Global solvability and asymptotical behavior in a two-species chemotaxis model with signal absorption
aa r X i v : . [ m a t h . A P ] F e b GLOBAL SOLVABILITY AND ASYMPTOTICAL BEHAVIOR INA TWO-SPECIES CHEMOTAXIS MODEL WITH SIGNALABSORPTION
GUOQIANG REN AND TIAN XIANG ∗ Abstract.
In this work, we study global existence, eventual smoothness andasymptotical behavior of positive solutions for the following two-species chemo-taxis consumption model: u t = ∆ u − χ ∇ · ( u ∇ w ) , x ∈ Ω , t > ,v t = ∆ v − χ ∇ · ( u ∇ w ) , x ∈ Ω , t > ,w t = ∆ w − ( αu + βv ) w, x ∈ Ω , t > , in a bounded smooth but not necessarily convex domain Ω ⊂ R n ( n = 2 , , , with nonnegative initial data u , v , w and homogeneous Neumann boundarydata. Here, the parameters χ , χ are positive and α, β are nonnegative.Under a smallness condition max { χ , χ }k w k L ∞ < π p /n , boundednessof classical solutions and stabilization to constant equilibrium have been shownin [47] . Here, without any smallness condition, we show global existence anduniform-in-time boundedness of classical solutions in 2D and global existence,eventual smoothness and asymptotical behavior (in convex domains) of weaksolutions in nD (n=3,4,5). Our findings also extend and improve the one-species chemotaxis-consumption model studied in [23, 27] . Introduction and sketch of the main results
In this project, we investigate the following Neumann initial-boundary valueproblem for a two-species chemotaxis system with consumption of chemoattractant: u t = ∆ u − χ ∇ · ( u ∇ w ) , x ∈ Ω , t > ,v t = ∆ v − χ ∇ · ( v ∇ w ) , x ∈ Ω , t > ,w t = ∆ w − ( αu + βv ) w, x ∈ Ω , t > , ∂u∂ν = ∂v∂ν = ∂w∂ν = 0 , x ∈ ∂ Ω , t > ,u ( x,
0) = u ( x ) , v ( x,
0) = v ( x ) , w ( x,
0) = w ( x ) , x ∈ Ω . (1.1)Hereafter, Ω ⊂ R n ( n ≥
1) is a bounded domain with a smooth boundary ∂ Ω and ∂∂ν denotes the outer normal derivative; the unknown variables u = u ( x, t ) and v = v ( x, t ) denote the population densities of two species and w represents theconcentration of the chemoattractant, χ , χ , α and β are positive constants and Mathematics Subject Classification.
Primary: 35K59, 35B65, 35B40, 35A09, 35K51; Sec-ondary: 35A01, 92D25.
Key words and phrases.
Two-species chemotaxis model, signal absorption, global existence,boundedness, asymptotics. ∗ Corresponding author. ∗ the given initial data are conveniently assumed throughout this paper to satisfy, forsome r > max { , n } , that( u , v , w ) ∈ C (Ω) × C (Ω) × W ,r (Ω) , u , v , w ≥ , . (1.2)The model (1.1) is used in mathematical biology to account the biased movement oftwo populations in respond to the concentration gradient of one common chemicalsignal. It is an obvious extension of the well-known Keller-Segel second model: u t = ∆ u − χ ∇ · ( u ∇ w ) , x ∈ Ω , t > ,w t = ∆ w − uw, x ∈ Ω , t > , (1.3)which and its variants have been studied mathematically in various contexts. Speak-ing of classical solutions, ifeither n ≤ χ k w k L ∞ (Ω) ≤ n + 1) , (1.4)then global boundedness of the solution ( u, w ) to (1.3) is ensured and further anyglobal bounded such solution converges uniformly according tolim t →∞ (cid:16) k u ( · , t ) − ¯ u k L ∞ (Ω) + k w ( · , t ) k L ∞ (Ω) (cid:17) = 0 , ¯ u = 1 | Ω | Z Ω u , (1.5)cf. [26, 41, 46]. Without smallness condition like (1.4), the problem possessesat least one global (certain) weak solution which is eventually smooth and enjoysthe convergence property (1.5) in 3D bounded convex domains [27]. As a simplestarting motivation, we are wondering whether such type weak solution contin-ues to exist in 4D or higher and, if so, whether it also enjoys (1.5). So far, it isstill widely open whether (1.3) possesses blow-ups in higher dimensions, only cer-tain blow-up properties of the local classical solutions to (1.3) are recently known[10]. For studies on chemotaxis-consumption systems with different boundary con-ditions, we refer the interested reader to the very recent works [5, 15]. For prop-erties of solutions in chemotaxis-consumption type models in more complex frame-work, for instance, with tensor-valued sensitivity, singular sensitivity, logistic source,predator-prey interaction or fluid interaction etc, we refer the interested reader to[1, 3, 11, 14, 36, 37, 38] and the references therein.It is well-known that logistic type source has an effective role in enhancing globalexistence, and boundedness in chemotaxis-involving systems. Indeed, a lot of studieshave been done to the IBVP (1.1) with Lotka-Volterra type competitive kinetics (thesame boundary and initial conditions are suspended): u t = ∆ u − χ ∇ · ( u ∇ w ) + µ u (1 − u − a v ) , x ∈ Ω , t > ,v t = ∆ v − χ ∇ · ( v ∇ w ) + µ v (1 − v − a u ) , x ∈ Ω , t > ,w t = ∆ w − ( αu + βv ) w, x ∈ Ω , t > . (1.6)Global boundedness of classical solutions to (1.6) are guaranteed undereither n ≤ { χ , χ } k w k L ∞ (Ω) < π √ n + 1 . WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 3
Such bounded solutions are known ([8, 12, 20, 21, 31]) to stabilize according to( u ( · , t ) , v ( · , t ) , w ( · , t )) in L ∞ (Ω) −→ (cid:16) − a − a a , − a − a a , (cid:17) , if a , a ∈ (0 , , (0 , , , if a ≥ > a > , (1 , , , if 0 < a < ≤ a . (1.7)Recently, global existence of generalized weak solutions and their long time behav-iors (similar to (1.7)) to (1.6) in nD are shown in [23] undermax { χ , χ } k w k L ∞ (Ω) < . Indeed, fluid interaction has been incorporated in (1.6), cf. [8, 12]. We alsomention that single or multiple-species signal-production type chemotaxis systemswith/without fluid interaction have been widely investigated also e.g. in [2, 4, 16,17, 18, 29, 30, 35, 42] and the references therein.Now, to formulate our main motivation of this project, we observe, without anydamping source, global existence and boundedness of classical solutions to the IBVP(1.1) in n D are obtained in [47] undermax { χ , χ } k w k L ∞ (Ω) < π r n . (1.8)Under this smallness condition, stabilization of solutions is also naturally derived:lim t →∞ (cid:16) k u ( · , t ) − ¯ u k L ∞ (Ω) + k v ( · , t ) − ¯ v k L ∞ (Ω) + k w ( · , t ) k L ∞ (Ω) (cid:17) = 0 . (1.9)Comparing these existing results, we find, even in the presence of Lotka-Volterratype competitive kinetics, certain smallness condition on initial data still needs to beimposed to have global existence, boundedness and convergence. A natural questionarises: whether and how far can we solve (1.1) globally without any damping source?More specifically, inspired from (1.4) and (1.8), we are wondering, first,(Q1) without the smallness condition in (1.8) with n = 2, can we still have 2Dglobal existence and boundedness of classical solutions to the IBVP (1.1)?Second, based on the existing knowledge about the one-species chemotaxis con-sumption model (1.3), cf. [23, 27, 38] and the references therein, we are wondering(Q2) without the smallness condition in (1.8), how far can we solve the two-species chemotaxis-consumption model (1.1) globally in a weak solutionsense in ≥
3D and, if so, how do such weak solutions behave after certainperhaps long waiting time?In this work, we shall answer (Q1) and (Q2) in a positive way for the IBVP (1.1):we show, without any smallness condition, global existence and uniform-in-timeboundedness of classical solutions in 2D and global existence, eventual smoothnessand asymptotical behavior (in convex domains) of weak solutions in n D ( n = 3 , , Theorem 1.1 ( Global dynamics for (1.1)) . Let χ , χ , α, β > and Ω ⊂ R n ( n ≤ be a bounded and smooth domain, and let u , v and w fulfill (1.2) .(B1) [ Global boundedness and convergence in D ] When n = 2 , the IBVP (1.1) has a unique global classical solution which is bounded on Ω × (0 , ∞ ) in the sense there exists C > such that k u ( · , t ) k L ∞ (Ω) + k v ( · , t ) k L ∞ (Ω) + k w ( · , t ) k W , ∞ (Ω) ≤ C, ∀ t > . (1.10) Naturally, such bounded solution converges according to (1.9) . GUOQIANG REN AND TIAN XIANG ∗ (B2) [ Global existence of weak solutions in , , D ] When n = 3 , , , thereexists at least one triple ( u, v, w ) of nonnegative functions satisfying u ∈ L n +2 n loc (Ω × [0 , ∞ )) ∩ L n +2 n +1 loc ([0 , ∞ ); W , n +2 n +1 (Ω)) ,v ∈ L n +2 n loc (Ω × [0 , ∞ )) ∩ L n +2 n +1 loc ([0 , ∞ ); W , n +2 n +1 (Ω)) and w ∈ L loc ([0 , ∞ ); W , (Ω)) , (1.11) which are a global weak solution of (1.1) in the sense of Definition 3.1 below.(B3) [ Eventual smoothness and convergence in convex domains ] When Ω ⊂ R n ( n = 3 , , is a smooth, bounded and convex domain, there exists T ∗ > such that the global weak solution obtained in (B2) is bounded,belongs to C , (Ω × [ T ∗ , ∞ )) and converges according to (1.9) . In 2D setting, our global boundedness of classical solutions in (B1) removes thesmallness condition (1.8) with n = 2 as required in [47]. In 3D setting, our globalexistence weak solutions relaxes the commonly used convexity assumption on Ωin the literature, cf. [23, 27, 38]. Moreover, the known eventual smoothness andlarge time behavior of weak solutions in 3D convex domains has been extended to4 and 5D and our findings also extend and improve the one-species chemotaxis-consumption model studied in [23, 27].The layout of this paper is structured as follows: In Sect. 2, we combine andextend existing technique to study local and global well-posedness with focus onboundedness and convergence in 2D for (1.1), which relies on a crucial evolutionidentity (2.9). In Sect. 3, we formulate the approximating system of (1.1), introducethe concept of weak solutions and derive basic properties of approximating solutions.In Sect. 4, we motivate and extend arguments mainly from [27, 38] to deriveglobal existence, eventual smoothness and convergence (in convex domains) of weaksolutions in 3, 4 and 5D, as detailed in Subsect. 4.2 and Subsect. 4.3. The convexityof domain Ω could be removed mainly because the boundary integral emerging fromthe identity (2.9) can be properly controlled in a manner as in (2.15).2. Boundedness and convergence in 2D
Basic facts and Local existence.
