ⁢ ∫ Ω u ε 2 ≤ c 1 ⁢ λ 2 2 ⁢ c 2 ⁢ ∫ Ω u ε α + c 1 ⁢ λ 2 ⁢ | Ω | 2 ⁢ c 2 for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Therefore, (3.31) entails that with c4:=max{c3,c1⁢λ22⁢c2,c1⁢λ2⁢|Ω|2⁢c2} we have

d d ⁢ t ⁢ ∫ Ω v ε 2 + d ⁢ ∫ Ω | ∇ ⁡ v ε | 2 ≤ c 4 ⁢ ∫ Ω u ε α + c 4 for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

which combined with Lemma 3.8 shows that if we let c5:=χ24⁢d⁢D, then once more due to Young’s inequality,

d d ⁢ t ⁢ { - ∫ Ω ln ⁡ u ε + c 5 ⁢ ∫ Ω v ε 2 } ≤ - D ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + χ ⁢ ∫ Ω ∇ ⁡ u ε u ε ⋅ ∇ ⁡ v ε - ρ ⁢ | Ω | + μ ⁢ ∫ Ω u ε α - 1 - c 5 ⁢ d ⁢ ∫ Ω | ∇ ⁡ v ε | 2 + c 4 ⁢ c 5 ⁢ ∫ Ω u ε α + c 4 ⁢ c 5 ≤ - ρ ⁢ | Ω | + μ ⁢ ∫ Ω u ε α - 1 + c 4 ⁢ c 5 ⁢ ∫ Ω u ε α + c 4 ⁢ c 5 ≤ ( c 4 ⁢ c 5 + μ ) ⁢ ∫ Ω u ε α + c 4 ⁢ c 5 + μ ⁢ | Ω | for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

and hence

- ∫ Ω ln ⁡ u ε ⁢ ( ⋅ , t ) + c 5 ⁢ ∫ Ω v ε 2 ⁢ ( ⋅ , t ) ≤ - ∫ Ω ln ⁡ u 0 + c 5 ⁢ ∫ Ω v 0 2 + ( c 4 ⁢ c 5 + μ ) ⁢ ∫ 0 t ∫ Ω u ε α + ( c 4 ⁢ c 5 + μ ⁢ | Ω | ) ⋅ t

for all t>0 and ε∈(0,1). In view of (2.4) and the fact that -∫Ωln⁡u0<∞ by (1.3), this readily yields the claim because ∫Ωln⁡uε⁢(⋅,t)≤∫Ωuε⁢(⋅,t)≤m for all t>0 and ε∈(0,1) thanks to (2.3). ∎

We are now in the position to derive the following consequence of (3.24).

Lemma 3.10.

Let α>1 be such that α≥2-2n, let D>0, d>0, ρ>0, μ>0, κ>0 and λ>0, and suppose that χ>0 satisfies (3.20). Then, given (u0,v0) fulfilling (1.3), one can find C=C⁢(u0,v0)>0 and T⋆=T⋆⁢(u0,v0)>0 such that

(3.32) ∫ Ω ln ⁡ u ε ⁢ ( ⋅ , t ) ≥ - C ⁢ for all ⁢ t > T ⋆ ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) ,

(3.33) ∫ T ⋆ ∞ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 ≤ C ⁢ for all ⁢ ε ∈ ( 0 , 1 )

as well as

(3.34) ∫ T ⋆ ∞ ∥ u ε ⁢ ( ⋅ , t ) - u ⋆ ∥ L p ⁢ ( Ω ) 2 ⁢ d ⁢ t ≤ C ⁢ for all ⁢ ε ∈ ( 0 , 1 ) ,

(3.35) ∫ T ⋆ ∞ ∥ v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ∥ L 2 ⁢ ( Ω ) 2 ⁢ d ⁢ t ≤ C ⁢ for all ⁢ ε ∈ ( 0 , 1 ) ,

where p>1 is taken from (3.10).

Proof.

According to Lemma 3.7, we can fix b>0,c1=c1⁢(u0,v0)>0 and T⋆=T⋆⁢(u0,v0)>0 such that, with (ℱε)ε∈(0,1) as defined in (3.22), we have

(3.36) ℱ ε ′ ⁢ ( t ) + c 1 ⋅ { ∥ ∇ ⁡ u ε ⁢ ( ⋅ , t ) u ε ⁢ ( ⋅ , t ) ∥ L 2 ⁢ ( Ω ) 2 + ∥ v ε ⁢ ( ⋅ , t ) - v ε ⁢ ( ⋅ , t ) ¯ ∥ L 2 ⁢ ( Ω ) 2 + ∥ u ε ⁢ ( ⋅ , t ) - u ⋆ ∥ L p ⁢ ( Ω ) 2 } ≤ 0

for all t>T⋆ and ε∈(0,1). Apart from that, again explicitly relying on the assumption α≥2-2n, we may invoke Lemma 3.9 along with (2.3) to find ci=ci⁢(u0,v0)>0, i∈{2,3,4}, such that

- ∫ Ω ln ⁡ u ε ⁢ ( ⋅ , T ⋆ ) ≤ c 2 , ∫ Ω v ε 2 ⁢ ( ⋅ , T ⋆ ) ≤ c 3 and ∫ Ω u ε ⁢ ( ⋅ , T ⋆ ) ≤ c 4 for all ⁢ ε ∈ ( 0 , 1 )

so that

ℱ ε ⁢ ( T ⋆ ) ≤ ∫ Ω u ε ⁢ ( ⋅ , T ⋆ ) - u ⋆ ⁢ ∫ Ω ln ⁡ u ε ⁢ ( ⋅ , T ⋆ ) + | Ω | ⁢ u ⋆ ⁢ | ln ⁡ u ⋆ | + b 2 ⁢ ∫ Ω v ε 2 ⁢ ( ⋅ , T ⋆ ) ≤ c 5 : = c 4 + c 2 u ⋆ + u ⋆ | ln u ⋆ | + b ⁢ c 3 2 for all ε ∈ ( 0 , 1 ) .

An integration of (3.36) therefore shows that, for all t>T⋆ and ε∈(0,1),

ℱ ε ⁢ ( t ) + c 1 ⁢ ∫ T ⋆ t { ∥ ∇ ⁡ u ε ⁢ ( ⋅ , s ) u ε ⁢ ( ⋅ , s ) ∥ L 2 ⁢ ( Ω ) 2 + ∥ v ε ⁢ ( ⋅ , s ) - v ε ⁢ ( ⋅ , s ) ¯ ∥ L 2 ⁢ ( Ω ) 2 + ∥ u ε ⁢ ( ⋅ , s ) - u ⋆ ∥ L p ⁢ ( Ω ) 2 } ⁢ d ⁢ s ≤ c 5 ,

and thereby entails (3.32), (3.33), (3.34) and (3.35) due to the fact that

ℱ ε ⁢ ( t ) ≥ - ( u ⋆ + u ⋆ ⁢ | ln ⁡ u ⋆ | ) ⁢ | Ω | - u ⋆ ⁢ ∫ Ω ln ⁡ u ε ⁢ ( ⋅ , t ) for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) . ∎

4 Persistence of Energy Decrease in the Limit Problem

We shall next address the question how far the monotonicity property in (3.24) persists in the limit of vanishing ε. Our crucial preparation in this respect relies on the integral estimate from Lemma 3.1.

Corollary 4.1.

Let α>1 be such that α≥2-2n, let D>0, d>0, ρ>0, μ>0, κ>0, λ>0 and χ>0 be such that (3.20) holds, and suppose that (1.3) is satisfied. Then there exist T⋆=T⋆⁢(u0,v0)>0 and C=C⁢(u0,v0)>0 such that

(4.1) ∫ t t + 1 ∫ Ω H q ⁢ ( u ε ) ≤ C for all ⁢ t > T ⋆ ⁢ 𝑎𝑛𝑑 ⁢ ε ∈ ( 0 , 1 ) ,

where H is taken from (3.1) and q:=max{α,2}.

Proof.

We employ Lemma 3.10 to find T⋆=T⋆⁢(u0,v0)>0, c1=c1⁢(u0,v0)>0 and c2=c2⁢(u0,v0)>0 such that

∫ Ω ln ⁡ u ε ⁢ ( ⋅ , t ) ≥ - c 1 for all ⁢ t > T ⋆ ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

∫ T ⋆ ∞ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 ≤ c 2 for all ⁢ ε ∈ ( 0 , 1 ) ,

and applying Lemma 2.1 yields c3=c3⁢(u0,v0)>0 and c4=c4⁢(u0,v0)>0 fulfilling

∫ Ω u ε ⁢ ( ⋅ , t ) ≤ c 3 for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

∫ t t + 1 ∫ Ω u ε α ≤ c 4 for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Since Lemma 3.1 provides c5=c5⁢(u0,v0)>0 such that

∫ Ω H q ⁢ ( u ε ) ≤ ∫ Ω u ε α + c 5 ⁢ ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + c 5 ⋅ { ∫ Ω | ln ⁡ u ε | } 2 + c 5 for all ⁢ t > 0 ,

and since, using the elementary inequality ln⁡ξ≤ξ for ξ>1, we see that herein

∫ Ω | ln ⁡ u ε | = 2 ⁢ ∫ { u ε > 1 } u ε - ∫ Ω ln ⁡ u ε ≤ 2 ⁢ c 3 + c 1 for all ⁢ t > T ⋆ ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

this implies that

∫ t t + 1 ∫ Ω H q ⁢ ( u ε ) ≤ ∫ t t + 1 ∫ Ω u ε α + c 5 ⁢ ∫ t t + 1 ∫ Ω | ∇ ⁡ u ε | 2 u ε 2 + c 5 ⁢ ∫ t t + 1 { ∫ Ω | ln ⁡ u ε | } 2 + c 5 ≤ c 4 + c 2 ⁢ c 5 + ( 2 ⁢ c 3 + c 1 ) 2 ⁢ c 5 + c 5 for all ⁢ t > T ⋆ ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

and hence establishes (4.1). ∎

Thanks to the latter, passing to the limit in the first integral appearing in (3.22) is therefore possible in the following sense.

Lemma 4.2.

Let α>1 be such that α≥2-2n, let D>0, d>0, ρ>0, μ>0, κ>0, λ>0 and χ>0 be such that (3.20) holds, and assume (1.3). Then there exist T⋆=T⋆⁢(u0,v0)>0 and a null set N=N⁢(u0,v0)⊂(T⋆,∞) such that, with (εj)j∈N and H taken from Proposition 1.1 and (3.1), we have

H ⁢ ( u ⁢ ( ⋅ , t ) ) ∈ L 1 ⁢ ( Ω ) for all ⁢ t ∈ ( T ⋆ , ∞ ) ∖ N

and

∫ Ω H ⁢ ( u ε ⁢ ( ⋅ , t ) ) → ∫ Ω H ⁢ ( u ⁢ ( ⋅ , t ) ) for all ⁢ t ∈ ( T ⋆ , ∞ ) ∖ N 𝑎𝑠 ⁢ ε = ε j ↘ 0 .

