Influence of flux limitation on large time behavior in a three-dimensional chemotaxis-Stokes system modeling coral fertilization
aa r X i v : . [ m a t h . A P ] F e b Influence of flux limitation on large time behavior in athree-dimensional chemotaxis-Stokes system modeling coralfertilization
Ji Liu ∗ College of Sciences, Nanjing Agricultural UniversityNanjing 210095, People’s Republic of China
Abstract:
In this paper, we consider the following system n t + u · ∇ n = ∆ n − ∇ · ( n S ( |∇ c | ) ∇ c ) − nm,c t + u · ∇ c = ∆ c − c + m,m t + u · ∇ m = ∆ m − mn,u t = ∆ u + ∇ P + ( n + m ) ∇ Φ , ∇ · u = 0which models the process of coral fertilization, in a smoothly three-dimensional boundeddomain, where S is a given function fulfilling |S ( σ ) | ≤ K S (1 + σ ) − θ , σ ≥ K S > . Based on conditional estimates of the quantity c and the gradients thereof,a relatively compressed argument as compared to that proceeding in related precedents showsthat if θ > , then for any initial data with proper regularity an associated initial-boundary problem un-der no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solutionwhich is globally bounded, and which also enjoys the stabilization features in the sense that k n ( · , t ) − n ∞ k L ∞ (Ω) + k c ( · , t ) − m ∞ k W , ∞ (Ω) + k m ( · , t ) − m ∞ k W , ∞ (Ω) + k u ( · , t ) k L ∞ (Ω) → t → ∞ with n ∞ := | Ω | (cid:8)R Ω n − R Ω m (cid:9) + and m ∞ := | Ω | (cid:8)R Ω m − R Ω n (cid:9) + . Key words:
Chemotaxis; Stokes; Flux limitation; Large time behavior : 35B40; 35K55; 35Q92; 35Q35; 92C17 ∗ Corresponding author.E-mail addresses: [email protected](J. Liu). Introduction
As shown in the experiments [10, 11, 28, 29], during the process of coral fertilization, chemotaxisphenomenon may occur in terms of oriented motion of sperms in response to some kind of chemicalsignal released by eggs. In mathematics, it can be modeled by n t + u · ∇ n = ∆ n − ∇ · ( n S ( n, c ) ∇ c ) − nm,c t + u · ∇ c = ∆ c − c + m,m t + u · ∇ m = ∆ m − mn,u t + κ ( u · ∇ ) u = ∆ u + ∇ P + ( n + m ) ∇ Φ , (1.1)where both n and m denote the densities of unfertilized sperms and eggs, respectively, c representsthe concentration of the signal, u stands for the velocity of the ambient ocean flow, P expressesthe pressure within the fluid and Φ is a given function used to denote the gravitational potential([14]).In the simplified case when n ≡ m, mathematical results on global dynamics of system (1.1) canbe found in [19, 20, 13], where, in particular, it is shown in [19, 20] that with given fluid velocity thethoroughness level of the fertilization depends on whether or not the role of chemotaxis comes intoplay, and where if the fluid flow is slow, i.e. κ = 0 in (1.1), but the velocity thereof is unknown, thensystem (1.1) is globally solvable in the weak sense ([13]). Whereas for the more realistic situationthat n m, complete consumption of eggs needs not only sufficiently high concentration of signalbut also adequately large initial densities for both sperms and eggs ([12]).Thereafter, the global dynamics of system (1.1) with n m and with an unknown fluid velocity u, which is referred as a more challenging case, becomes an interesting subject. Specially, in spatiallytwo-dimensional setting, it is proved that system (1.1) with κ = 1 possesses a unique classicalsolution which is globally bounded and approaches to some constant equilibrium as time goes toinfinity ([14]). While for physically most relevant three-dimensional case, the corresponding globalsolvability of (1.1) in the context of Stokes-fluid or Navier–Stokes-fluid requires some smallnesshypothesis for initial data ([22, 27, 7, 18]), or necessary aids of some type of nonlinear mechanism,such as porous medium diffusion ([26]), or signal-dependent sensitivity ([21]), or saturation effectsof cells ([22, 18, 24, 38]), or p -Laplace diffusion of cells ([25]).Recently, from some refined models proposed in [1, 5, 6], it can be observed that the migration ofcells rests with some gradient-dependent limitations, which inspires investigations of global dynam-ics on associated initial-boundary problems. In particular, it is uncovered that global solvabilitycan be enforced by appropriately strong p-Laplace type cell diffusion ([4, 23, 32, 33, 39, 25]), andthat the precise parameters which describe the critical interaction of flux-limited cross diffusion andrelativistic heat equation type diffusion can be achieved by constructions of suitable sub-solutionson the basis of a comparison argument ([2, 3, 30, 8]).When merely flux limitation of signal is taken into account, that is the function S in (1.1) issupposed to generalize the prototype S ( σ ) = K S (1 + σ ) − θ , for any σ ≥ K S > θ > , the simplified parabolic-elliptic Keller–Segel system under homogeneousNeumann boundary conditions is globally solvable with a bounded solution in the classical senseunder the assumptions that |S ( σ ) | ≤ K S (1 + σ ) − θ , for any σ ≥ θ = θ c := N − N − , where N ≥ S ( σ ) > k S (1 + σ ) − θ , for any σ ≥ k S > θ < θ c , spontaneous aggregation in the sense of unbounded density mayemerge at some finite time for N ≥ θ for global solvability coincides with (1.3) inspatially three-dimensional setting, i.e. θ > . In view of the above results, we intend to discover that to which extent the saturation effectsinduced by S fulfilling (1.3) can prevent the phenomenon of explosion to occur spontaneously toan associated initial-boundary problem with the coral fertilization model (1.1). Particularly, theprecise problem we consider herein is n t + u · ∇ n = ∆ n − ∇ · ( n S ( |∇ c | ) ∇ c ) − nm, x ∈ Ω , t > ,c t + u · ∇ c = ∆ c − c + m, x ∈ Ω , t > ,m t + u · ∇ m = ∆ m − mn, x ∈ Ω , t > ,u t = ∆ u + ∇ P + ( n + m ) ∇ Φ , ∇ · u = 0 , x ∈ Ω , t > , ∂n∂ν = ∂c∂ν = ∂m∂ν = 0 , u = 0 , x ∈ ∂ Ω , t > ,n ( x,
0) = n ( x ) , c ( x,
0) = c ( x ) , m ( x,
0) = m ( x ) , u ( x,
0) = u ( x ) , x ∈ Ω , (1.4)where Ω ⊂ R is a general bounded domain with smooth boundary, where the gravitational potentialΦ complies with Φ ∈ W , ∞ (Ω) , (1.5)and where for certain K > n , c , m , u ) fulfills n ∈ C ( ¯Ω) such that n ≥ k n k C (Ω) ≤ K ,c ∈ W , ∞ (Ω) such that c ≥ k c k W , ∞ (Ω) ≤ K ,m ∈ W , ∞ (Ω) such that m ≥ k m k W , ∞ (Ω) ≤ K ,u ∈ S α ∈ ( , D ( A α ) such that k A α u k L (Ω) ≤ K (1.6)with A := −P ∆ standing for the realization of the Stokes operator whose domain is D ( A ) := W , (Ω; R ) T W , ,σ (Ω) , where P denotes the Helmholtz projection on L (Ω; R ) , and where W , ,σ (Ω) := W , (Ω; R ) T L σ (Ω) with L σ (Ω) := { ω ∈ L (Ω; R ) |∇ · ω = 0 } . Based on well-established conditional regularity features of the fluid field u ([36]), a suitableapplication of smoothness estimates for Neumann heat semigroup can establish an inequality whichshows a relationship between uniform L ∞ estimates of the signal gradient and the conditional L p n, which underlies the derivation of the following results on global boundedness (inSection 5). Theorem 1.1
Assume that both (1.5) and (1.6) are valid. Let
S ∈ C ([0 , ∞ )) fulfill |S ( σ ) | ≤ K S (1 + σ ) − θ , for any σ ≥ with certain K S > and θ > . Then one can find a unique quadruple of functions n ∈ C ( ¯Ω × [0 , ∞ )) T C , ( ¯Ω × (0 , ∞ )) ,c ∈ T l> C ([0 , ∞ ); W ,l (Ω)) T C , ( ¯Ω × (0 , ∞ )) ,m ∈ T r> C ([0 , ∞ ); W ,r (Ω)) T C , ( ¯Ω × (0 , ∞ )) ,u ∈ S α ∈ ( , C ([0 , ∞ ); D ( A α )) T C , ( ¯Ω × (0 , ∞ ); R ) (1.8) with n ≥ , c ≥ and m ≥ in Ω × (0 , ∞ ) such that combined with some P ∈ C , (Ω × (0 , ∞ )) , thequintuple ( n, c, m, u, P ) constitutes a classical solution of (1.4) in Ω × (0 , ∞ ) , and is also boundedin line with k n ( · , t ) k L ∞ (Ω) + k c ( · , t ) k W , ∞ (Ω) + k m ( · , t ) k W , ∞ (Ω) + k A α u ( · , t ) k L (Ω) ≤ C for all t > with some α ∈ ( , and C > . Relying on the boundedness features in various forms with regards to each component of thesolution and on the basic relaxation properties exhibited in Lemma 5.1 below, we are also ableto achieve the following statement on the convergence for each component of the solution as timebecomes arbitrarily large.
Theorem 1.2
The global classical solution constructed in Theorem 1.1 has the property that k n ( · , t ) − n ∞ k L ∞ (Ω) + k c ( · , t ) − m ∞ k W , ∞ (Ω) + k m ( · , t ) − m ∞ k W , ∞ (Ω) + k u ( · , t ) k L ∞ (Ω) → as t → ∞ , (1.10) where n ∞ := | Ω | (cid:8)R Ω n − R Ω m (cid:9) + and m ∞ := | Ω | (cid:8)R Ω m − R Ω n (cid:9) + . More precisely, with the aids of the basic relaxation features in Lemma 5.1 as well as the uniform L ∞ bounds of n provided by Theorem 1.1, some constant equilibrium can be detected at first forthe solution which converges in space with lower regularity, such as L space (see subsection 5.1).In subsequence, by means of an associated Ehrling lemma, this in conjunction with the H¨olderregularities of both n and u as well as of the gradients of both m and c improves the regularity ofthe space, in which the solution stabilizes towards the constant equilibrium, to the desired level asasserted in Theorem 1.2 (see subsection 5.2). In light of the treatments for closely related problems ([37]), the following assertions on localsolvability and extensibility of (1.4) are valid, though accounted for the evolution of one morequantity m. emma 2.1 Let the initial data ( n , c , m , u ) comply with (1.6), and let both (1.5) and (1.7)hold. Then one can find T max ∈ (0 , + ∞ ] and a unique quadruple of functions ( n, c, m, u ) satisfying n ∈ C ( ¯Ω × [0 , T max )) T C , ( ¯Ω × (0 , T max )) ,c ∈ T l> C ([0 , T max ); W ,l (Ω)) T C , ( ¯Ω × (0 , T max )) ,m ∈ T r> C ([0 , T max ); W ,r (Ω)) T C , ( ¯Ω × (0 , T max )) ,u ∈ S α ∈ ( , C ([0 , T max ); D ( A α )) T C , ( ¯Ω × (0 , T max ); R ) , (2.1) and n ≥ , c ≥ in Ω × (0 , T max ) , such that there exists some P ∈ C , (Ω × (0 , T max )) which alongwith ( n, c, m, u ) solves (1.4) classically in Ω × (0 , T max ) , and thateither T max = ∞ or for any α ∈ ( 34 , , lim t ր T max sup n k n ( · , t ) k L ∞ (Ω) + k c ( · , t ) k W ,l (Ω) + k m ( · , t ) k W ,r (Ω) + k A α u ( · , t ) k L (Ω) o = ∞ (2.2) is valid. Proof.