In the subsequent analysis, we shall needthe well-known Gagliardo-Nirenberg interpolation inequality, we list it here for con-venience of reference.
Lemma 2.1. (Gagliardo-Nirenberg interpolation inequality [6, 19] ) Let p ≥ and q ∈ (0 , p ) . Then there exists a positive constant C GN = C p,q such that k w k L p (Ω) ≤ C GN (cid:16) k∇ w k δL (Ω) k w k (1 − δ ) L q (Ω) + k w k L r (Ω) (cid:17) , ∀ w ∈ H (Ω) ∩ L q (Ω) , where r > is arbitrary and δ is given by p = δ ( 12 − n ) + 1 − δq ⇐⇒ δ = q − p q + n − ∈ (0 , . Next, we state the following well-established local solvability, extendibility andbasic estimates of solutions to the IBVP (1.1).
Lemma 2.2.
Let χ , χ , α, β > and Ω ⊂ R n ( n ≥ be a bounded and smoothdomain, and let u , v and w fulfill (1.2) with r > max { , n } . Then there is a WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 5 unique, positive and classical maximal solution ( u, v, w ) of the IBVP (1.1) on somemaximal interval [0 , T m ) with < T m ≤ ∞ such that ( u, v ) ∈ (cid:0) C (cid:0) Ω × [0 , T m ) (cid:1) ∩ C , (Ω × (0 , T m )) (cid:1) ,w ∈ C (cid:0) Ω × [0 , T m ) (cid:1) ∩ C , (Ω × (0 , T m )) ∩ W , ∞ loc ((0 , T m ) , W ,r (Ω)) . If T m < ∞ , then the following extensibility criterion holds: lim sup t ր T m (cid:0) k u ( · , t ) k L ∞ (Ω) + k v ( · , t ) k L ∞ (Ω) + k w ( · , t ) k W ,r (Ω) (cid:1) = ∞ . Furthermore, u and v have conservation of mass within (0 , T m ) : k u ( · , t ) k L (Ω) = k u k L (Ω) , k v ( · , t ) k L (Ω) = k v k L (Ω) , (2.1) and, for any p ∈ [1 , ∞ ] , the L p -norm of w is non-increasing: t
7→ k w ( · , t ) k L p (Ω) is non-increasing in [0 , T m ); (2.2) in particular, k w ( · , t ) k L p (Ω) ≤ k w k L p (Ω) . (2.3) Proof.
The local existence, regularity and extendibility of classical solutions to theIBVP (1.1) are based on contraction mapping argument and parabolic regularitytheory of parabolic systems, which can be found in [3, 25, 26, 32, 34, 39]. Theconservations of u and v in (2.1) follows upon integration by parts due to the no-flux boundary conditions, and the positivity of solution, (2.2) and hence (2.3) with p = ∞ follows from an application of the (strong) maximum principle. For the case p ∈ [1 , ∞ ), testing the w -equation by w p − and integrating by parts, we get ddt Z Ω w p = − ( p − Z Ω w p − |∇ w | − Z Ω ( αu + βv ) w ≤ , which upon being integrated from s to t entails (2.2). (cid:3) Henceforth, we will denote by C i various constants which may vary line by line.2.2. Uniform boundedness and global existence in 2D.
In this subsection, weshall extend the idea in [41] to show 2 D boundedness and global existence of classicalsolutions to the IBVP (1.1) without any smallness condition, thus proving (B1). Togain our gaol, we first establish a series of important a-priori estimates; we statethem in n dimensional setting since they are valid in arbitrary space dimensionsand they are quite convenient in subsequent sections.To move from L -boundness obtained in (2.1) and (2.2) to higher order L p -regularity, we first compute from the u - and v -equations in (1.1) that ddt R Ω u ln u + R Ω |∇ u | u = χ R Ω ∇ u ∇ w, ∀ t ∈ (0 , T m ) , ddt R Ω v ln v + R Ω |∇ v | v = χ R Ω ∇ v ∇ w, ∀ t ∈ (0 , T m ) . (2.4)To cancel out exactly the chemotaxis involving terms on the right-hand side of (2.4),inspired from [34, 41], we compute the following time evolution:12 ddt Z Ω |∇ w | w = Z Ω ∇ w ∇ w t w − Z Ω |∇ w | w w t . GUOQIANG REN AND TIAN XIANG ∗ For the first term, an integration by parts along with the w -equation entails that Z Ω ∇ w ∇ w t w = Z Ω w ∇ w · ∇ ∆ w − Z Ω ( αu + βv ) w |∇ w | − Z Ω ( α ∇ u + β ∇ v ) ∇ w = 12 Z ∂ Ω w ∂ |∇ w | ∂ν + 12 Z Ω ∇ w · ∇|∇ w | w − Z Ω | D w | w − Z Ω ( αu + βv ) w |∇ w | − α Z Ω ∇ u ∇ w − β Z Ω ∇ v ∇ w, where we have applied the following point-wise identity2 ∇ w · ∇ ∆ w = ∆ |∇ w | − | D w | , | D w | = n X i,j =1 | w x i x j | . In the same sprit, for the second term, one has12 Z Ω |∇ w | w w t = Z Ω |∇ w | w − Z Ω ∇ w · ∇|∇ w | w − Z Ω ( αu + βv ) w |∇ w | . Collecting these equalities together, we end up with12 ddt Z Ω |∇ w | w + (cid:18)Z Ω | D w | w − Z Ω ∇ w · ∇|∇ w | w + Z Ω |∇ w | w (cid:19) = 12 Z ∂ Ω w ∂ |∇ w | ∂ν − Z Ω ( αu + βv ) w |∇ w | − α Z Ω ∇ u ∇ w − β Z Ω ∇ v ∇ w. (2.5)For later use, cf. Lemma 2.3, it is a good place to notice (cf. [41, (3.9)]) that Z Ω w | D ln w | = Z Ω | D w | w − Z Ω ∇ w · D w · ∇ ww + Z Ω |∇ w | w . (2.6)Then the elementary inequality − ab ≥ − a − b implies Z Ω w | D ln w | ≥ Z Ω | D w | w − Z Ω |∇ w | w . (2.7)Plugging (2.6) into (2.5) and noticing ∇ w ∇|∇ w | = 2 ∇ w · D w · ∇ w , we get that12 ddt Z Ω |∇ w | w + Z Ω w | D ln w | + 12 Z Ω ( αu + βv ) w |∇ w | = 12 Z ∂ Ω w ∂ |∇ w | ∂ν − α Z Ω ∇ u ∇ w − β Z Ω ∇ v ∇ w. (2.8)Combining (2.4) with (2.8), we conclude an important identity for (1.1) as follows: ddt (cid:18)Z Ω αχ u ln u + βχ v ln v + χ χ |∇ w | w (cid:19) + αχ Z Ω |∇ u | u + βχ Z Ω |∇ v | v + χ χ Z Ω ( αu + βv ) w |∇ w | + χ χ Z Ω w | D ln w | = χ χ Z ∂ Ω w ∂ |∇ w | ∂ν . (2.9)Thanks to this crucial evolution identity, we then improve the basic estimates in(2.1) and (2.2) to ( L , L , L )-boundedness of ( u ln u, v ln v, ∇ w ) as follows. WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 7
Lemma 2.3.
Under the basic conditions in Lemma 2.2, for any τ ∈ (0 , T m ) , thereexists C = C ( u , v , w , τ, | Ω | ) > such that the solution ( u, v, w ) of (1.1) verifies Z Ω (cid:0) | u ln u | + | v ln v | + |∇ w | (cid:1) ( · , t ) ≤ C, ∀ t ∈ ( τ, T m ) . (2.10) Proof.
In the case that Ω is convex (as the case in [27]), notice that ∂∂ν |∇ w | ≤ ∂w∂ν = 0, and so the boundary integral on the right-hand side of (2.9) isnon-positive. Then a simple integration of (2.9) from τ to t gives rise to Z Ω (cid:18) αχ u ln u + βχ v ln v + χ χ |∇ w | w (cid:19) ( · , t ) ≤ Z Ω (cid:18) αχ u ln u + βχ v ln v + χ χ |∇ w | w (cid:19) ( · , τ ) , ∀ t ∈ ( τ, T m ) . (2.11)On the other hand, the algebraic fact that − z ln z ≤ e − for all z > Z Ω ( αχ | u ln u | + βχ | v ln v | ) ( · , t )= αχ Z Ω u ln u − Z {
0, there exists C ǫ > χ χ Z ∂ Ω w ∂ |∇ w | ∂ν ≤ ǫ Z Ω (cid:18) | D w | w + |∇ w | w (cid:19) + C ǫ Z Ω |∇ w | w ≤ ǫ Z Ω (cid:18) | D w | w + |∇ w | w (cid:19) + C ǫ Z Ω w ≤ ǫ (cid:8) (cid:2) (2 + √ n ) + 1 (cid:3) + (2 + √ n ) (cid:9) Z Ω w | D ln w | + C ǫ Z Ω w , (2.15) GUOQIANG REN AND TIAN XIANG ∗ where, from the first to the second inequality, we have used H¨older’s inequality toestimate: for any η >
0, there exists C η > Z Ω |∇ w | w ≤ η Z Ω |∇ w | w + C η Z Ω w. (2.16)Now, choosing ǫ = min (cid:26) , χ χ √ n ) + 1] + (2 + √ n ) (cid:27) in (2.15), and then, substituting (2.15) into (2.9), we obtain a key ordinary differ-ential inequality (ODI) as follows: ddt (cid:18)Z Ω αχ u ln u + βχ v ln v + χ χ |∇ w | w (cid:19) + αχ Z Ω |∇ u | u + βχ Z Ω |∇ v | v + χ χ Z Ω w | D ln w | ≤ C . (2.17)By the Gagliardo-Nirenberg inequality (cf. Lemma 2.1) and the mass conservationsof u and v in (2.1), one can easily show (cf. [41, (3.36)], for any η > Z Ω u ln u ≤ η Z Ω |∇ u | u + C η , Z Ω v ln v ≤ η Z Ω |∇ v | v + C η . (2.18)On the other hand, by (2.13) and (2.16), we readily infer that Z Ω |∇ w | w ≤ η Z Ω w | D ln w | + C η . (2.19)Combining (2.18) and (2.19) with (2.17) and choosing sufficiently small η > ddt (cid:18)Z Ω αχ u ln u + βχ v ln v + χ χ |∇ w | w (cid:19) + C (cid:18)Z Ω αχ u ln u + βχ v ln v + χ χ |∇ w | w (cid:19) ≤ C . (2.20)Solving the Gronwall inequality (2.20), using the trick as in (2.12) and noticing thefact w ≤ k w k L ∞ (Ω) , we finally end up with the desired estimate (2.10). (cid:3) In signal production single species chemotaxis models, the boundedness informa-tion provided in (2.10) is quite known to allow one to infer 2D global boundedness,cf. [3, 40, 44]. Here, in signal consumption multi-species cases, instead of usingthe technique used in [27], we shall also show that the compound boundedness in(2.10) enable us to derive first the ( L , L , L )-boundedness of ( u, v, ∇ w ) and then( L ∞ , L ∞ , W , ∞ )-boundedness of ( u, v, w ), which clarifies the boundedness proof forthe case of a ≤ t ]. Lemma 2.4.