Proof.

We take T⋆=T⋆⁢(u0,v0)>0 as given by Corollary 4.1 and hence infer from the latter that, for each T>T⋆, the family (H⁢(uε))ε∈(0,1) is bounded in Lq⁢(Ω×(T⋆,T)) for q=min⁡{α,2}. As q>1, this implies that, for any such T, (H⁢(uε))ε∈(0,1) is uniformly integrable over Ω×(T⋆,T) so that the existence of a null set N=N⁢(u0,v0)⊂(T⋆,∞) with the claimed properties readily results upon an application of the Vitali convergence theorem, relying on the fact that uε→u a.e. in Ω×(T⋆,∞) as ε=εj↘0 according to Proposition 1.1. ∎

Along with a similar but in fact more straightforward property of the respective second summands, this ensures that we may indeed pass to the limit in (3.22) and (3.24) to achieve the following.

Corollary 4.3.

Let α>1 satisfy α≥2-2n, let D>0, d>0, ρ>0, μ>0, κ>0, λ>0 and χ>0 be such that (3.20) is valid, let b>0 be as in Lemma 3.7, and assume (1.3). Then there exist T⋆=T⋆⁢(u0,v0)>0 and a null set N⋆=N⋆⁢(u0,v0)⊂(T⋆,∞) such that, letting

(4.2) ℱ ( t ) : = ∫ Ω ( u ( ⋅ , t ) - u ⋆ - u ⋆ ln u ⁢ ( ⋅ , t ) u ⋆ ) + b 2 ∫ Ω ( v ( ⋅ , t ) - v ⁢ ( ⋅ , t ) ¯ ) 2 , t ∈ ( T ⋆ , ∞ ) ∖ N ⋆ ,

defined a function F:(T⋆,∞)∖N⋆→[0,∞) satisfying

(4.3) ℱ ⁢ ( t ) ≤ ℱ ⁢ ( t 0 ) for all ⁢ t 0 ∈ ( T ⋆ , ∞ ) ∖ N ⋆ ⁢ and each ⁢ t ∈ ( t 0 , ∞ ) ∖ N ⋆ .

Proof.

We combine Lemmas 4.2 and 3.7 to find T⋆=T⋆⁢(u0,v0)>0 and a null set N1=N1⁢(u0,v0)⊂(T⋆,∞) such that the function H defined in (3.1) satisfies H⁢(u⁢(⋅,t))∈L1⁢(Ω) for all t∈(T⋆,∞)∖N1, that taking (εj)j∈ℕ from Proposition 1.1, we have

(4.4) ∫ Ω H ⁢ ( u ε ⁢ ( ⋅ , t ) ) → ∫ Ω H ⁢ ( u ⁢ ( ⋅ , t ) ) for all ⁢ t ∈ ( T ⋆ , ∞ ) ∖ N 1 as ⁢ ε = ε j ↘ 0 ,

and that moreover

(4.5) ℱ ε ⁢ ( t ) ≤ ℱ ε ⁢ ( t 0 ) for all ⁢ t 0 ∈ ( T ⋆ , ∞ ) , any ⁢ t > t 0 ⁢ and each ⁢ ε ∈ ( 0 , 1 ) ,

where (ℱε)ε∈(0,1) is as in (3.22). Apart from that, taking a null set N2=N2⁢(u0,v0)⊂(0,∞) such that, in accordance with Proposition 1.1, we have

(4.6) v ε ⁢ ( ⋅ , t ) → v ⁢ ( ⋅ , t ) in ⁢ L 2 ⁢ ( Ω ) for all ⁢ t ∈ ( 0 , ∞ ) ∖ N 2 as ⁢ ε = ε j ↘ 0 ,

and that hence clearly also

(4.7) v ε ⁢ ( ⋅ , t ) ¯ → v ⁢ ( ⋅ , t ) ¯ for all ⁢ t ∈ ( 0 , ∞ ) ∖ N 2 as ⁢ ε = ε j ↘ 0 ,

we conclude that if we let N⋆:=N1∪N2, then for all t∈(T⋆,∞)∖N⋆, the definition in (4.2) introduces a real-valued and nonnegative function ℱ on (T⋆,∞)∖N⋆ which, due to (4.5), (4.4), (4.6) and (4.7), indeed satisfies (4.3). ∎

5 Convergence of (u,v) in the Large Time Limit

5.1 Stabilization of ∫Ωv

Next, concerned with possible implications of (4.3) on the large time behavior of u and v, we first address the averages v¯ appearing therein, and proceed to make sure that these stabilize toward a limit compatible with the claim in (1.12). Our verification of this will utilize the following elementary convergence feature.

Lemma 5.1.

If g∈Lloc1⁢((0,∞)) and g∞∈R are such that

∫ t t + 1 | g ⁢ ( s ) - g ∞ | ⁢ d ⁢ s → 0 𝑎𝑠 ⁢ t → ∞ ,

then, for each a>0,

∫ 0 t e - a ⁢ ( t - s ) ⁢ g ⁢ ( s ) ⁢ d ⁢ s → g ∞ a 𝑎𝑠 ⁢ t → ∞ .