Lemma 2.1 follows from an appropriate modification of the reasoning of [37, Lemma2.1].Thanks to the nonnegativity of both n and m, we also have the basic estimates as follows. Lemma 2.2
Suppose ( n, c, m, u ) is the solution as provided in Lemma 2.1. We have Z Ω n ( · , t ) ≤ Z Ω n , Z Ω m ( · , t ) ≤ Z Ω m for each t ∈ (0 , T max ) (2.3) and k m ( · , t ) k L ∞ (Ω) ≤ k m k L ∞ (Ω) for each t ∈ (0 , T max ) (2.4) as well as k c ( · , t ) k L ∞ (Ω) ≤ max {k c k L ∞ (Ω) , k m k L ∞ (Ω) } for each t ∈ (0 , T max ) . (2.5) Proof.
In view of the nonnegativity of n and m, (2.3) follows from straightforward integrationsof n -equation and m -equation in (1.4), respectively. Also due to n ≥ m ≥ , (2.4) can beinferred from the maximum principle. By virtue of (2.4), (2.5) is a consequence of the comparisonprinciple.The following conditional estimates of the fluid velocity, which are derived from well-establishedreasoning frameworks in [36, Proposition 1.1 and Corollary 2.1], will play a helpful role in theanalysis of Section 3. Lemma 2.3
Let ( n, c, m, u ) be a solution constructed in Lemma 2.1. Then for some α ∈ ( , , p ≥ , q > and δ > , there exist K = K ( α, p, q, δ ) > and K = K ( α, p, q, δ ) > withthe properties that k A α u ( · , t ) k L (Ω) ≤ K · ( s ∈ (0 ,t ) k n ( · , s ) k L p (Ω) ) pp − · ( α − + δ ) for any t ∈ (0 , T max ) (2.6)5 nd ,whereafter, that k u ( · , t ) k L q (Ω) ≤ K · ( s ∈ (0 ,t ) k n ( · , s ) k L p (Ω) ) pp − · (cid:16) q − q + δ (cid:17) for any t ∈ (0 , T max ) . (2.7) Proof.
Based on u -equation in (1.4), it follows from the reasoning of [36, Proposition 1.1] that k A α u ( · , t ) k L (Ω) ≤ C · ( s ∈ (0 ,t ) k n ( · , s ) + m ( · , s ) k L p (Ω) ) pp − · ( α − + δ ) (2.8)with some C = C ( K , | Ω | , p, α, δ ) > ∈ (0 , T max ) . From (1.6) and (2.4), we deduce that k n ( · , t ) + m ( · , t ) k L p (Ω) ≤k n ( · , t ) k L p (Ω) + k m ( · , t ) k L p (Ω) ≤k n ( · , t ) k L p (Ω) + K | Ω | p (2.9)for any t ∈ (0 , T max ) . Substituting (2.9) into (2.8) thus shows that k A α u ( · , t ) k L (Ω) ≤ C · ( K | Ω | p + sup s ∈ (0 ,t ) k n ( · , s ) k L p (Ω) ) pp − · ( α − + δ ) ≤ C · ( s ∈ (0 ,t ) k n ( · , s ) k L p (Ω) ) pp − · ( α − + δ )with C := C · (1 + K | Ω | p ) for any t ∈ (0 , T max ) , which implies (2.6). According to the argumentsof [36, Corollary 2.1], there exists some C = C ( K , | Ω | , p, q, δ ) > k u ( · , t ) k L q (Ω) ≤ C · ( s ∈ (0 ,t ) k n ( · , s ) + m ( · , s ) k L p (Ω) ) pp − · (cid:16) q − q + δ (cid:17) for any t ∈ (0 , T max ) , from which and (2.9) we can establish (2.7) in a similar manner. The aim of this section is to establish a temporally independent L ∞ bound for ∇ c, subject to aconditional estimate appearing as that on the right hand sides of (2.6) and (2.7). Recalling thearguments pursuing in [36], it is essential to resort to higher order conditional estimates as compareto W , ∞ -topology, which is based on a combination of the estimates provided by Lemma 2.2 withproper applications of the L p - L q estimates of the sectorial operator ([17]). Here and throughoutthe sequel, we abbreviate B := B l to stand for the sectorial operator − ∆ + 1 under homogeneousNeumann boundary conditions in T l> L l (Ω) , and use ( B µ ) µ> to denote the family of positivefractional powers B µ = B µl . Moreover, in order to express in more consice forms, we let H p ( t ) := 1 + sup s ∈ (0 ,t ) k n ( · , s ) k L p (Ω) , t ∈ (0 , T max ) (3.1)6nd J l,µ ( t ) := 1 + sup s ∈ (0 ,t ) (cid:13)(cid:13) B µ ( c ( · , s ) − e − sB c ) (cid:13)(cid:13) L l (Ω) , t ∈ (0 , T max ) . (3.2) Lemma 3.1
Suppose that µ ∈ ( , and l > . Then for all δ > there exists some C = C ( µ, l, δ ) > such that (cid:13)(cid:13) ∇ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L ∞ (Ω) ≤ C · ( s ∈ (0 ,t ) (cid:13)(cid:13) B µ ( c ( · , s ) − e − sB c ) (cid:13)(cid:13) L l (Ω) ) l +32 µl + δ (3.3) for all t ∈ (0 , T max ) . Proof.
Since µ ∈ ( ,
1) allows for choices of l > δ > − l +32 µl > δ < − l +32 µl , we can thus take ν ( δ ) := l + 32 l + δµ < µ. (3.4)Invoking the interpolation inequality established in [15, Theorem 2.14.1] for fractional powers ofsectorial operators, we can find C = C ( µ, l, δ ) > (cid:13)(cid:13) B ν ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L l (Ω) ≤ C (cid:13)(cid:13) B µ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) νµ L l (Ω) (cid:13)(cid:13) c ( · , t ) − e − sB c (cid:13)(cid:13) µ − νµ L l (Ω) ≤ C n K | Ω | l o − δ − l +32 µl (cid:13)(cid:13) B µ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) l +32 µl + δL l (Ω) with K > t ∈ (0 , T max ) , which combined with the continuous embedding D ( B ν ) ֒ → W , ∞ (Ω) ([16]) provides C = C ( µ, l, δ ) > (cid:13)(cid:13) ∇ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L ∞ (Ω) ≤ C (cid:13)(cid:13) B ν ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L l (Ω) ≤ C C n K | Ω | l o − δ − l +32 µl (cid:13)(cid:13) B µ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) l +32 µl + δL l (Ω) for all t ∈ (0 , T max ) , and thus (3.3) follows with C := C C n K | Ω | l o − δ − l +32 µl . Relying on Lemma 3.1, we can achieve the following conditional estimates in topology of D ( B µ ) . Lemma 3.2
Let l > , p ≥ and µ ∈ ( , . Then for any δ > one can find C = C ( l, p, µ, δ ) > satisfying (cid:13)(cid:13) B µ ( c ( · , s ) − e − sB c ) (cid:13)(cid:13) L l (Ω) ≤ C · ( s ∈ (0 ,t ) k n ( · , s ) k L p (Ω) ) pp − · ( µ + δ ) (3.5) for all t ∈ (0 , T max ) . Proof.