Let Ω ⊂ R be bounded and smooth. Then for any τ ∈ (0 , T m ) , thereexists C = C ( u , v , w , τ, | Ω | ) > such that the solution ( u, v, w ) of (1.1) verifies Z Ω (cid:0) u + v + |∇ w | (cid:1) ( · , t ) ≤ C, ∀ t ∈ ( τ, T m ) . (2.21) WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 9
Proof.
Integrating by parts from (1.1) and using Cauchy-Schwarz inequality, we get ddt R Ω u + R Ω |∇ u | ≤ χ R Ω u |∇ w | , ∀ t ∈ (0 , T m ) , ddt R Ω v + R Ω |∇ v | ≤ χ R Ω v |∇ w | , ∀ t ∈ (0 , T m ) . (2.22)Similarly, taking gradient of the w -equation and then multiplying it by |∇ w | ∇ w and finally integrating over Ω by parts, we derive that ddt Z Ω |∇ w | + 2 Z Ω |∇|∇ w | | + 4 Z Ω |∇ w | | D w | = 2 Z ∂ Ω |∇ w | ∂∂ν |∇ w | + 4 Z Ω ( αu + βv ) w ∇|∇ w | ∇ w + 4 Z Ω ( αu + βv ) w |∇ w | ∆ w. (2.23)Next, based on (2.22) and (2.23), we estimate the terms on the right-hand side of(2.23). For the boundary integral, one can use (cf. [42, 43, 44]) the boundary traceembedding to bound it in terms of the boundedness of k∇ w k L in (2.10) to infer,for any ǫ >
0, there exists C ǫ > Z ∂ Ω |∇ w | ∂∂ν |∇ w | ≤ ǫ Z Ω |∇|∇ w | | + C ǫ (cid:18)Z Ω |∇ w | (cid:19) ≤ ǫ Z Ω |∇|∇ w | | + C ǫ , ∀ ǫ > . (2.24)Noticing k w k L ∞ ≤ k w k L ∞ , we deduce that4 Z Ω ( αu + βv ) w ∇|∇ w | ∇ w ≤ Z Ω |∇|∇ w | | + 8 k w k L ∞ Z Ω ( αu + βv ) |∇ w | ≤ Z Ω |∇|∇ w | | + 16 α k w k L ∞ Z Ω u |∇ w | + 16 β k w k L ∞ Z Ω v |∇ w | (2.25)and, similarly, since | ∆ w | ≤ | D w | , we obtain that4 Z Ω ( αu + βv ) w |∇ w | ∆ w ≤ Z Ω |∇ w | | D w | + 4 α k w k L ∞ Z Ω u |∇ w | + 4 β k w k L ∞ Z Ω v |∇ w | . (2.26)Combining the estimates (2.22), (2.23), (2.24) with ǫ = , (2.25) and (2.26), weconclude a Key ODI as follows, for t ∈ (0 , T m ), ddt Z Ω (cid:0) u + v + |∇ w | (cid:1) + Z Ω (cid:0) |∇ u | + |∇ v | + |∇|∇ w | | (cid:1) ≤ (cid:0) χ + 20 α k w k L ∞ (cid:1) Z Ω u |∇ w | + (cid:0) χ + 20 β k w k L ∞ (cid:1) Z Ω v |∇ w | . (2.27)Now, the Young’s inequality with epsilon shows, for any ǫ >
0, that Z Ω u |∇ w | + Z Ω v |∇ w | ≤ ǫ Z Ω |∇ w | + 23 √ ǫ Z Ω u + 23 √ ǫ Z Ω v . (2.28)Now, thanks to the compound boundedness information in (2.10), using the usual2D GN inequality as in Lemma 2.1 and its extended version involving logarithmic ∗ functions (cf. [28, Lemma A. 5] or [45, Lemma 3.4]), we can easily deduce, for any ǫ , ǫ , there exist C ǫ , C ǫ > C > R Ω u + R Ω u ≤ ǫ R Ω |∇ u | + C ǫ , R Ω v + R Ω v ≤ ǫ R Ω |∇ v | + C ǫ , R Ω |∇ w | ≤ C R Ω |∇|∇ w | | + C. (2.29)Substituting (2.28) and (2.29) into (2.27) and then choosing sufficiently small ǫ i > ddt Z Ω (cid:0) u + v + |∇ w | (cid:1) + Z Ω (cid:0) u + v + |∇ w | (cid:1) ≤ C, which immediately entails (2.21). (cid:3) Proof of 2D global existence, boundedness and convergence.
In light of thegained ( L , L , L )-boundedness of ( u, v, ∇ w ) in (2.21) and the L ∞ -boundedness of w in (2.3), using semigroup type arguments to w -equation, one can easily derivefirst W ,q -boundedness of w for any finite q , and then, testing the u, v -equations toderive ( L , L )-boundedness of ( u, v ), and then, using semigroup type argumentsto w -equation again to derive W , ∞ -boundedness of w , and finally, applying semi-group type arguments to u, v -equations to derive ( L ∞ , L ∞ , W , ∞ )-boundedness of( u, v, w ) as in (1.10), see details in e.g. [3, 45, 47], for instance. The convergence in(1.9) goes in the same way as [27, 47]. (cid:3) Preliminaries on weak solutions in 3D or higher dimensions
In this section, we first introduce the concept of weak solutions, and then, westate some useful lemmas for later use.
Definition 3.1.
By a global weak solution of (1.1) , we mean a triple ( u, v, w ) ofnonnegative functions n = 3 : u ∈ L loc ([0 , ∞ ); L (Ω)) ,v ∈ L loc ([0 , ∞ ); L (Ω)) and w ∈ L loc ([0 , ∞ ); W , (Ω)) ,n = 4 , u ∈ L loc ([0 , ∞ ); W , (Ω)) ,v ∈ L loc ([0 , ∞ ); W , (Ω)) and w ∈ L ∞ loc (Ω × [0 , ∞ )) ∩ L loc ([0 , ∞ ); W , (Ω)) , such that uw, vw, u ∇ w and v ∇ w belong to L loc ([0 , ∞ ); L (Ω)) and that the following identities motivated from integration by parts − Z ∞ Z Ω uϕ t − Z Ω u ϕ ( · ,
0) = Z ∞ Z Ω ∇ u · ∇ ϕ + χ Z ∞ Z Ω u ∇ w · ∇ ϕ, (3.1) − Z ∞ Z Ω vϕ t − Z Ω v ϕ ( · ,
0) = − Z ∞ Z Ω ∇ v · ∇ ϕ + χ Z ∞ Z Ω v ∇ w · ∇ ϕ (3.2) and − Z ∞ Z Ω wϕ t − Z Ω w ϕ ( · ,
0) = − Z ∞ Z Ω ∇ w · ∇ ϕ − Z ∞ Z Ω ( αu + βv ) wϕ (3.3) hold for all ϕ ∈ C ∞ (Ω × [0 , ∞ )) . WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 11
In order to achieve global solvability within this framework through an appro-priate regularization process, for ε ∈ (0 , F ε : [0 , ∞ ) R + by F ε ( s ) := ε ln(1 + εs ) , if n = 3 , s εs , if n = 4 , . (3.4)It then follows easily that the C ∞ ([0 , ∞ ))-family ( F ε ) ε ∈ (0 , has the properties that F ε (0) = 0 , F ε ( s ) → s as ε ց < F ′ ε ( s ) ≤ s ≥ , (3.5)and that, for any s ≥ ≤ F ′ ε ( s ) ր ε ց , ≤ sF ′ ε ( s ) ≤ ε , ≤ − sF ′′ ε ( s ) ≤ , ∀ ε ∈ (0 , . (3.6)Then, for ε ∈ (0 , u εt = ∆ u ε − χ ∇ · ( u ε F ′ ε ( u ε ) ∇ w ε ) , x ∈ Ω , t > ,v εt = ∆ v ε − χ ∇ · ( v ε F ′ ε ( v ε ) ∇ w ε ) , x ∈ Ω , t > ,w εt = ∆ w ε − ( αF ε ( u ε ) + βF ε ( v ε )) w ε , x ∈ Ω , t > , ∂u ε ∂ν = ∂v ε ∂ν = ∂w ε ∂ν = 0 , x ∈ ∂ Ω , t > ,u ε ( x,
0) = u ( x ) , v ε ( x,
0) = v ( x ) , w ε ( x,
0) = w ( x ) , x ∈ Ω . (3.7)The framework of contraction mapping argument first allows one to concludelocal well-posedness on (0 , T m,ε ), and then, given the basic estimates in Lemmas3.3 and 3.4 on (0 , T m,ε ) and the choice of F ε in (3.4), using Neumann semigroupestimates to the w -equation in (3.7), one can easily see that k∇ w ε k L ∞ is uniformlybounded on (0 , T m,ε ), and this allows one to conclude finally T m,ε = ∞ , namely, theglobal existence of classical solution to the approximating system (3.7); see, quitedetailed display of similar reasonings in related circumstances [9, 26, 27, 34, 38]. Lemma 3.2.