Proof.

We first note that, since ∫0te-a⁢(t-s)⁢d⁢s=1a⁢(1-e-a⁢t)→1a as t→∞, upon replacing g by g-g∞ if necessary, we may assume that g∞=0. Then, given η>0, we can first pick t1=t1⁢(η)>1 large such that

∫ t - 1 t | g ⁢ ( s ) | ⁢ d ⁢ s ≤ ( 1 - e - a ) ⁢ η 2 for all ⁢ t > t 1 ,

and then choose t2=t2⁢(η)>t1 in such a way that

(5.1) | ∫ 0 t 1 e a ⁢ s ⁢ g ⁢ ( s ) ⁢ d ⁢ s | ⋅ e - a ⁢ t 2 ≤ η 2 .

For fixed t>t2, we now rely on (5.1) in estimating

(5.2) | ∫ 0 t e - a ⁢ ( t - s ) ⁢ g ⁢ ( s ) ⁢ d ⁢ s | = | e - a ⁢ t ⁢ ∫ 0 t 1 e a ⁢ s ⁢ g ⁢ ( s ) ⁢ d ⁢ s + ∫ t 1 t e - a ⁢ ( t - s ) ⁢ g ⁢ ( s ) ⁢ d ⁢ s | ≤ η 2 + ∫ t 1 t e - a ⁢ ( t - s ) ⁢ | g ⁢ ( s ) | ⁢ d ⁢ s ,

where, taking k=k⁢(t)∈ℕ such that t-k≤t1<t-k+1, we see that

∫ t 1 t e - a ⁢ ( t - s ) ⁢ | g ⁢ ( s ) | ⁢ d ⁢ s ≤ ∫ t - k t e - a ⁢ ( t - s ) ⁢ | g ⁢ ( s ) | ⁢ d ⁢ s = ∑ j = 0 k - 1 ∫ t - j - 1 t - j e - a ⁢ ( t - s ) ⁢ | g ⁢ ( s ) | ⁢ d ⁢ s ≤ ∑ j = 0 k - 1 e - a ⁢ j ⁢ ∫ t - j - 1 t - j | g ⁢ ( s ) | ⁢ d ⁢ s ≤ { ∑ j = 0 k - 1 e - a ⁢ j } ⋅ ( 1 - e - a ) ⁢ η 2 = 1 - e - k ⁢ a 1 - e - a ⋅ ( 1 - e - a ) ⁢ η 2 ≤ η 2 .

Combined with (5.2), this yields the claim due to the fact that η>0 was arbitrary. ∎

In fact, due to a basic mass evolution property in (2.1), the latter warrants the following.

Lemma 5.2.

Let α>1 be such that α≥2-2n, let D>0, d>0, ρ>0, μ>0, κ>0, λ>0 and χ>0 be such that (3.20) holds, and assume (1.3). Then there exists a null set N=N⁢(u0,v0)⊂(T⋆,∞) such that

(5.3) ∫ Ω v ⁢ ( ⋅ , t ) → λ ⁢ u ⋆ ⁢ | Ω | κ 𝑎𝑠 ⁢ ( 0 , ∞ ) ∖ N ∋ t → ∞ .

Proof.

An integration of the second equation in (2.1) shows that

d d ⁢ t ⁢ ∫ Ω v ε + κ ⁢ ∫ Ω v ε = λ ⁢ ∫ Ω u ε 1 + ε ⁢ u ε for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) ,

and that hence

(5.4) ∫ Ω v ε ⁢ ( ⋅ , t ) = e - κ ⁢ t ⁢ ∫ Ω v 0 + λ ⁢ ∫ 0 t e - κ ⁢ ( t - s ) ⋅ { ∫ Ω u ε ⁢ ( ⋅ , s ) 1 + ε ⁢ u ε ⁢ ( ⋅ , s ) } ⁢ d ⁢ s for all ⁢ t > 0 ⁢ and ⁢ ε ∈ ( 0 , 1 ) .

Now, according to Proposition 1.1, we can find a null set N=N⁢(u0,v0)⊂(0,∞) such that, with (εj)j∈ℕ as given there, we have

(5.5) u ε → u ⁢ in ⁢ L 1 ⁢ ( Ω × ( 0 , t ) ) ⁢ and a.e. in ⁢ Ω × ( 0 , t ) ⁢ ⁢ for all ⁢ t > 0 ,

(5.6) v ε ⁢ ( ⋅ , t ) → v ⁢ ( ⋅ , t ) ⁢ in ⁢ L 1 ⁢ ( Ω ) ⁢ ⁢ for all ⁢ t ∈ ( 0 , ∞ ) ∖ N

and ε=εj↘0. Since (5.5) together with the dominated convergence theorem entails that, for all t>0, we also have

u ε 1 + ε ⁢ u ε = u ε - ε ⁢ u ε 2 1 + ε ⁢ u ε → u in ⁢ L 1 ⁢ ( Ω × ( 0 , t ) ) as ⁢ ε = ε j ↘ 0

due to the fact that ε⁢uε21+ε⁢uε→0 a.e. in Ω×(0,∞) as ε=εj↘0 and 0≤ε⁢uε21+ε⁢uε≤uε for all ε∈(0,1), we see that, on the right-hand side of (5.4),