Taking δ > δ < min (cid:26) − l + 32 µl , l (1 − µ ) (cid:27) , (3.6)we let q := 3 l l (1 − µ ) − δ . (3.7)7hen thanks to δ < l (1 − µ ) and to l > l − lµ + 2 lµδ > l − lµ implied by (3.6), onecan see that l = 3 l l − lµ − l (1 − µ ) > q = 3 l l − lµ − δ > l l − lµ > . (3.8)Now, applying B µ to both sides of the variation-of-constants representation c ( · , t ) − e − tB c = Z t e − B ( t − s ) { m ( · , s ) − u ( · , s ) ∇ c ( · , s ) } ds for all t ∈ (0 , T max ) , we obtain (cid:13)(cid:13) B µ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L l (Ω) ≤ Z t (cid:13)(cid:13)(cid:13) B µ e − B ( t − s ) m ( · , s ) (cid:13)(cid:13)(cid:13) L l (Ω) ds + Z t (cid:13)(cid:13)(cid:13) B µ e − B ( t − s ) u ( · , s ) ∇ c ( · , s ) (cid:13)(cid:13)(cid:13) L l (Ω) ds for all t ∈ (0 , T max ) , where in accordance with the L p - L q estimates of the sectorial operator ([17]),it follows from (1.6), (2.5), (2.7), (3.1), (3.2) and (3.3) that Z t (cid:13)(cid:13)(cid:13) B µ e − B ( t − s ) m ( · , s ) (cid:13)(cid:13)(cid:13) L l (Ω) ds ≤ C Z t (cid:0) t − s ) − µ (cid:1) e − ( t − s ) k m ( · , s ) k L l (Ω) ds ≤ C K | Ω | l Z t (cid:0) t − s ) − µ (cid:1) e − ( t − s ) ds ≤ C for all t ∈ (0 , T max ) with some C > C := C K | Ω | l R ∞ (1 + ρ − µ ) e − ρ dρ < ∞ due to µ ∈ ( , , and that Z t (cid:13)(cid:13)(cid:13) B µ e − B ( t − s ) u ( · , s ) ∇ c ( · , s ) (cid:13)(cid:13)(cid:13) L l (Ω) ds ≤ C Z t (cid:16) t − s ) − µ − ( q − l ) (cid:17) e − ( t − s ) k u ( · , s ) ∇ c ( · , s ) k L q (Ω) ds ≤ C Z t (cid:16) t − s ) − µ − ( q − l ) (cid:17) e − ( t − s ) k u ( · , s ) k L q (Ω) k∇ c ( · , s ) k L ∞ (Ω) ds ≤ C Z t (cid:16) t − s ) − µ − ( q − l ) (cid:17) e − ( t − s ) k u ( · , s ) k L q (Ω) · n(cid:13)(cid:13) ∇ ( c ( · , s ) − e − tB c ) (cid:13)(cid:13) L ∞ (Ω) + k∇ e − tB c k L ∞ (Ω) (cid:9) ds ≤ C K Z t (cid:16) t − s ) − µ − ( q − l ) (cid:17) e − ( t − s ) ds · H pp − · (cid:16) q − q + δ (cid:17) p ( t ) · (cid:26) C J l +32 µl + δl,µ ( t ) + C k∇ c k L ∞ (Ω) (cid:27) ≤ C H pp − · (cid:16) q − q + δ (cid:17) p ( t ) · J l +32 µl + δl,µ ( t )for all t ∈ (0 , T max ) , where C := C K ( C + C K ) R ∞ (cid:16) ρ − µ − ( q − l ) (cid:17) e − ρ dρ < ∞ thanks to(3.8). Thereupon, (cid:13)(cid:13) B µ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L l (Ω) ≤ C + C H pp − · (cid:16) q − q + δ (cid:17) p ( t ) · J l +32 µl + δl,µ ( t ) (3.9)for all t ∈ (0 , T max ) . Observing from (3.6) that δ + l + 32 µl < ,
8e make use of Young’s inequality to attain (cid:13)(cid:13) B µ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L l (Ω) ≤ C + 12 J l,µ ( t ) + C H pp − · (cid:16) q − q + δ (cid:17) · µl µl − l − − µlδ p ( t )with certain C = C ( µ, l, p, q, δ ) for all t ∈ (0 , T max ) , which further implies J l,µ ( t ) ≤ C + 12 J l,µ ( t ) + C H pp − · (cid:16) q − q + δ (cid:17) · µl µl − l − − µlδ p ( t ) , that is J l,µ ( t ) ≤ C ) + 2 C H pp − · (cid:16) q − q + δ (cid:17) · µl µl − l − − µlδ p ( t ) (3.10)for all t ∈ (0 , T max ) . Letting φ (˜ δ ) := pp − · µl − l − δ l + ˜ δ ! · µl µl − l − − µl ˜ δ , we find that φ (˜ δ ) ց pp − · µ as ˜ δ ց , whence for some chosen δ > δ ′ ∈ (cid:16) , min n − l +32 µl , l (1 − µ ) o(cid:17) such that φ ( δ ′ ) ≤ pp − · µ δ. Thus, upon an elementary calculation, we draw on (3.7) to have pp − · (cid:18) q − q + δ ′ (cid:19) · µl µl − l − − µlδ ′ = pp − · (cid:18) µl − l − δ ′ l + δ ′ (cid:19) · µl µl − l − − µlδ ′ = φ ( δ ′ ) ≤ pp − · µ δ, which along with (3.10) and (3.1) implies (3.5).Based on a well-known continuous embedding, a combination of Lemma 3.1 with Lemma 3.2provides the desired uniform L ∞ conditional estimates of ∇ c as follows. Lemma 3.3
Assume that p ≥ and δ > . Then there exists C ( p, δ ) > such that k∇ c ( · , t ) k L ∞ (Ω) ≤ C · ( s ∈ (0 ,t ) k n ( · , s ) k L p (Ω) ) pp − · ( + δ ) (3.11) for all t ∈ (0 , T max ) . Proof.