Let χ , χ > and α, β > and Ω ⊂ R n ( n ≥ be a bounded andsmooth domain, and, let F ε be defined by (3.4) and initial data ( u , v , w ) satisfy (1.2) . Then for each ε ∈ (0 , , the system (3.7) admits a global classical solution ( u ε , v ε , w ε ) such that u ε > , v ε > and w ε > on ¯Ω × (0 , ∞ ) . Lemma 3.3.
For all ε ∈ (0 , , the solution of (3.7) satisfies, for t > , k u ε ( · , t ) k L (Ω) = k u k L (Ω) , k v ε ( · , t ) k L (Ω) = k v k L (Ω) . (3.8) Proof.
Integrating the first and second equations in (3.7) and the no-flux boundaryconditions, we immediately obtain (3.8). (cid:3)
Lemma 3.4.
Let ε ∈ (0 , and p ∈ [1 , ∞ ] . Then the solution of (3.7) verifies t
7→ k w ε ( · , t ) k L p (Ω) is nonincreasing in [0 , ∞ ) . (3.9) In particular, k w ε ( · , t ) k L p (Ω) ≤ k w k L p (Ω) . (3.10) Proof.
Notice that F ε and α, β, u ε , v ε , w ε are nonnegative, we have w εt ≤ ∆ w ε , andthus, using the maximum principle and energy estimate, we readily derive (3.9) and(3.10), see details in Lemma 2.2. (cid:3) ∗ Lemma 3.5.
For all ε ∈ (0 , . we have α Z ∞ Z Ω F ε ( u ε ) w ε + β Z ∞ Z Ω F ε ( v ε ) w ε ≤ Z Ω w . (3.11) In particular, the limit triple ( u, v, w ) defined by Lemma 4.8 satisfies α Z ∞ Z Ω uw + β Z ∞ Z Ω vw ≤ Z Ω w . (3.12) Proof.
Integrating the third equation in (3.7) we get Z Ω w ε ( · , t ) + α Z t Z Ω F ε ( u ε ) w ε + β Z t Z Ω F ε ( v ε ) w ε = Z Ω w for all t ≥
0. Due to w ε ≥
0, it immediately derives (3.11). Applying the Fatou’slemma, we also readily conclude (3.12). This completes the proof. (cid:3) Global dynamics of weak solutions in 3,4 and 5D
In this section, we first establish important a-priori ε -independent estimates forclassical solutions to (3.7) and then we pass to the limit as ε → , , Lemma 4.1.
There exists positive constant K = K ( k u k L , k v k L , k w k L ) > such that the global solution of (3.7) fulfills, for all ε ∈ (0 , and t > , ddt Z Ω (cid:18) αχ u ε ln u ε + βχ v ε ln v ε + χ χ |∇ w ε | w ε (cid:19) + Z Ω (cid:18) αχ u ε ln u ε + βχ v ε ln v ε + χ χ |∇ w ε | w ε (cid:19) + αχ Z Ω |∇ u ε | u ε + βχ Z Ω |∇ v ε | v ε + χ χ Z Ω w ε | D ln w ε | + χ χ Z Ω ( αF ε ( u ε ) + βF ε ( v ε )) |∇ w ε | w ε ≤ , if Ω is convex ,K , if Ω is non-convex . (4.1) Hence, there exists K := K ( u , v , w ) > such that Z Ω (cid:18) | u ε ln u ε | + | v ε ln v ε | + |∇ w ε | + |∇ w ε | w ε (cid:19) ( · , t ) ≤ K , ∀ t ∈ (0 , ∞ ) . (4.2) WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 13
Proof.
Conducting similar computations leading to (2.9), we calculate that ddt Z Ω (cid:18) αχ u ε ln u ε + βχ v ε ln v ε + χ χ |∇ w ε | w ε (cid:19) + Z Ω (cid:18) αχ u ε ln u ε + βχ v ε ln v ε + χ χ |∇ w ε | w ε (cid:19) + αχ Z Ω |∇ u ε | u ε + βχ Z Ω |∇ v ε | v ε + χ χ Z Ω w ε | D ln w ε | + χ χ Z Ω ( αF ε ( u ε ) + βF ε ( v ε )) |∇ w ε | w ε = Z Ω (cid:18) αχ u ε ln u ε + βχ v ε ln v ε + χ χ |∇ w ε | w ε (cid:19) + χ χ Z ∂ Ω w ∂ |∇ w | ∂ν . (4.3)By the L -boundedness of u ε and v ε in (3.8), an straightforward application of(2.18) shows that αχ Z Ω u ε ln u ε ≤ αχ Z Ω |∇ u ε | u ε + C ( k u k L ) (4.4)and βχ Z Ω v ε ln v ε ≤ βχ Z Ω |∇ v ε | v ε + C ( k v k L ) . (4.5)Also, thanks to (3.10), a simple use of (2.19) entails χ χ Z Ω |∇ w ε | w ε ≤ χ χ Z Ω w ε | D ln w ε | + C ( k w k L ) (4.6)and, in the case that Ω is non-convex, an easy use of (2.15) shows that χ χ Z ∂ Ω w ε ∂ |∇ w ε | ∂ν ≤ , if Ω is convex , χ χ R Ω w ε | D ln w ε | + C ( k w k L ) , if Ω is non-convex . (4.7)Substituting (4.4), (4.5),(4.6) and (4.7) into (4.3), we derive (4.1). Since ( u ε , v ε , w ε )satisfies an ODI of the form (2.20), and so (4.2) follows similarly as (2.10). (cid:3) ε -independent estimates for the regularized problem.Lemma 4.2. There exists K = K ( u , v , w ) > such that the global solution of (3.7) fulfills, for all ε ∈ (0 , , Z ∞ Z Ω (cid:18) |∇ u ε | u ε + |∇ v ε | v ε + | D w ε | + |∇ w ε | (cid:19) + Z ∞ Z Ω (cid:0) F ε ( u ε ) |∇ w ε | + F ε ( v ε ) |∇ w ε | (cid:1) ≤ K , if Ω is convex. (4.8) There exists K = K ( u , v , w ) > such that the global solution of (3.7) fulfills,for any ε ∈ (0 , and t ∈ [0 , ∞ ) , Z t +1 t Z Ω (cid:18) |∇ u ε | u ε + |∇ v ε | v ε + | D w ε | + |∇ w ε | (cid:19) + Z t +1 t Z Ω (cid:0) F ε ( u ε ) |∇ w ε | + F ε ( v ε ) |∇ w ε | (cid:1) ≤ K , if Ω is non-convex . (4.9) ∗ Proof.
In the case that Ω is convex, integrating (4.3) with respect to t ∈ (0 , ∞ ) andusing the fact that − z ln z ≤ e − for all z >
0, we have Z t Z Ω (cid:18) αχ |∇ u ε | u ε + βχ |∇ v ε | v ε + χ χ w ε | D ln w ε | (cid:19) + χ χ Z t Z Ω ( αF ε ( u ε ) + βF ε ( v ε )) |∇ w ε | w ε ≤ Z Ω (cid:18) αχ u ln u + βχ v ln v + χ χ |∇ w | w (cid:19) − Z Ω ( αχ u ε ln u ε + βχ v ε ln v ε ) ≤ Z Ω (cid:18) αχ u ln u + βχ v ln v + χ χ |∇ w | w (cid:19) + ( αχ + βχ ) e − | Ω | . (4.10)In light of (2.13) and (2.14) and the fact w ε ≤ k w k L ∞ , we infer that1 k w k L ∞ Z t Z Ω |∇ w ε | ≤ Z t Z Ω |∇ w ε | w ε ≤ (2 + √ n ) Z t Z Ω w ε | D ln w ε | (4.11)and 1 k w k L ∞ Z t Z Ω | D w ε | ≤ (cid:2) (2 + √ n ) + 1 (cid:3) Z t Z Ω w ε | D ln w ε | . (4.12)Combining (4.11) and (4.12) into (4.10) and sending t → ∞ , we derive (4.8).Similarly, when Ω is nonconvex, substituting (4.7) into (4.3), we obtain that ddt Z Ω (cid:18) αχ u ε ln u ε + βχ v ε ln v ε + χ χ |∇ w ε | w ε (cid:19) + αχ Z Ω |∇ u ε | u ε + βχ Z Ω |∇ v ε | v ε + 3 χ χ Z Ω w ε | D ln w ε | + χ χ Z Ω ( αF ε ( u ε ) + βF ε ( v ε )) |∇ w ε | w ε ≤ C ( k w k L ) . (4.13)Integrating (4.13) from t to t + 1 and using (4.2), we achieve (4.9). (cid:3) Lemma 4.3.
There exists K = K ( u , v , w ) > such that the global solution of (3.7) fulfills, for all ε ∈ (0 , and for any t ∈ [0 , ∞ ) , Z t +1 t Z Ω (cid:16) u n +2 n ε + v n +2 n ε + |∇ u ε | n +2 n +1 + |∇ v ε | n +2 n +1 (cid:17) ≤ K . (4.14) Proof.
Given the mass conservation of u ε in (3.8), the Gagliardo-Nirenberg inequal-ity (cf. Lemma 2.1) allows us to deduce that Z Ω u n +2 n ε = k√ u ε k n +2) n L n +2) n ≤ C k∇√ u ε k L k√ u ε k n L + C k√ u ε k n +2) n L = C k u k n L Z Ω |∇ u ε | u ε + C k u k n +2 n L . (4.15)Likewise, one also that Z Ω v n +2 n ε ≤ C k v k n L Z Ω |∇ v ε | v ε + C k v k n +2 n L . (4.16) WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 15
Applying the Young inequality, we obtain Z Ω |∇ u ε | n +2 n +1 = Z Ω (cid:18) |∇ u ε | u ε (cid:19) n +22( n +1) · u n +22( n +1) ε ≤ Z Ω |∇ u ε | u ε + Z Ω u n +2 n ε . (4.17)Similarly, Z Ω |∇ v ε | n +2 n +1 ≤ Z Ω |∇ v ε | v ε + Z Ω v n +2 n ε . (4.18)For any t ≥
0, integrating (4.15), (4.16), (4.17) and (4.18) from t to t + 1, thenusing (4.8) or (4.9), we readily conclude (4.14). (cid:3) Lemma 4.4.