| λ ⁢ ∫ 0 t e - κ ⁢ ( t - s ) ⋅ { ∫ Ω u ε ⁢ ( ⋅ , s ) 1 + ε ⁢ u ε ⁢ ( ⋅ , s ) } ⁢ d ⁢ s - λ ⁢ ∫ 0 t e - κ ⁢ ( t - s ) ⋅ { ∫ Ω u ⁢ ( ⋅ , s ) } ⁢ d ⁢ s | = λ ⋅ | ∫ 0 t e - κ ⁢ ( t - s ) ⋅ { ∫ Ω ( u ε ⁢ ( ⋅ , s ) 1 + ε ⁢ u ε ⁢ ( ⋅ , s ) - u ⁢ ( ⋅ , s ) ) } ⁢ d ⁢ s | ≤ λ ⁢ ∫ 0 t ∫ Ω | u ε 1 + ε ⁢ u ε - u | → 0 for all ⁢ t > 0 as ⁢ ε = ε j ↘ 0 .

Consequently, (5.4) and (5.6) imply that

(5.7) ∫ Ω v ⁢ ( ⋅ , t ) = e - κ ⁢ t ⁢ ∫ Ω v 0 + λ ⁢ ∫ 0 t e - κ ⁢ ( t - s ) ⋅ { ∫ Ω u ⁢ ( ⋅ , s ) } ⁢ d ⁢ s for all ⁢ t ∈ ( 0 , ∞ ) ∖ N ,

where clearly

(5.8) e - κ ⁢ t ⁢ ∫ Ω v 0 → 0 as ⁢ t → ∞ .

Furthermore, an application of Lemma 5.1 to a:=κ and g(t):=λ∫Ωu(⋅,t), t>0, shows that

(5.9) λ ⁢ ∫ 0 t e - κ ⁢ ( t - s ) ⋅ { ∫ Ω u ⁢ ( ⋅ , s ) } ⁢ d ⁢ s → λ ⁢ u ⋆ ⁢ | Ω | κ as ⁢ t → ∞

because, by the Cauchy–Schwarz inequality,

∫ t t + 1 | g ( s ) - λ u ⋆ | Ω | | d s = λ ∫ t t + 1 | ∫ Ω ( u ( ⋅ , s ) - u ⋆ ) ) | d s ≤ λ ⁢ ∫ t t + 1 ∥ u ⁢ ( ⋅ , s ) - u ⋆ ∥ L 1 ⁢ ( Ω ) ⁢ d ⁢ s ≤ λ ⋅ { ∫ t t + 1 ∥ u ⁢ ( ⋅ , s ) - u ⋆ ∥ L 1 ⁢ ( Ω ) 2 ⁢ d ⁢ s } 1 2 ≤ λ ⋅ { ∫ t ∞ ∥ u ⁢ ( ⋅ , s ) - u ⋆ ∥ L 1 ⁢ ( Ω ) 2 ⁢ d ⁢ s } 1 2 for all ⁢ t > 0 ,

and because Lemma 3.10 obviously entails that

∫ t ∞ ∥ u ⁢ ( ⋅ , s ) - u ⋆ ∥ L 1 ⁢ ( Ω ) 2 → 0 as ⁢ t → ∞ .

In summary, from (5.7), (5.8) and (5.9), we obtain (5.3). ∎

5.2 Decay of ℱ along a Subsequence

Thanks to the weak convergence properties implied by Lemma 5.2 and Lemma 3.10, the two summands constituting ℱ become arbitrarily small at least along suitable sequences of times.

Lemma 5.3.

Let α>1 be such that α≥2-2n, let D>0, d>0, ρ>0, μ>0, κ>0, λ>0 and χ>0 be such that (3.20) holds, and given (u0,v0) fulfilling (1.3), let N⋆=N⋆⁢(u0,v0) be as in Corollary 4.3. Then there exists (tk)k∈N⊂(T⋆,∞)∖N⋆ such that tk→∞ as k→∞ and that, with H taken from (3.1), we have

(5.10) H ⁢ ( u ⁢ ( ⋅ , t k ) ) → 0 ⁢ 𝑖𝑛 ⁢ L 1 ⁢ ( Ω ) ⁢ ⁢ 𝑎𝑠 ⁢ k → ∞ ,

(5.11) v ⁢ ( ⋅ , t k ) → v ⋆ ⁢ 𝑖𝑛 ⁢ L 2 ⁢ ( Ω ) ⁢ ⁢ 𝑎𝑠 ⁢ k → ∞ .

Proof.

We first invoke Lemma 5.2 to identify a null set N1⊃N⋆ such that

(5.12) v ⁢ ( ⋅ , t ) ¯ → λ ⁢ u ⋆ κ = v ⋆ as ⁢ ( 0 , ∞ ) ∖ N 1 ∋ t → ∞ ,

and once more use that the exponent p appearing in Lemma 3.10 satisfies p≥1 in finding T1=T1⁢(u0,v0)>0, c1=c1⁢(u0,v0)>0 and c2=c2⁢(u0,v0)>0 fulfilling

(5.13) ∫ T 1 ∞ ∥ u ε ⁢ ( ⋅ , t ) - u ⋆ ∥ L 1 ⁢ ( Ω ) 2 ⁢ d ⁢ t ≤ c 1 ⁢ for all ⁢ ε ∈ ( 0 , 1 ) ,

(5.14) ∫ T 1 ∞ ∫ Ω ( v ε ⁢ ( x , t ) - v ε ⁢ ( ⋅ , t ) ¯ ) 2 ⁢ d ⁢ x ⁢ d ⁢ t ≤ c 2 ⁢ for all ⁢ ε ∈ ( 0 , 1 ) .