For given δ > , we can find l > l < δ, and whereby l + 33 l <
13 + δ. Let ϕ (˜ δ ) := (cid:18) l + 32 µl + ˜ δ (cid:19) · (cid:18) µ δ (cid:19) , ˜ δ > .
9t is evident that ϕ (˜ δ ) ց l + 32 µl · µ l + 33 l <
13 + δ as ˜ δ ց , which allows for a choice of δ ′′ = δ ′′ ( δ ) > ϕ ( δ ′′ ) ≤
13 + δ. (3.12)Combining Lemma 3.1 with Lemma 3.2 provides some C = C ( p, l, µ, δ ′′ ) > (cid:13)(cid:13) ∇ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L ∞ (Ω) ≤ C H pp − · ( µ + δ ′′ ) · (cid:16) l +32 µl + δ ′′ (cid:17) p (3.13)for all t ∈ (0 , T max ) . Moreover, from the regularity properties of the Neumman heat semigroup([16, 35]), we have (cid:13)(cid:13) ∇ e − tB c (cid:13)(cid:13) L ∞ (Ω) ≤ C k∇ c k L ∞ (Ω) (3.14)for all t ∈ (0 , T max ) . Therefore, a collection of (3.1), (3.12), (3.13) and (3.14) entails k∇ c ( · , t ) k L ∞ (Ω) ≤ (cid:13)(cid:13) ∇ ( c ( · , t ) − e − sB c ) (cid:13)(cid:13) L ∞ (Ω) + (cid:13)(cid:13) ∇ e − sB c (cid:13)(cid:13) L ∞ (Ω) ≤ C H pp − · ( µ + δ ′′ ) · (cid:16) l +32 µl + δ ′′ (cid:17) p ( t ) + C k∇ c k L ∞ (Ω) ≤ C H pp − · ( µ + δ ′′ ) · (cid:16) l +32 µl + δ ′′ (cid:17) p ( t )= C H pp − · ϕ ( δ ′′ ) p ( t ) ≤ C H pp − · ( + δ ) p ( t )with C := C + C k∇ c k L ∞ (Ω) for all t ∈ (0 , T max ) , which shows (3.11). With the aids of Lemma 3.3, a standard testing procedure as used in the arguments of [36, Lemma3.2] can yield the L p bounds for the component n. Lemma 4.1
Let both (1.5) and (1.6) hold. If (1.7) is fulfilled with some K S > and θ > , then one can find some C = C ( p ) > such that sup t ∈ (0 ,T max ) k n ( · , t ) k L p (Ω) ≤ C for any p > . (4.1) Proof.
Without loss of generality, we suppose θ < , then it is possible to take δ > δ )(1 − θ ) < . (4.2)Testing n -equation in (1.4) by n p − and integrating by parts, we obtain from ∇ · u = 0 , Young’sinequality and the nonnegativity of both n and m that1 p ddt Z Ω n p + ( p − Z Ω n p − |∇ n | =( p − Z Ω n p − S ( |∇ c | ) ∇ c · ∇ n − Z Ω n p m ≤ ( p − Z Ω n p − S ( |∇ c | ) ∇ c · ∇ n ≤ p − Z Ω n p − |∇ n | + p − Z Ω n p S ( |∇ c | ) |∇ c | t ∈ (0 , T max ) , which is actually ddt Z Ω n p + 2( p − p Z Ω (cid:12)(cid:12)(cid:12) ∇ n p (cid:12)(cid:12)(cid:12) ≤ p ( p − K S Z Ω n p |∇ c | − θ (4.3)for all t ∈ (0 , T max ) . Thanks to 2(1 − θ ) > , we draw on Lemma 3.2 to gain p ( p − K S Z Ω n p |∇ c | − θ ≤ p ( p − K S k∇ c k − θL ∞ (Ω) Z Ω n p ≤ C H pp − · ( + δ ) · (2 − θ ) p ( t ) · Z Ω n p (4.4)for all t ∈ (0 , T max ) , where C = C ( p ) > . Along with (2.3), an application of the Gagliardo–Nirenberg inequality provides C = C ( p ) > C = C ( p ) > (cid:26)Z Ω n p (cid:27) p − p − = k n p k p − p − L (Ω) ≤ C k∇ n p k L (Ω) k n p k p − L p (Ω) + C k n p k p − p − L p (Ω) ≤ C k∇ n p k L (Ω) + C (4.5)for all t ∈ (0 , T max ) . With λ := 3 p − p − , (4.6)(4.5) implies C · (cid:26)Z Ω n p (cid:27) λ − p − p ≤ p − p Z Ω (cid:12)(cid:12)(cid:12) ∇ n p (cid:12)(cid:12)(cid:12) (4.7)for all t ∈ (0 , T max ) , where C := p − pC . Inserting (4) and (4.4) into (4.3) and employing Young’sinequality yield ddt Z Ω n p + C · (cid:26)Z Ω n p (cid:27) λ ≤ C H pp − · ( + δ ) · (2 − θ ) p ( T ) · Z Ω n p + 2( p − p ≤ C · (cid:26)Z Ω n p (cid:27) λ + (cid:18) C (cid:19) λ − · C λλ − · H λλ − · pp − · ( + δ ) · (2 − θ ) p ( T )+ 2( p − p ≤ C · (cid:26)Z Ω n p (cid:27) λ + C · H λλ − · pp − · ( + δ ) · (2 − θ ) p ( T )for each T ∈ (0 , T max ) with C := (cid:16) C (cid:17) λ − · C λλ − + p − p due to H p ≥ , and thus ddt Z Ω n p + C · (cid:26)Z Ω n p (cid:27) λ ≤ C · H λλ − · pp − · ( + δ ) · (2 − θ ) p ( T )for all t ∈ (0 , T ) . By means of an ODE comparison argument, this further entails Z Ω n p ( · , t ) ≤ max (Z Ω n p , (cid:26) C C · H λλ − · pp − · ( + δ ) · (2 − θ ) p ( T ) (cid:27) λ ) (4.8)11or all t ∈ (0 , T ) . In view of (3.1), we infer from (4.8) that H p ( T ) ≤ ( k n k L p (Ω) , (cid:18) C C (cid:19) pλ · H ̺p ( T ) ) ≤ C H ̺p ( T ) (4.9)for any T ∈ (0 , T max ) , where ̺ := λ − · p − · (cid:0) + δ (cid:1) · (2 − θ ) and C := 1+max (cid:26) k n k L p (Ω) , (cid:16) C C (cid:17) pλ (cid:27) . It is clear from (4.6) and (4.2) that ̺ = 3( p − · p − · (cid:18)