There exists K = K ( u , v , w ) > such that for all ε ∈ (0 , , theglobal solution of (3.7) fulfills Z ∞ (cid:16) k u ε − ¯ u k L nn − + k v ε − ¯ v k L nn − (cid:17) ≤ K , if Ω is convex ; (4.19) and, there exists K = K ( u , v , w ) > such that, for any ε ∈ (0 , and t ≥ , Z t +1 t (cid:16) k u ε − ¯ u k L nn − + | v ε − ¯ v k L nn − (cid:17) ≤ K , if Ω is non-convex . (4.20) Proof.
The Cauchy-Schwarz inequality entails that (cid:18)Z Ω |∇ u ε | (cid:19) + (cid:18)Z Ω |∇ v ε | (cid:19) ≤ k u k L Z Ω |∇ u ε | u ε + k v k L Z Ω |∇ v ε | v ε . Notice from (3.8) that ¯ u ε = ¯ u and ¯ v ε = ¯ v ; we then we infer from the Sobolevembedding W , (Ω) ֒ → L nn − (Ω) and the Poincare inequality that (cid:16) k u ε − ¯ u k L nn − + k v ε − ¯ v k L nn − (cid:17) ≤ C (cid:18)Z Ω |∇ u ε | (cid:19) + C (cid:18)Z Ω |∇ v ε | (cid:19) . Integrating those inequalities from 0 to t (if Ω is convex) or from t and t + 1 (if Ω isnon-convex) and making use of (4.8) or (4.9), we readily infer (4.19) and (4.20). (cid:3) In the sequel, for our subsequent compactness argument, we study the space-timeregularity of the time derivatives of solutions to the regularized system (3.7).
Lemma 4.5.
There exists K = K ( u , v , w ) > such that for all ε ∈ (0 , , theglobal solution of (3.7) fulfills Z ∞ Z Ω w εt ≤ K , if Ω is convex ; (4.21) and, there exists K = K ( u , v , w ) > such that, for any ε ∈ (0 , and t ≥ , Z t +1 t Z Ω w εt ≤ K , if Ω is non-convex . (4.22) ∗ Proof.
Multiplying the third equation in (3.7) by 2 w εt and then integrating over Ωby parts, we obtain that2 Z Ω w εt + ddt Z Ω |∇ w ε | = − α Z Ω F ε ( u ε )( w ε ) t − β Z Ω F ε ( v ε )( w ε ) t = − α ddt Z Ω F ε ( u ε ) w ε + α Z Ω F ′ ε ( u ε ) w ε u εt − β ddt Z Ω F ε ( v ε ) w ε + β Z Ω F ′ ε ( v ε ) w ε v εt ;that is, 2 Z Ω w εt + ddt Z Ω (cid:0) |∇ w ε | + αF ε ( u ε ) w ε + βF ε ( v ε ) w ε (cid:1) = α Z Ω F ′ ε ( u ε ) w ε u εt + β Z Ω F ′ ε ( v ε ) w ε v εt . (4.23)Using the first equation in (3.7) and integrating by parts, we get Z Ω F ′ ε ( u ε ) w ε u εt = − Z Ω F ′′ ε ( u ε ) w ε |∇ u ε | − Z Ω F ′ ε ( u ε ) w ε ∇ u ε · ∇ w ε + χ Z Ω F ′ ε ( u ε ) F ′′ ε ( u ε ) u ε w ε ∇ u ε · ∇ w ε + 2 χ Z Ω ( F ′ ε ( u ε )) u ε w ε |∇ w ε | =: H + H + H + H . (4.24)Since 0 ≤ − sF ′′ ε ( s ) ≤ w ε ≤ k w k L ∞ due to (3.6) and 3.10, we estimate H ≤ k w k L ∞ Z Ω u ε | F ′′ ε ( u ε ) | · |∇ u ε | u ε ≤ k w k L ∞ Z Ω |∇ u ε | u ε . (4.25)Similarly, by Young’s inequality, we estimate H as follows: H ≤ Z Ω F ε ( u ε ) |∇ w ε | + k w k L ∞ Z Ω u ε ( F ′ ε ( u ε )) F ε ( u ε ) · |∇ u ε | u ε ≤ Z Ω F ε ( u ε ) |∇ w ε | + k w k L ∞ Z Ω |∇ u ε | u ε , (4.26)where we used the following fact due to the definition of F ε in (3.4):0 ≤ s ( F ′ ε ( s )) F ε ( s ) = εs (1+ εs ) ln(1+ εs ) , if n = 3 , εs ) , if n = 4 , ≤ . (4.27)Analogously, the term H is bounded according to H ≤ k w k L ∞ Z Ω F ε ( u ε ) |∇ w ε | + χ k w k L ∞ Z Ω u ε ( F ′ ε ( u ε ) F ′′ ε ( u ε )) F ε ( u ε ) · |∇ u ε | u ε ≤ k w k L ∞ Z Ω F ε ( u ε ) |∇ w ε | + χ k w k L ∞ Z Ω |∇ u ε | u ε , (4.28) WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 17 where we used the following fact due to the definition of F ε in (3.4):0 ≤ s ( F ′ ε ( s ) F ′′ ε ( s )) F ε ( s ) = ( εs ) (1+ εs ) ln(1+ εs ) , if n = 3 , εs (1+ εs ) , if n = 4 , ≤ . Finally, we use (4.27) to bound H as H ≤ χ k w k L ∞ Z Ω u ε ( F ′ ε ( u ε )) F ε ( u ε ) · F ε ( u ε ) |∇ w ε | ≤ χ k w k L ∞ Z Ω F ε ( u ε ) |∇ w ε | . (4.29)Collecting (4.24),(4.25), (4.26), (4.28) and (4.29), we obtain that α Z Ω F ′ ε ( u ε ) w ε u εt ≤ (cid:0) χ (cid:1) α k w k L ∞ Z Ω |∇ u ε | u ε + (cid:0) χ k w k L ∞ + k w k L ∞ (cid:1) α Z Ω F ε ( u ε ) |∇ w ε | . (4.30)In a similar manner, one can show that β Z Ω F ′ ε ( v ε ) w ε v εt ≤ (cid:0) χ (cid:1) β k w k L ∞ Z Ω |∇ v ε | v ε + (cid:0) χ k w k L ∞ + k w k L ∞ (cid:1) β Z Ω F ε ( v ε ) |∇ w ε | . (4.31)For any 0 ≤ s ≤ t , integrating (4.23) from s to t and combining (4.30) with (4.31),we end up with2 Z ts Z Ω w εt ≤ Z Ω (cid:0) |∇ w ε | + αF ε ( u ε ) w ε + βF ε ( v ε ) w ε (cid:1) ( · , s )+ (cid:0) χ (cid:1) α k w k L ∞ Z ts Z Ω |∇ u ε | u ε + (cid:0) χ (cid:1) β k w k L ∞ Z ts Z Ω |∇ v ε | v ε + (cid:0) χ k w k L ∞ + k w k L ∞ (cid:1) α Z ts Z Ω F ε ( u ε ) |∇ w ε | + (cid:0) χ k w k L ∞ + k w k L ∞ (cid:1) β Z ts Z Ω F ε ( v ε ) |∇ w ε | . (4.32)Using the boundedness of k∇ w ε k L in (4.2) and the conservations of u ε and v ε ,0 ≤ F ε ( s ) ≤ s and w ε ≤ k w k L ∞ , cf. (3.6), 3.10 and (3.8), we see that Z Ω (cid:0) |∇ w ε | + αF ε ( u ε ) w ε + βF ε ( v ε ) w ε (cid:1) ( · , s ) ≤ K + ( α k u k L + β k v k L ) k w k L ∞ . Inserting this into (4.32) and using (4.8) or (4.9), we accomplish (4.21) or (4.22). (cid:3)
Lemma 4.6.
For m > n + 1 , there exists K = K ( u , v , w ) > such that theglobal solution of (3.7) fulfills, for all ε ∈ (0 , , Z ∞ (cid:16) k u εt ( · , t ) k W m, ) ∗ + k v εt ( · , t ) k W m, ) ∗ (cid:17) ≤ K , if Ω is convex ; (4.33) ∗ and, there exists K = K ( u , v , w ) > such that, for any ε ∈ (0 , and t ≥ , Z t +1 t (cid:16) k u εt ( · , s ) k W m, ) ∗ + k v εt ( · , s ) k W m, ) ∗ (cid:17) ≤ K , if Ω is non-convex . (4.34) Proof.
For given ϕ ∈ W m, , we multiply the first equation in (3.7) by ϕ , andintegrate over Ω by parts and use (4.27) to get (cid:12)(cid:12)(cid:12)(cid:12)Z Ω u εt ϕ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − Z Ω ∇ u ε · ∇ ϕ + χ Z Ω u ε F ′ ε ( u ε ) ∇ w ε · ∇ ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) Z Ω |∇ u ε | u ε (cid:17) · (cid:16) Z Ω u ε |∇ ϕ | (cid:17) + χ (cid:16) Z Ω F ε ( u ε ) |∇ w ε | (cid:17) · (cid:16) Z Ω u ε F ′ ε ( u ε ) F ε ( u ε ) |∇ ϕ | (cid:17) ≤ (cid:18)Z Ω |∇ u ε | u ε (cid:19) k u k L k∇ ϕ k L ∞ + χ (cid:18)Z Ω F ε ( u ε ) |∇ w ε | (cid:19) k u k L k∇ ϕ k L ∞ . Hence, the Sobolev embedding W m, (Ω) ֒ → W , ∞ (Ω) due to m > n shows that k u εt ( · , t ) k W m, ) ∗ ≤ C Z Ω |∇ u ε | u ε + C Z Ω F ε ( u ε ) |∇ w ε | , ∀ t > . Likewise, k v εt ( · , t ) k W m, ) ∗ ≤ C Z Ω |∇ v ε | v ε + C Z Ω F ε ( v ε ) |∇ w ε | , ∀ t > . By these two inequalities, we readily conclude (4.33) or (4.34) from (4.8) or (4.9). (cid:3)
Lemma 4.7.
For n ≤ , there exists K = K ( u , v , w ) > such that the globalsolution of (3.7) satisfies, for any ε ∈ (0 , and t > , Z t +1 t (cid:0) k u εt ( · , s ) k ( W , ∞ ) ∗ + k v εt ( · , s ) k ( W , ∞ ) ∗ + k w εt ( · , s ) k ( W , ∞ ) ∗ (cid:1) ≤ K . (4.35) Proof.