Since Proposition 1.1 ensures that, with (εj)j∈ℕ as given there, we have ∥uε⁢(⋅,t)-u⋆∥L1⁢(Ω)→∥u⁢(⋅,t)-u⋆∥L1⁢(Ω) for a.e. t>T1 and ε=εj↘0, through Fatou’s lemma, we infer from (5.13) that

(5.15) ∫ T 1 ∞ ∥ u ⁢ ( ⋅ , t ) - u ⋆ ∥ L 1 ⁢ ( Ω ) 2 ⁢ d ⁢ t < ∞ ,

and similarly, we conclude from (5.14) that

(5.16) ∫ T 1 ∞ ∫ Ω ( v ⁢ ( x , t ) - v ⁢ ( ⋅ , t ) ¯ ) 2 ⁢ d ⁢ x ⁢ d ⁢ t < ∞

because vε→v a.e. in Ω×(T1,∞) as ε=εj↘0 by (1.5), and because vε⁢(⋅,t)¯→v⁢(⋅,t)¯ for a.e. t>T1 and ε=εj↘0 due to (1.6).

Apart from that, we may employ Corollary 4.1 to obtain T2=T2⁢(u0,v0)>T1 and c3=c3⁢(u0,v0)>0 such that, again with q=min⁡{α,2}, we have

∫ t t + 1 ∫ Ω H q ⁢ ( u ε ) ≤ c 3 for all ⁢ t > T 2 ,

which once more by means of Fatou’s lemma entails that

(5.17) ∫ t t + 1 ∫ Ω H q ⁢ ( u ) ≤ c 3 for all ⁢ t > T 2

because H⁢(uε)→H⁢(u) a.e. in Ω×(T2,∞) as ε=εj↘0 according to (1.4).

Now a combination of (5.15), (5.16) and (5.17) enables us to pick (tk)k∈ℕ⊂(T2,∞)∖N1 with the properties that, as k→∞, we have tk→∞ and

(5.18) ∥ u ⁢ ( ⋅ , t k ) - u ⋆ ∥ L 1 ⁢ ( Ω ) → 0 ,

∥ v ⁢ ( ⋅ , t k ) - v ⁢ ( ⋅ , t k ) ¯ ∥ L 2 ⁢ ( Ω ) → 0 ,

and that

(5.19) ∫ Ω H q ⁢ ( u ⁢ ( ⋅ , t k ) ) ≤ c 3 for all ⁢ k ∈ ℕ ,

where, on the basis of (5.18), we may assume upon extracting a subsequence if necessary that also

u ⁢ ( ⋅ , t k ) → u ⋆ a.e. in ⁢ Ω as ⁢ k → ∞ .

Since the latter clearly entails that H⁢(u⁢(⋅,tk))→H⁢(u⋆)=0 a.e. in Ω as k→∞, and since (5.19) along with the inequality q>1 warrants equi-integrability of (H⁢(u⁢(⋅,tk)))k∈ℕ over Ω, an application of the Vitali convergence theorem shows that indeed (5.10) is valid, whereas (5.11) results from (5.10) due to the fact that v⁢(⋅,tk)¯→v⋆ as k→∞ by (5.12). ∎

5.3 Convergence. Proof of Theorem 1.2

We now only need to make use of the downward monotonicity of ℱ outside N⋆ to obtain its genuine decay in the following flavor.

Corollary 5.4.

Let α>1 be such that α≥2-2n, let D>0, d>0, ρ>0, μ>0, κ>0, λ>0 and χ>0 be such that (3.20) holds, and given (u0,v0) fulfilling (1.3), let N⋆=N⋆⁢(u0,v0), T⋆=T⋆⁢(u0,v0)>0 and F be as in Corollary 4.3. Then

(5.20) ℱ ⁢ ( t ) → 0 𝑎𝑠 ⁢ ( T ⋆ , ∞ ) ∖ N ⋆ ∋ t → ∞ .

Proof.

We only need to observe that, according to Lemma 5.3, given any η>0, we can choose

t 0 = t 0 ⁢ ( η ) ∈ ( T ⋆ , ∞ ) ∖ N ⋆

suitably large such that

∫ Ω H ⁢ ( u ⁢ ( ⋅ , t 0 ) ) ≤ η 2 ,

and that, with b as in Corollary 4.3, we have

∫ Ω ( v ⁢ ( ⋅ , t ) - v ⁢ ( ⋅ , t ) ¯ ) 2 ≤ η b ,

and that thus, by (4.2),

ℱ ⁢ ( t 0 ) ≤ η 2 + b 2 ⋅ η b = η .

Therefore, namely, (4.3) ensures that ℱ⁢(t)≤η for all t∈(t0,∞)∖N⋆ and that therefore (5.20) holds. ∎

Thanks to the auxiliary statement on H from Lemma 3.2, this can be turned into our main result asserting convergence of u and v in the claimed Lebesgue space topologies.

Proof of Theorem 1.2.

With K1>0 and K2>0 taken from Lemmas 3.5 and 3.7, we define C(Ω):=1K1⁢K2, and assuming (1.10), we let T⋆>0 and N⋆⊂(T⋆,∞) be as given by Corollary 4.3 to then infer from Corollary 5.4 that the global generalized solution (u,v) of (1.2) from Proposition 1.1 satisfies

(5.21) ∫ Ω H ⁢ ( u ⁢ ( ⋅ , t ) ) → 0 ⁢ as ⁢ ( T ⋆ , ∞ ) ∖ N ⋆ ∋ t → ∞ ,

(5.22) ∫ Ω ( v ⁢ ( ⋅ , t ) - v ⁢ ( ⋅ , t ) ¯ ) 2 → 0 ⁢ as ⁢ ( T ⋆ , ∞ ) ∖ N ⋆ ∋ t → ∞ .

Apart from that taking a null set N1⊂(0,∞) such that, in accordance with Lemma 5.2, we have

v ⁢ ( ⋅ , t ) ¯ → v ⋆ as ⁢ ( 0 , ∞ ) ∖ N 1 ∋ t → ∞ ,

from (5.22), we infer that (1.12) holds if we let T:=T⋆ and N:=N⋆∪N1. Since (5.21) together with Lemma 3.2 ensures that

∫ Ω | u ⁢ ( ⋅ , t ) - u ⋆ | → 0 as ⁢ ( T ⋆ , ∞ ) ∖ N ⋆ ∋ t → ∞ ,

we furthermore see that also (1.11) is valid for this choice of T and N. ∎

Communicated by Sergio Solimini

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: 411007140

Funding statement: The author moreover acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Emergence of structures and advantages in cross-diffusion systems (No. 411007140, GZ: WI 3707/5-1).

A The Underlying Solution Concept

The following definition essentially adapts the one underlying the existence theory from [47] to the case of arbitrary positive parameters D, d, χ, κ and λ. We accordingly may refrain from detailing a discussion about how far this concept is consistent with that of classical solvability here, and rather refer the reader to [47] and the related precedent in [20] instead.

Definition A.1.

Let

(A.1) { u ∈ L loc α ⁢ ( Ω ¯ × [ 0 , ∞ ) ) , v ∈ L loc 1 ⁢ ( [ 0 , ∞ ) ; W 1 , 1 ⁢ ( Ω ) )

be nonnegative. Then we call (u,v) a global generalized solution of (1.2) if

(A.2) - ∫ 0 ∞ ∫ Ω v ⁢ φ t - ∫ Ω v 0 ⁢ φ ⁢ ( ⋅ , 0 ) = - d ⁢ ∫ 0 ∞ ∫ Ω ∇ ⁡ v ⋅ ∇ ⁡ φ - κ ⁢ ∫ 0 ∞ ∫ Ω v ⁢ φ + λ ⁢ ∫ 0 ∞ ∫ Ω u ⁢ φ

for all φ∈c0∞⁢(Ω¯×[0,∞)), if

(A.3) ∫ Ω u ⁢ ( ⋅ , t ) ≤ ∫ Ω u 0 + ∫ 0 t ∫ Ω ( ρ ⁢ u - μ ⁢ u α ) for a.e. ⁢ t > 0 ,

and if there exist functions ϕ∈C2⁢([0,∞)), ψ∈C2⁢([0,∞)) and Φ∈C2⁢([0,∞)) such that

(A.4) ϕ ′ < 0 , ψ > 0 and ϕ ′′ > 0 on ⁢ [ 0 , ∞ ) ,

that

(A.5) Φ ′ = ϕ ′′ on ⁢ [ 0 , ∞ ) ,

that

(A.6) { d ⁢ ϕ ⁢ ( u ) ⁢ ψ ′′ ⁢ ( v ) - ( D + d ) 2 4 ⁢ D ⁢ ϕ ′ ⁣ 2 ⁢ ( u ) ϕ ′′ ⁢ ( u ) ⋅ ψ ′ ⁣ 2 ⁢ ( v ) ψ ⁢ ( v ) - χ 2 4 ⁢ D ⁢ u 2 ⁢ ϕ ′′ ⁢ ( u ) ⁢ ψ ⁢ ( v ) + ( d - D ) ⁢ χ 2 ⁢ D ⁢ u ⁢ ϕ ′ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) } ⁢ | ∇ ⁡ v | 2 , u ⁢ ϕ ′ ⁢ ( u ) ⁢ ψ ⁢ ( v ) ⁢ | ∇ ⁡ v | , ϕ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) ⁢ | ∇ ⁡ v | and Φ ⁢ ( u ) ⁢ ϕ ′ ⁢ ( u ) ϕ ′′ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) ⁢ | ∇ ⁡ v | as well as u α ⁢ ϕ ′ ⁢ ( u ) ⁢ ψ ⁢ ( v ) , v ⁢ ϕ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) and u ⁢ ϕ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) belong to ⁢ L loc 1 ⁢ ( Ω ¯ × [ 0 , ∞ ) ) ,

that

(A.7) Φ ⁢ ( u ) ⁢ ψ ⁢ ( v ) ∈ L loc 2 ⁢ ( [ 0 , ∞ ) ; W 1 , 2 ⁢ ( Ω ) ) ,

and that, for each nonnegative φ∈C0∞⁢(Ω¯×[0,∞)), the inequality

- ∫ 0 ∞ ∫ Ω ϕ ⁢ ( u ) ⁢ ψ ⁢ ( v ) ⁢ φ t - ∫ Ω ϕ ⁢ ( u 0 ) ⁢ ψ ⁢ ( v 0 ) ⁢ φ ⁢ ( ⋅ , 0 )