13 + δ (cid:19) · (2 − θ ) = (1 + 3 δ )(1 − θ ) < , whence (4.9) shows that H p ( T ) ≤ C − ̺ for any T ∈ (0 , T max ) , which implies (4.1) by letting T ր T max . Now, we are in the position to pursue the boundedness of each quantity on the left hand sideof (2.2). In particular, the quantities associated with the components of both c and u can beestimated straightforwardly by a collection of Lemmas 2.2–2.3, Lemma 3.3 and Lemma 4.1. For m, the corresponding boundedness needs to be verified by similar strategies as performed in Lemmas3.2–3.3. In the final, aided by the bounds established for the quantities related to c, m and u, thetemporally independent L ∞ bounds of n can be achieved through an appropriate application ofheat semigroup theories as done in [36]. Lemma 4.2
Suppose both (1.5) and (1.6) are valid. Let (1.7) hold with K S > and θ > . Then there exists
C > with the properties that k c ( · , t ) k W , ∞ (Ω) ≤ C for any t ∈ (0 , T max ) (4.10) and k m ( · , t ) k W , ∞ (Ω) ≤ C for any t ∈ (0 , T max ) (4.11) as well as k A α u ( · , t ) k L (Ω) ≤ C for any t ∈ (0 , T max ) (4.12) with certain α ∈ ( , . Proof.
Thanks to Lemma 4.1, both (4.10) and (4.12) are immediate consequences of Lemmas2.2–2.3 and Lemma 3.3. Specially, due to α ∈ ( , , the continuous embedding together with(4.12) provides C > C > k u ( · , t ) k L ∞ (Ω) ≤ C k A α u ( · , t ) k L (Ω) ≤ C (4.13)for any t ∈ (0 , T max ) . For each given r > , we let β ∈ ( ,
1) fulfill β > r + 32 r , ϑ ∈ ( ,
1) satisfying β > ϑ > r + 32 r . (4.14)Now, applying B β on both sides of the Duhamel representation of m and invoking the L p - L q estimates of the sectorial operator ([17]), we obtain C > (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) L r (Ω) ≤ Z t (cid:13)(cid:13)(cid:13) B β e − B ( t − s ) u ( · , s ) ∇ m ( · , s ) (cid:13)(cid:13)(cid:13) L r (Ω) ds + Z t (cid:13)(cid:13)(cid:13) B β e − B ( t − s ) m ( · , s )(1 − n ( · , s )) (cid:13)(cid:13)(cid:13) L r (Ω) ds ≤ C Z t (cid:16) t − s ) − β (cid:17) e − ( t − s ) k u ( · , s ) ∇ m ( · , s ) k L r (Ω) ds + C Z t (cid:16) t − s ) − β (cid:17) e − ( t − s ) k m ( · , s )(1 − n ( · , s )) k L r (Ω) ds (4.15)for all t ∈ (0 , T max ) . Since (4.14) ensures the inequality (cid:13)(cid:13) ∇ (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13) L r (Ω) ≤ C (cid:13)(cid:13)(cid:13) B ϑ (cid:0) ( m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) L r (Ω) with some C > D ( B β ) ֒ → W ,r (Ω) for any t ∈ (0 , T max ) , and also allows foran application of the following interpolation features of the fractional power of sectorial operators([15, Theorem 2.14.1]) (cid:13)(cid:13)(cid:13) B ϑ (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) L r (Ω) ≤ (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) ϑβ L r (Ω) (cid:13)(cid:13) m ( · , t ) − e − Bt m (cid:13)(cid:13) β − ϑβ L r (Ω) for all t ∈ (0 , T max ) , we recall (1.6), (2.4), (4.14) and make use of the regularity properties of( e − tB ) t ≥ ([35]) to have k u ( · , t ) ∇ m ( · , t ) k L r (Ω) ≤ k u ( · , t ) k L ∞ (Ω) · n(cid:13)(cid:13) ∇ (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13) L r (Ω) + (cid:13)(cid:13) ∇ e − Bt m (cid:13)(cid:13) L r (Ω) o ≤ C C (cid:13)(cid:13)(cid:13) B ϑ ( m ( · , t ) − e − Bt m ) (cid:13)(cid:13)(cid:13) L r (Ω) + C C k∇ m k L r (Ω) ≤ C C (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) ϑβ L r (Ω) (cid:16) k m ( · , t ) k L r (Ω) + k m k L r (Ω) (cid:17) β − ϑβ + C C K | Ω | r ≤ C C (cid:16) K | Ω | r (cid:17) β − ϑβ (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) ϑβ L r (Ω) + C C K | Ω | r ≤ C (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) ϑβ L r (Ω) + C (4.16)with some C > C := max (cid:26) C C (cid:16) K | Ω | r (cid:17) β − ϑβ , C C K | Ω | r (cid:27) for all t ∈ (0 , T max ) . k m ( · , t )(1 − n ( · , t )) k L r (Ω) ≤ k m ( · , t ) k L ∞ (Ω) · (cid:16) | Ω | r + k n ( · , t ) k L r (Ω) (cid:17) ≤ k m k L ∞ (Ω) · (cid:16) | Ω | r + C (cid:17) ≤ K · (cid:16) | Ω | r + C (cid:17) =: C (4.17)for all t ∈ (0 , T max ) , where C := C ( r ) > . Inserting (4.16) and (4.17) into (4.15) entails (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) L r (Ω) ≤ C ( C + C ) · Z t (cid:16) t − s ) − β (cid:17) e − ( t − s ) ds + C C · sup τ ∈ (0 ,t ) (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , τ ) − e − Bτ m (cid:1)(cid:13)(cid:13)(cid:13) ϑβ L r (Ω) · Z t (cid:16) t − s ) − β (cid:17) e − ( t − s ) ds ≤ C + C · sup τ ∈ (0 ,t ) (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , τ ) − e − Bτ m (cid:1)(cid:13)(cid:13)(cid:13) ϑβ L r (Ω) (4.18)with C := C ( C + C ) R ∞ (cid:0) ρ − β (cid:1) e − ρ dρ < ∞ thanks to β < t ∈ (0 , T max ) . Define M β,r ( t ) := 1 + sup s ∈ (0 ,t ) (cid:13)(cid:13)(cid:13) B β (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13)(cid:13) L r (Ω) for all t ∈ (0 , T max ) . Then (4.18) implies M β,r ( t ) ≤ C · M ϑβ β,r ( t ) ≤ C · M ϑβ β,r ( t )for all t ∈ (0 , T max ) , where C := 1 + 2 C , and thus from (4.14), M β,r ( t ) ≤ C ββ − ϑ (4.19)for all t ∈ (0 , T max ) , which combined with the embedding D ( B β ) ֒ → W , ∞ (Ω) provides C = C ( r, β, K , | Ω | ) > (cid:13)(cid:13) ∇ (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13) L ∞ (Ω) ≤ C for all t ∈ (0 , T max ) . As a result, by (1.6) and the heat semigroup estimates ([35]), we achieve k∇ m ( · , t ) k L ∞ (Ω) ≤ (cid:13)(cid:13) ∇ (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13) L ∞ (Ω) + (cid:13)(cid:13) ∇ e − Bt m (cid:13)(cid:13) L ∞ (Ω) ≤ C + C k∇ m k L ∞ (Ω) ≤ C + C K with some C > t ∈ (0 , T max ) . This together with (2.4) yields (4.11).14 emma 4.3
If both (1.5) and (1.6) are satisfied, and if (1.7) is fulfilled with some K S > and θ > , then one can find C > such that k n ( · , t ) k L ∞ (Ω) ≤ C for each t ∈ (0 , T max ) . (4.20) Proof.
Let g := n S ( |∇ c | ) ∇ c + un and g := n (1 − m ) . Then for each ι > C = C ( ι ) > C = C ( ι ) > k g ( · , t ) k L ι (Ω) ≤ C and k g ( · , t ) k L ι (Ω) ≤ C for each t ∈ (0 , T max ) . Thereupon, proceeding along a similar reasoning as that of [36, Lemma 3.4], we establish (4.20).
Proof of Theorem 1.1.
In view of the blow-up criterion (2.2), Theorem 1.1 is a directconsequence of Lemmas 4.2–4.3. L (Ω) As the cornerstone of this section, the following assertions on decay properties of nm and ∇ m aswell as on the stabilization of spatial L -integrals of both n and m follow from a straightforwardtesting procedure together with an argument analogous to that of [14, Lemma 4.2]. Lemma 5.1
The components n and m of the solutions constructed in Theorem 1.1 have theproperties that Z ∞ Z Ω nm < ∞ , Z ∞ Z Ω |∇ m | < ∞ , (5.1) and Z Ω n ( · , t ) → (cid:26)Z Ω n − Z Ω m (cid:27) + , Z Ω m ( · , t ) → (cid:26)Z Ω m − Z Ω n (cid:27) + (5.2) as t → ∞ . Proof.
Integrating n -equation and m -equation over Ω × (0 , t ) for any t > , respectively, weobtain from n ≥ m ≥ Z t Z Ω nm ≤ min (cid:26)Z Ω n , Z Ω m (cid:27) , which implies the first inequality in (5.1). Testing m -equation by m and integrating the resultedequation on (0 , t ) yield12 Z Ω m ( · , t ) + Z t Z Ω |∇ m | = 12 Z Ω m − Z t Z Ω nm for any t > , whence the second inequality in (5.1) holds thanks to the nonnegativity of n and m. Relying on(5.1), the convergence in (5.2) can be obtained in accordance with the reasoning of [14, Lemma4.2].Besides the decay features and the convergence involved in Lemma 5.1, some higher order esti-mates, such as the H¨older estimates, are also essential for the derivation of the desired convergence.15 emma 5.2
For components n and m, one can find some γ ∈ (0 , and C > such that k n k C γ, γ (¯Ω × [ t,t +1]) for all t > and k m k C γ, γ (¯Ω × [ t,t +1]) for all t > . (5.4) Proof.
Denoting ξ := n S ( |∇ c | ) ∇ c + un and ξ := nm, n -equation in (1.4) can be rewrittenas n t = ∆ n − ∇ · ξ ( x, t ) − ξ ( x, t ) , x ∈ Ω , t > , thanks to ∇ · u = 0 , where from (4.20), (4.10), (1.7), (4.13) and (2.5), we can infer that both ξ and ξ are bounded in Ω × (0 , ∞ ) , which implies (5.3) according to [31, Theeorem 1.3]. Similarly,we let ζ := mu, then again by ∇ · u = 0 another equivalent expression of m -equation appears as m t = ∆ m − ∇ · ζ ( x, t ) − ξ ( x, t ) , x ∈ Ω , t > . It is evident from (2.4) and (4.13) that ζ is bounded in Ω × (0 , ∞ ) , whereupon combining with theboundedness feature of ξ we conclude that (5.4) also holds.In light of Lemmas 5.1–5.2, let us provide the uniform L (Ω)-convergence of m at first. Lemma 5.3
For component m, we have m ( · , t ) → | Ω | (cid:26)Z Ω m − Z Ω n (cid:27) + in L (Ω) as t → ∞ . (5.5) Proof.