For any given ψ ∈ W , ∞ with k ψ k W , ∞ ≤
1, notice that 0 ≤ F ′ ≤ n +22 ≤ n ≤
6, and so by Young’s inequality, we get that (cid:12)(cid:12)(cid:12) Z Ω u εt ψ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) − Z Ω ∇ u ε · ∇ ϕ + χ Z Ω u ε F ′ ε ( u ε ) ∇ w ε · ∇ ψ (cid:12)(cid:12)(cid:12) ≤ Z Ω |∇ u ε | + χ Z Ω u ε |∇ w ε |≤ Z Ω |∇ u ε | n +2 n +1 + | Ω | + χ Z Ω u n +2 n ε + χ Z Ω |∇ w ε | n +22 ≤ Z Ω |∇ u ε | n +2 n +1 + (1 + χ ) | Ω | + χ Z Ω u n +2 n ε + χ Z Ω |∇ w ε | , which, upon being integrated from t to t + 1, gives rise to Z t +1 t k u εt k ( W , ∞ ) ∗ ≤ C (cid:18) Z t +1 t Z Ω (cid:16) u n +2 n ε + |∇ u ε | n +2 n +1 + |∇ w ε | (cid:17)(cid:19) . (4.36) WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 19
In the same manner, we have Z t +1 t k v εt k ( W , ∞ ) ∗ ≤ C (cid:18) Z t +1 t Z Ω (cid:16) v n +2 n ε + |∇ v ε | n +2 n +1 + |∇ w ε | (cid:17)(cid:19) , (4.37)and, using the third equation in (3.7) and the facts that k u ε k L = k u k L , k v ε k L = k v k L , 0 ≤ F ε ( s ) ≤ s and w ε ≤ k w k L ∞ , cf. (3.6), 3.10 and (3.8), we deduce that (cid:12)(cid:12)(cid:12) Z Ω w εt ψ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) − Z Ω ∇ w ε · ∇ ϕ − α Z Ω F ε ( u ε ) w ε ψ − β Z Ω F ε ( v ε ) w ε ψ (cid:12)(cid:12)(cid:12) ≤ Z Ω |∇ w ε | + α Z Ω u ε w ε + β Z Ω v ε w ε ≤ Z Ω |∇ w ε | + | Ω | + ( α k u k L + β k v k L ) k w k L ∞ , entailing Z t +1 t k w εt k ( W , ∞ ) ∗ ≤ C (cid:18) Z t +1 t Z Ω |∇ w ε | (cid:19) . (4.38)The desired estimate (4.35) follows from a combination of (4.36), (4.37) and (4.38)with Lemmas 4.2 and 4.3. (cid:3) Global existence of weak solutions in 3,4,5 D.
With the estimates gainedin last subsection, we can first derive strong compactness properties by means ofthe Aubin-Lions and then obtain the existence of weak solutions in in 3, 4 and 5Dvia extraction procedure, cf. [27, 34, 38].
Lemma 4.8.
For n ∈ { , , } , there exist ( ε j ) j ∈ N ⊂ (0 , and nonnegative func-tions u, v and w satisfying (1.11) such that ε j ց as j → ∞ and that the globalsolution of (3.7) satisfies, as ε = ε j ց , that ( u ε , v ε , w ε ) → ( u, v, w ) a.e. in Ω × (0 , ∞ ) , (4.39)( u ε , F ε ( u ε ) , u ε F ′ ε ( u ε )) → ( u, u, u ) in L p loc (Ω × [0 , ∞ )) , ∀ p ∈ [1 , n + 2 n ) , (4.40)( v ε , F ε ( v ε ) , v ε F ′ ε ( v ε )) → ( v, v, v ) in L p loc (Ω × [0 , ∞ )) , ∀ p ∈ [1 , n + 2 n ) , (4.41)( ∇ u ε , ∇ v ε ) ⇀ ( ∇ u, ∇ v ) in L n +2 n +1 loc (Ω × [0 , ∞ )) , (4.42) w ε ( · , t ) → w ( · , t ) in L q (Ω) f or a.e. t ∈ (0 , ∞ ) , (4.43) w ε ∗ ⇀ w in L ∞ loc (Ω × [0 , ∞ )) , ∇ w ε ⇀ ∇ w in L loc (Ω × [0 , ∞ )) and (4.44)( u ε F ′ ε ( u ε ) ∇ w ε , v ε F ′ ε ( v ε ) ∇ w ε ) ⇀ ( u ∇ w, v ∇ w ) in L loc (Ω × [0 , ∞ )) . (4.45) Here, the exponent q is related to the Sobolev conjugate number n ( n − + and satisfies q ∈ [1 , ∞ ] if n = 3; q ∈ [1 , ∞ ) if n = 4; q ∈ [1 , if n = 5 . (4.46) Moreover, ( u, v, w ) is a global weak solution of (1.1) in the sense of Definition 3.1.Proof. By Lemmas 3.3, 3.4, 4.1, 4.2, 4.3 and 4.7, for each
T >
0, we know that( u ε ) ε ∈ (0 , and ( v ε ) ε ∈ (0 , are bounded in L n +2 n +1 ((0 , T ); W , n +2 n +1 (Ω))and ( u εt ) ε ∈ (0 , and ( v εt ) ε ∈ (0 , are bounded in L ((0 , T ); ( W , ∞ (Ω)) ∗ )as well as ( w ε ) ε ∈ (0 , is bounded in L ((0 , T ); W , (Ω)) ∗ and ( w εt ) ε ∈ (0 , is bounded in L ((0 , T ); ( W , ∞ (Ω)) ∗ ) . By the compact embeddings W , n +2 n +1 (Ω) ֒ → L (Ω) and W , (Ω) ֒ → L q (Ω) for all q ∈ [1 , ∞ ] with 1 − n > − nq , twice direct applications of the Aubin-Lions lemma [24],we see there exist ( ε j ) j ∈ N ⊂ (0 ,
1) and nonnegative functions u, v and w fulfilling(1.11) and such that, for any such q and ε = ε j ց u ε → u and v ε → v in L (Ω × (0 , T )) (4.47)as well as w ε → w in L ((0 , T ); L q (Ω)) . (4.48)Now, based on (4.47) and (4.48), upon passing to a subsequence if necessary, we caninfer that (4.39) and (4.43) hold, and also that (4.42) and (4.44) hold. Using the a.e.convergence in (4.39), the boundedness in (4.14) and the properties of F ε in (3.5) and(3.6), we use the Vitali convergence theorem (roughly, a.e. convergence plus uniformintegrability imply convergence) to infer (4.40) and (4.41). Using the convergencefeatures in (4.40), (4.41) and (4.44) and noting the fact that nn +2 + < n ≤ (cid:3) Proof of 3,4, 5D global existence of weak solutions.
The statement on 3,4and 5D global existence of weak solutions has been fully contained in Lemma 4.8. (cid:3)
Large time behavior of global weak solutions in convex domains.
Inthis subsection, we focus on the eventual smoothness and stabilization of globalweak solutions to (1.1), that is, the limiting functions ( u, v, w ) of ( u ε , v ε , w ε ). Lemma 4.9.
For n ∈ { , , } , there exists ( ε j ) j ∈ N ⊂ (0 , of numbers ε j ց such that as ε = ε j ց , the global solution of (3.7) fulfill the following properties: w ∈ C ([0 , ∞ ); L (Ω)) (4.49) and w ε → w in L ∞ loc ([0 , ∞ ); L (Ω)) . (4.50) Proof.
In light of the ε -independent estimates provided in Lemmas 3.4 and 4.5, weshall adapt the arguments in [27, Corollary 5.3] to derive (4.49) and (4.50). As amatter of fact, for any T >
0, it follows from Lemmas 3.4 and 4.5 that ( w ε ) ε ∈ (0 , isbounded in L ∞ ((0 , T ); L ∞ (Ω)) and that ( w εt ) ε ∈ (0 , is bounded in L ((0 , T ); L (Ω)).Then, arguing as [27, Corollary 5.3], we see thatsup t ∈ (0 ,T ) k w ε ( · , t ) k L + sup t = s, t,s ∈ (0 ,T ) |k w ε ( · , t ) k L − k w ε ( · , s ) k L || t − s | ≤ C , that is, ( w ε ) ε ∈ (0 , is bounded in C ([0 , T ]; L (Ω)), and thus is relatively compactin C ([0 , T ]; L (Ω)) by the Arzela and Ascoli compactness theorem. This establishes(4.50) and thus (4.49). (cid:3) In the sequel, we fix ( ε = ε j ) j ∈ N so that Lemmas 4.8 and 4.9 hold. With thestrong convergence provided in Lemma 4.14 below, the following lemma follows. Lemma 4.10.
With ( u ε , v ε , w ε ) replaced by ( u, v, w ) constructed in Lemma 4.8, theconclusions of Lemmas 3.3, 3.4 and 4.2 through 4.7 still hold. WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 21
Lemma 4.11.
The global weak solution of (1.1) constructed in Lemma 4.8 fulfills w ( · , t ) → in L ∞ (Ω) as t → ∞ . (4.51) Proof.