≤ - D ∫ 0 ∞ ∫ Ω | ∇ ( Φ ( u ) ψ ⁢ ( v ) ) + { D + d 2 ⁢ D ϕ ′ ⁢ ( u ) ϕ ′′ ⁢ ( u ) ⋅ ψ ′ ⁢ ( v ) ψ ⁢ ( v ) - 1 2 Φ ( u ) ψ ′ ⁢ ( v ) ψ ⁢ ( v )

- χ 2 ⁢ D u ϕ ′′ ⁢ ( u ) ⋅ ψ ⁢ ( v ) } ∇ v | 2 φ

- ∫ 0 ∞ ∫ Ω { d ϕ ( u ) ψ ′′ ( v ) - ( D + d ) 2 4 ⁢ D ϕ ′ ⁣ 2 ⁢ ( u ) ϕ ′′ ⁢ ( u ) ⋅ ψ ′ ⁣ 2 ⁢ ( v ) ψ ⁢ ( v )

- χ 2 4 ⁢ D u 2 ϕ ′′ ( u ) ψ ( v ) + ( d - D ) ⁢ χ 2 ⁢ D u ϕ ′ ( u ) ψ ′ ( v ) } ⋅ | ∇ v | 2 φ

- D ⁢ ∫ 0 ∞ ∫ Ω ϕ ′ ⁢ ( u ) ϕ ′′ ⁢ ( u ) ⁢ ψ ⁢ ( v ) ⁢ ∇ ⁡ ( Φ ⁢ ( u ) ⁢ ψ ⁢ ( v ) ) ⋅ ∇ ⁡ φ

+ ∫ 0 ∞ ∫ Ω { χ ⁢ u ⁢ ϕ ′ ⁢ ( u ) ⁢ ψ ⁢ ( v ) - d ⁢ ϕ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) + D 2 ⁢ Φ ⁢ ( u ) ⁢ ϕ ′ ⁢ ( u ) ϕ ′′ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) } ⁢ ∇ ⁡ v ⋅ ∇ ⁡ φ

(A.8) + ∫ 0 ∞ ∫ Ω { ( ρ ⁢ u - μ ⁢ u α ) ⁢ ϕ ′ ⁢ ( u ) ⁢ ψ ⁢ ( v ) - κ ⁢ v ⁢ ϕ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) + λ ⁢ u ⁢ ϕ ⁢ ( u ) ⁢ ψ ′ ⁢ ( v ) } ⋅ φ

holds.

According to the analysis detailed in [47], with some appropriately chosen (εj)j∈ℕ⊂(0,1) fulfilling εj↘0 as j→∞, and with some null set N⊂(0,∞), the corresponding solutions (uε,vε) of (2.1) indeed satisfy (1.4)–(1.6) with some pair (u,v) which complies with (A.1)–(A.8) if we let

ϕ ( s ) : = ( s + 1 ) - r , Φ ( s ) : = - 2 r + 1 r ( s + 1 ) - r 2 and ψ ( s ¯ ) : = e - θ ⁢ s ¯ , s ≥ 0 , s ¯ ≥ 0 ,

and herein firstly take r>0 suitably small and then θ>0 adequately large such that

{ d - ( D + d ) 2 4 ⁢ D ⋅ r r + 1 } ⋅ θ 2 - ( D - d ) ⁢ χ ⁢ r 2 ⁢ D ⋅ θ - r ⁢ ( r + 1 ) 4 ⁢ D > 0 .

Acknowledgements

The author is very grateful to the anonymous reviewers, especially one of whom gave numerous useful suggestions that lead to a substantial improvement of this manuscript.

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Received: 2020-04-09

Revised: 2020-08-04

Accepted: 2020-08-22

Published Online: 2020-09-03

Published in Print: 2020-11-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

Abstract

1 Introduction

Asymptotics in Weakly Dampened Chemotaxis-Growth Systems. Analysis beyond Blow-up

Proposition 1.1.

Theorem 1.2.

Remark.

2 Preliminaries

2.1 Approximation of Generalized Solutions

2.2 Basic Bounds for uε. Absorbing Sets in L1

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

3 Construction of Eventual Energy Functionals at the Approximate Level

3.1 The Time Evolution of ∫Ω(uε-u⋆-u⋆⁢ln⁡uεu⋆)

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

3.2 Estimating ∫Ω(uεα-1-u⋆α-1)⋅(uε-u⋆) from below

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

3.3 The Evolution of ∫Ω(vε-vε¯)2

Lemma 3.6.

Proof.

3.4 An Eventual Lyapunov Property for Small χ

Lemma 3.7.

Proof.

Lemma 3.8.

Proof.

Lemma 3.9.

Proof.

Lemma 3.10.

Proof.

4 Persistence of Energy Decrease in the Limit Problem

Corollary 4.1.

Proof.

Lemma 4.2.

Proof.

Corollary 4.3.

Proof.

5 Convergence of (u,v) in the Large Time Limit

5.1 Stabilization of ∫Ωv

Lemma 5.1.

Proof.

Lemma 5.2.

Proof.

5.2 Decay of ℱ along a Subsequence

Lemma 5.3.

Proof.

5.3 Convergence. Proof of Theorem 1.2

Corollary 5.4.

Proof.

Proof of Theorem 1.2.

A The Underlying Solution Concept

Definition A.1.

Acknowledgements

References

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