We shall extend the arguments in [27, Lemma 5.4] (with a small flaw) for n = 3 to higher dimensional cases. Notice from Lemmas 3.4 and 4.2 that k w k L ∞ (Ω × (0 , ∞ )) + Z t +1 t Z Ω |∇ w | ≤ C , ∀ t ≥ . This ensures the existence of a sequence times t k → ∞ with 1 + t k ≤ t k +1 ≤ t k such that ( w ( · , t k )) k ∈ N is bounded in W , (Ω). This together with the compactembedding from W , (Ω) into L q (Ω) for q satisfying (4.46) allows us to deduce, upto a subsequence, for some nonnegative function w ∞ , that w ( · , t k ) → w ∞ in L q (Ω) as k → ∞ . (4.52)Using Cauchy-Schwarz inequality, we infer that Z t k t k Z Ω | w ε ( · , t ) − w ε ( · , t k ) | = Z t k t k Z Ω (cid:18)Z tt k w εt ( · , s ) (cid:19) ≤ Z t k t k Z Ω w εt ( · , t ) , ∀ ε ∈ (0 , . In view of Lemma 4.5, upon passing to the limit ε = ε j ց
0, this gives rise to Z t k t k Z Ω | w ( · , t ) − w ( · , t k ) | ≤ Z t k t k Z Ω w t ( · , t ) → k → ∞ . This together with (4.52) with q = 2 implies that12 Z t k t k Z Ω | w ( · , t ) − w ∞ | ≤ k w ( · , t k ) − w ∞ k L + Z t k t k Z Ω | w ( · , t ) − w ( · , t k ) | → k → ∞ . (4.53)Extracting Lemma 4.4 from Lemma 4.10, we have Z t k t k (cid:16) k u − ¯ u k L nn − + k v − ¯ v k L nn − (cid:17) → k → ∞ . (4.54) ∗ Since w ε ≤ k w k L ∞ , we use H¨older’s inequality to estimate, for all k ∈ N , that Z t k t k Z Ω | ( αu + βv ) w − ( α ¯ u + β ¯ v ) w ∞ |≤ Z t k t k Z Ω ( α | u − ¯ u | + β | v − ¯ v | ) w + ( α ¯ u + β ¯ v ) Z t k t k Z Ω | w − w ∞ |≤ " α (cid:18)Z t k t k k u − ¯ u k L nn − (cid:19) + β (cid:18)Z t k t k k v − ¯ v k L nn − (cid:19) t k t k k w k L n (cid:19) + ( α ¯ u + β ¯ v ) | Ω | (cid:18)Z t k t k k w − w ∞ k L (cid:19) ≤ " α (cid:18)Z t k t k k u − ¯ u k L nn − (cid:19) + β (cid:18)Z t k t k k v − ¯ v k L nn − (cid:19) k w k L ∞ | Ω | + ( α ¯ u + β ¯ v ) | Ω | (cid:18)Z t k t k k w − w ∞ k L (cid:19) , and so, we obtain from (4.53) and (4.54) that Z t k t k Z Ω ( αu + βv ) w → ( α ¯ u + β ¯ v ) Z Ω w ∞ as k → ∞ . (4.55)On the other hand, by (3.12) and the fact 1 + t k ≤ t k +1 , we have ∞ X k =1 Z t k t k Z Ω ( αu + βv ) w ≤ Z ∞ Z Ω ( αu + βv ) w < + ∞ . Since ¯ u , ¯ v >
0, this couples with (4.55) implies w ∞ ≡
0, and so (4.52) becomes w ( · , t k ) → in L q (Ω) as k → ∞ . This together with the fact that t
7→ k w ( · , t ) k L q is non-increasing by Lemma 3.4shows actually that w ( · , t ) → in L q (Ω) as k → ∞ . (4.56)Since k w ( · , t ) k L ∞ is non-increasing in t and is nonnegative, we see, as t → ∞ , that k w ( · , t ) k L ∞ converges decreasingly to some a ≥
0. If a >
0, then, for any η ∈ (0 , a )and t >
0, we define Ω( t ) = { x ∈ Ω : w ( x, t ) ≥ a − η } , and thus we get k w ( · , t ) k L q = (cid:18)Z Ω w q ( · , t ) (cid:19) q ≥ ( a − η ) | Ω( t ) | q , which couples with (4.56) immediately yields lim t →∞ | Ω( t ) | = 0. While, this isincompatible with the fact that k w ( · , t ) k L ∞ ≥ a for all t ≥
0. Therefore, we musthave a = 0, and so (4.51) follows. (cid:3) Lemma 4.12.
For n ∈ { , , } and for any δ > , there exist t ( δ ) > and ε ( δ ) ∈ (0 , such that for all ε ∈ ( ε j ) j ∈ N satisfying ε < ε ( δ ) , the w -component ofthe global classical solution of (3.7) fulfills ≤ w ε ≤ δ on Ω × ( t ( δ ) , ∞ ) . (4.57) WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 23
Proof.
For given δ > q satisfying (4.46), it follows from Lemma 4.11 thereexists ˆ t > w defined by Lemma (4.8) fulfills k w ( · , t ) k L q ≤ δ/ t > ˆ t . Thanks to (4.43), there exists t ∈ (ˆ t , ˆ t + 1) such that w ε ( · , t ) → w ( · , t ) in L q (Ω) as ε = ε j ց
0. This joined with the fact from Lemma 3.4 that t
7→ k w ( · , t ) k L q is non-increasing ensures there exists ε ( δ ) > t ≥ t and ε ∈ ( ε j ) j ∈ N with ε < ε ( δ ), k w ε ( · , t ) k L q ≤ k w ε ( · , t ) k L q ≤ k w ε ( · , t ) − w ( · , t ) k L q + k w ( · , t ) k L q ≤ δ k w ( · , t ) k L q ≤ δ. (4.58)Now, let a = lim sup ( ε,t ) → (0 , ∞ ) k w ε ( · , t ) k L ∞ ; if a >
0, then, for any η ∈ (0 , a ), wedefine E εt = { x ∈ Ω : w ε ( x, t ) ≥ a − η } , and then we derive k w ε ( · , t ) k L q = (cid:18)Z Ω w qε ( · , t ) (cid:19) q ≥ ( a − η ) | E εt | q , which couples with (4.58) immediately yields lim sup ( ε,t ) → (0 , ∞ ) | E εt | = 0. While,this is contradictory to the definitions of a and E εt , and therefore, we must have a = 0, and so the decay estimate (4.57) follows. (cid:3) Lemma 4.13.
For n ∈ { , , } and p ∈ (1 , ∞ ) , there exist t ( p ) > , ε ( p ) ∈ (0 , and K ( p ) = K ( u , v , w , p ) > such that for ε ∈ ( ε j ) j ∈ N with ε < ε ( p ) , theglobal classical solution of (3.7) satisfies Z Ω u pε ( · , t ) + Z Ω v pε ( · , t ) ≤ K ( p ) , ∀ t ≥ t ( p ) . (4.59) Proof.
With Lemmas 4.11 and 4.12 at hand, we can extend the arguments in [27,Lemma 6.2] for n = 3 to its higher dimensional cases (4.59). To this end, for p ∈ (1 , ∞ ) and for r ∈ (0 , p − (cid:26) ≤ z < r + 1 p , p − − p r · r + ( p − z r + 1 − pz > (cid:27) ⇐⇒ ≤ z < p − − r ) √ pr ( p − p (cid:16)p ( p − r + 1) + √ pr (cid:17) and 2( p − − r ) √ pr ( p − p (cid:16)p ( p − r + 1) + √ pr (cid:17) > p − − r ) rp ( p + p ) = ( p − − r ) rp , and thus, we see, for δ = min (cid:26) , χ , χ (cid:27) min (cid:26) ( p − − r ) r p , r + 12 p (cid:27) = min (cid:26) , χ , χ (cid:27) ( p − − r ) r p > , (4.60)we have that A i := p (2 δ ) − r (cid:26) p − − p r · r + ( p − (2 δχ i ) r + 1 − δpχ i (cid:27) > , i = 1 , . (4.61)With these preparations, we now specify φ as follows: φ ( s ) = (2 δ − s ) − r , s ∈ [0 , δ ) . ∗ Now, for ε ∈ ( ε j ) j ∈ N satisfying ε < ε ( δ ) and t > t ( δ ) with ε ( δ ) and t ( δ )prescribed by Lemma 4.12, we see that φ ( w ε ) is well-defined, and, moreover, we useintegration by parts to compute from (3.7) that ddt Z Ω ( u pε + v pε ) φ ( w ε )= − ( p − p Z Ω (cid:0) u p − ε |∇ u ε | + v p − ε |∇ v ε | (cid:1) φ ( w ε ) − Z Ω [ φ ′′ ( w ε ) − pχ F ′ ε ( u ε ) φ ′ ( w ε )] u pε |∇ w ε | − Z Ω [ φ ′′ ( w ε ) − pχ F ′ ε ( v ε ) φ ′ ( w ε )] v pε |∇ w ε | + p Z Ω u p − ε [ − φ ′ ( w ε ) + ( p − χ F ′ ε ( u ε ) φ ( w ε )] ∇ u ε ∇ w ε + p Z Ω v p − ε [ − φ ′ ( w ε ) + ( p − χ F ′ ε ( v ε ) φ ( w ε )] ∇ v ε ∇ w ε − Z Ω [ αF ε ( u ε ) + βF ε ( v ε )] ( u pε + v pε ) φ ′ ( w ε )=: I + I + I + I + I + I . (4.62)By the nonnegativity of u ε , v ε , w ε , F ε , φ ′ , we first see obviously that I ≤
0, andthen, by the facts 0 ≤ w ε ≤ δ in (4.57), 0 ≤ F ′ ε ( s ) ≤ δ in (4.60), we infer that φ ′′ ( w ε ) − pχ F ′ ε ( u ε ) φ ′ ( w ε ) ≥ φ ′′ ( w ε ) − pχ φ ′ ( w ε ) ≥ r (2 δ ) − r − ( r + 1 − pχ δ ) > ,φ ′′ ( w ε ) − pχ F ′ ε ( v ε ) φ ′ ( w ε ) ≥ φ ′′ ( w ε ) − pχ φ ′ ( w ε ) ≥ r (2 δ ) − r − ( r + 1 − pχ δ ) > . Hence, we employ Young’s inequality to estimate I as I ≤ − I + p Z Ω [ − φ ′ ( w ε ) + ( p − χ F ′ ε ( u ε ) φ ( w ε )] φ ′′ ( w ε ) − pχ φ ′ ( w ε ) u p − ε |∇ u ε | , (4.63)and then, based on (4.62), we further use the facts 0 ≤ F ′ ε ( s ) ≤
1, the choices of δ and A in (4.60) and (4.61) to estimate, for ( x, t, s ) ∈ Ω × ( t ( δ ) , ∞ ) × [0 , δ ], B ( x, t, s ) := p ( p − φ ( s ) − p · [ − φ ′ ( s ) + ( p − χ F ′ ε ( u ε ) φ ( s )] φ ′′ ( s ) − pχ φ ′ ( s ) ≥ p ( p − φ ( s ) − p · φ ′ ( s ) + ( p − χ φ ( s ) φ ′′ ( s ) − pχ φ ′ ( s )= p (2 δ − s ) − r (cid:26) p − − p r · r + ( p − (2 δ − s ) χ r + 1 − (2 δ − s ) pχ (cid:27) ≥ p (2 δ ) − r (cid:26) p − − p r · r + ( p − (2 δχ ) r + 1 − δpχ (cid:27) = A > . Combining this with (4.63), we conclude that I ≤ − I + ( p − p Z Ω φ ( w ε ) u p − ε |∇ u ε | − A Z Ω u p − ε |∇ u ε | . (4.64)In the same reasoning, we readily estimate the term I as I ≤ − I + ( p − p Z Ω φ ( w ε ) v p − ε |∇ v ε | − A Z Ω v p − ε |∇ v ε | . (4.65) WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 25
Substituting (4.63), (4.64) and (4.65) into (4.62) and recalling I ≤
0, we infer that ddt Z Ω ( u pε + v pε ) φ ( w ε ) ≤ − A Z Ω u p − ε |∇ u ε | − A Z Ω v p − ε |∇ v ε | = − A p Z Ω |∇ u p ε | − A p Z Ω |∇ v p ε | . (4.66)Next, due to the mass conservations of u ε and v ε in (3.8), by the GN inequality (cf.Lemma 2.1) and the fact that φ ( w ε ) ≤ δ − r , we infer that Z Ω u pε φ ( w ε ) ≤ δ − r k u p ε k L ≤ C k∇ u p ε k n ( p − p − n +2 L k u p ε k p − n +2 L p + C k u p ε k L p ≤ C k∇ u p ε k n ( p − p − n +2 L + C and, similarly, that Z Ω v pε φ ( w ε ) ≤ C k∇ v p ε k n ( p − p − n +2 L + C . Setting y ε ( t ) = R Ω ( u pε + v pε ) φ ( w ε )( · , t ), we derive from (4.66) an ODI as follows: y ′ ε ( t ) ≤ − C ( y ε ( t ) − ( p − n +2 n ( p − + ⇐⇒ (cid:20) ( y ε ( t ) − − p − n + (cid:21) ′ ≥ C ( p − n , t > t ( δ ) . An integration enables us to deduce that y ε ( t ) ≤ (cid:18) C ( p − n ( t − t ( δ )) (cid:19) − ( p − n , t > t ( δ ) . Recalling the definition of y ε and φ ( w ε ) ≥ (2 δ ) − r , we immediately arrive at Z Ω ( u pε + v pε ) ≤ (2 δ ) r y ε ( t ) ≤ (2 δ ) r (cid:18) C ( p − n (cid:19) − ( p − n , t ≥ t ( δ ) , yielding our desired estimate (4.59) upon setting t ( δ ) = 1 + t ( δ ). (cid:3) With the uniform eventual L p -boundedness of u ε and v ε in Lemma 4.13, it isquite standard via bootstrap argument or semigroup technique (cf. [9, 27] to obtainthe uniform eventual C -boundedness of weak solutions. Lemma 4.14.
For n ∈ { , , } , there exist T > and K = K ( u , v , w ) > such that for ε ∈ ( ε j i ) i ∈ N of ( ε j ) j ∈ N , the global classical solution of (3.7) fulfills k u ε ( · , t ) k C (Ω) + k v ε ( · , t ) k C (Ω) ≤ K , ∀ t ≥ T, (4.67) and such that u ε → u, v ε → v and w ε → w in C , loc (Ω × [ T, ∞ )) as ε = ε j i ց . (4.68) Proof.
By Lemma 4.13, for p > n , there exist t = t ( p ) > ε ( p ) ∈ (0 , ε ∈ ( ε j ) j ∈ N with ε < ε ( p ), the solution of (3.7) satisfies k u ε ( · , t ) k L p (Ω) + k v ε ( · , t ) k L p (Ω) ≤ C , ∀ t ≥ t . (4.69)We use the variation-of-constants formula for w ε to write w ε ( · , t ) = e t (∆ − w ε ( · , t ) − Z tt e ( t − s )(∆ − ( αF ε ( u ε ) + βF ε ( v ε ) − w ε ( · , s ) ds, t ≥ t . ∗ Since | F ε ( s ) | ≤ | s | and w ε ≤ k w k L ∞ (Ω) , we employ the well-known smoothing L p - L q -estimates for the Neumann heat semigroup { e t ∆ } t ≥ (c.f. [9, 33]) to infer k∇ w ε ( · , t ) k L p (Ω) ≤ C k w ε ( · , t ) k L ∞ (Ω) + C Z tt h t − s ) − i e − ( t − s ) × ( α k u ε ( · , s ) k L p + β k v ε ( · , s ) k L p + 1) ds ≤ C , t ≥ t + 1 . (4.70)Given (4.69) and (4.70), H¨older’s inequality enables us to infer that k u ε ( · , t ) ∇ w ε ( · , t ) k L p + k v ε ( · , t ) ∇ w ε ( · , t ) k L p ≤ ( k u ε ( · , t ) k L p + k v ε ( · , t ) k L p ) k∇ w ε ( · , t ) k L p ≤ C , t ≥ t := t + 1 . (4.71)Applying the variation-of-constants formula for u ε , we have u ε ( · , t ) = e t ∆ u ε ( · , t ) − χ Z tt e ( t − τ )∆ ∇ · ( u ε F ′ ε ( u ε ) ∇ w ε )( · , τ ) dτ, t ≥ t . This together with (4.69) and (4.71) allows us to find θ ∈ (0 , γ ∈ (0 ,
1) and q > θ − nq > k A θ u ε ( · , t ) k L q ≤ C , ∀ t > t := t + 1 (4.72)and, for any t, s ≥ t such that | t − s | ≤ k A θ u ε ( · , t ) − A θ u ε ( · , s ) k L q ≤ C | t − s | γ , (4.73)where A θ denotes the fractional power of the realization of − ∆ + 1 in L q (Ω) un-der homogeneous Neumann boundary conditions. By the continuous embedding D ( A θ ) ֒ → C σ for all σ ∈ (0 , θ − nq ) (cf. [7, 9, 40]), (4.72) and (4.73), we knowthat ( u ε ) ε ∈ ( ε j ) j ∈ N is bounded in both L ∞ (Ω × ( t , ∞ )) and in C σ, σ loc (Ω × [ t , ∞ ))for some σ ∈ (0 , v ε ) ε ∈ ( ε j ) j ∈ N is bounded in both L ∞ (Ω × ( t , ∞ )) and in C σ, σ loc (Ω × [ t , ∞ )) for some σ ∈ (0 , w ε ) ε ∈ ( ε j ) j ∈ N in both L ∞ (( t , ∞ ); C σ (Ω)) and in C σ, σ loc (Ω × [ t , ∞ ))with t = t + 1. This, in turn, by a similar argument, we also obtain theboundedness of ( u ε ) ε ∈ ( ε j ) j ∈ N and ( v ε ) ε ∈ ( ε j ) j ∈ N in both L ∞ (( t , ∞ ); C σ ′ (Ω)) andin C σ ′ , σ ′ loc (Ω × [ t , ∞ )) for some σ ′ ∈ (0 ,
1) and t = t + 1, and thus (4.67)follows. Finally, an application of the Arzel`a-Ascoli theorem implies (4.68). (cid:3) Lemma 4.15.
For n ∈ { , , } , the global weak solution of (1.1) constructed fromLemma 4.8 satisfies u ( · , t ) → ¯ u and v ( · , t ) → ¯ v in L ∞ (Ω) as t → ∞ , (4.74) where ¯ u and ¯ v are the average of u and v over Ω , respectively.Proof. With the information provided by Lemmas 4.4, 4.6 and 4.14, we can readilyextend and adapt the arguments in [27, Lemma 7.2] to derive (4.74). Indeed, let usassume to the contrary there exists a sequence of t k → ∞ such that d := inf k ∈ N k u ( · , t k ) − ¯ u k L ∞ > , (4.75)where we with no loss of generality can assume that all t k > T and 1 + t k ≤ t k +1 as T provided by Lemma 4.14. In light of (4.67) and (4.68), ( u ( · , t k )) k ∈ N is relatively WO-SPECIES CHEMOTAXIS MODEL WITH SIGNAL ABSORPTION 27 compact, and then by the Arzel`a-Ascoli theorem, we may assume for convenience,for some nonnegative function u ∞ , that u ( · , t k ) → u ∞ in L ∞ (Ω) as k → ∞ . (4.76)In the sprit of Lemma 4.11, we use the Cauchy-Schwarz inequality to estimate that Z t k t k k u ε ( · , t ) − u ε ( · , t k ) k W m, ) ∗ = Z t k t k (cid:13)(cid:13)(cid:13)(cid:13)Z tt k u εt ( · , s ) (cid:13)(cid:13)(cid:13)(cid:13) W m, ) ∗ ≤ Z t k t k k u εt ( · , s ) k W m, ) ∗ , ∀ ε ∈ (0 , . In view of (4.33) in Lemma 4.6, upon passing to the limit ε = ε j ց
0, this yields Z t k t k k u ( · , t ) − u ( · , t k ) k W m, ) ∗ ≤ Z t k t k k u t ( · , s ) k → k → ∞ . Since L ∞ (Ω) ֒ → (cid:0) W m, (Ω) (cid:1) ∗ due to m > n +1, it then follows from (4.76) that Z t k t k k u ( · , t k ) − u ∞ k W m, ) ∗ → k → ∞ . Hence, we use triangle inequality to deduce from the above two estimates that Z t k t k k u ( · , t ) − u ∞ k W m, ) ∗ → k → ∞ . (4.77)On the other hand, notice also that L nn − (Ω) ֒ → (cid:0) W m, (Ω) (cid:1) ∗ due to m > n +1; Thecontent of Lemma 4.4 from Lemma 4.10 ensures that Z t k t k k u ( · , t ) − ¯ u k W m, ) ∗ → k → ∞ . (4.78)By uniqueness, it follows from (4.77) and (4.78) that u ∞ ≡ ¯ u , which is impossibleby (4.75) and (4.76). This contradiction says that the u -limit in (4.74) is true;similarly, the v -limits follows in the same way. (cid:3) Proof of eventual smoothness and convergence in convex domains.
Theeventual smoothness and boundedness of weak solutions result immediately from(4.67) and (4.68) in Lemma 4.14. The convergence of weak solutions as in (1.9)follows from Lemmas 4.14 and 4.15. (cid:3)
Acknowledgments
G. Ren was supported by the National Natural ScienceFoundation of China (No.12001214) and the Postdoctoral Science Foundation (Nos.2020M672319, 2020TQ0111). T. Xiang was funded by the National Natural ScienceFoundation of China (Nos. 12071476 and 11871226).
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School of Mathematics and Statistics, Huazhong University of Science and Technol-ogy, Wuhan, 430074, Hubei, P. R. China; Hubei Key Laboratory of Engineering Model-ing and Scientific Computing, Huazhong University of Science and Technology, Wuhan,430074, Hubei, P. R. China
Email address : Institute for Mathematical Sciences, Renmin University of China, Bejing, 100872,China
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