From (5.1) and the Poincar´e inequality, we have Z ∞ k m ( · , t ) − m ( · , t ) k L (Ω) dt ≤ C Z ∞ Z Ω |∇ m | < ∞ , (5.6)where C > m ( · , t ) := | Ω | R Ω m ( · , t ) for t ≥ . Moreover, the H¨older continuity of m in (5.4)implies the uniform continuity of 0 ≤ t → k m ( · , t ) − m ( · , t ) k L (Ω) . Thus, by means of a reasoningsimilar to that of [9, Theorem 1.1], we conclude from (5.6) that k m ( · , t ) − m ( · , t ) k L (Ω) → t → ∞ , whence there exists some t > ε > k m ( · , t ) − m ( · , t ) k L (Ω) < ε t > t . (5.7)Apart from that, the convergence in (5.2) enable us to choose some t > (cid:12)(cid:12)(cid:12)(cid:12) m ( · , t ) − | Ω | (cid:26)Z Ω m − Z Ω n (cid:27) + (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε | Ω | for all t > t . (5.8)Setting t := max { t , t } and m ∞ := | Ω | (cid:8)R Ω m − R Ω n (cid:9) + , we obtain from (5.7) and (5.8) that Z Ω | m ( · , t ) − m ∞ | ≤ Z Ω | m ( · , t ) − m ( · , t ) | + 2 Z Ω | m ( · , t ) − m ∞ | ≤ ε ε ε for all t > t , n, m and u in L (Ω) can be achieved by virtue ofstandard testing procedures as used in [14]. Lemma 5.4
For n, c and u, we have n ( · , t ) → | Ω | (cid:26)Z Ω n − Z Ω m (cid:27) + in L (Ω) as t → ∞ (5.9) and c ( · , t ) → | Ω | (cid:26)Z Ω m − Z Ω n (cid:27) + in L (Ω) as t → ∞ (5.10) as well as u ( · , t ) → in L (Ω) as t → ∞ . (5.11) Proof.
The convergence in (5.10) is a immediate consequence of [14, Lemma 4.7]. Let ¯ n ( · , t ) := | Ω | R Ω n ( · , t ) for t ≥ . Then from n -equation and ∇ · u = 0 , we integrate by parts to obtain12 ddt Z Ω ( n ( · , t ) − ¯ n ( · , t )) = Z Ω ( n ( · , t ) − ¯ n ( · , t )) · ( n t − (¯ n ) t )= Z Ω ( n − ¯ n ) · (cid:0) ∆ n − ∇ · ( n S ( |∇ c | ) ∇ c ) − nm − u · ∇ n + nm (cid:1) = − Z Ω |∇ n | + Z Ω n S ( |∇ c | ) ∇ c · ∇ n − Z Ω ( n − ¯ n ) · nm + Z Ω ( n − ¯ n ) · nm (5.12)for each t > , where combined with (1.7) and (4.20) an application of Young’s inequality entails Z Ω n S ( |∇ c | ) ∇ c · ∇ n ≤ Z Ω |∇ n | + K S Z Ω n |∇ c | ≤ Z Ω |∇ n | + C Z Ω |∇ c | with some C > t > . Thereafter, (5.9) follows from the arguments of [14, Lemma 4.8].In the final, with the aids of (5.9) and (5.10), one can see that (5.11) is actually valid according tothe reasoning of [14, Lemma 4.9].
Based on the boundedness properties in spaces with higher order regularity and on the convergencepreviously achieved in L (Ω) , we can make use of an Ehrling type lemma to improve the regu-larity of the spaces, in which each component of the solution converges to corresponding constantequilibrium, to the level as claimed in Theorem 1.2. Proof of Theorem 1.2
In view of (5.3) and (4.12) with α ∈ ( , , there exist γ ∈ (0 , α − )and C > k n ( · , t ) − n ∞ k C γ (¯Ω) ≤ C for any t > k u ( · , t ) k C γ (¯Ω) ≤ C for any t > D ( A α ) ֒ → C γ ( ¯Ω) ([16]), where n ∞ := | Ω | (cid:8)R Ω n − R Ω m (cid:9) + . Moreover, from (3.5), (4.1) and (4.19), we can infer that for some r > µ ∈ ( r +32 r , C = C ( r ) > (cid:13)(cid:13) B µ (cid:0) c ( · , t ) − e − Bt c (cid:1)(cid:13)(cid:13) L r (Ω) ≤ C for any t > (cid:13)(cid:13) B µ (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13) L r (Ω) ≤ C for any t > , and whereby combining with (1.6), we can find C > k B µ c ( · , t ) k L r (Ω) ≤ (cid:13)(cid:13) B µ (cid:0) c ( · , t ) − e − Bt c (cid:1)(cid:13)(cid:13) L r (Ω) + (cid:13)(cid:13) B µ e − Bt c (cid:13)(cid:13) L r (Ω) ≤ C + t − µ e − t k c k L r (Ω) ≤ C for any t > , (5.15)and similarly, k B µ m ( · , t ) k L r (Ω) ≤ (cid:13)(cid:13) B µ (cid:0) m ( · , t ) − e − Bt m (cid:1)(cid:13)(cid:13) L r (Ω) + (cid:13)(cid:13) B µ e − Bt m (cid:13)(cid:13) L r (Ω) ≤ C + t − µ e − t k m k L r (Ω) ≤ C for any t > . (5.16)Since µ ∈ ( r +32 r ,
1) admits the continuity of the embedding D ( B µ ) ֒ → C γ ( ¯Ω) for γ ∈ (0 , µ − r +3 r ) , both (5.15) and (5.16) imply the existence of C > k c ( · , t ) k C γ (¯Ω) ≤ C and k m ( · , t ) k C γ (¯Ω) ≤ C for any t > . Therefore, with m ∞ := | Ω | (cid:8)R Ω m − R Ω n (cid:9) + , we can infer that k c ( · , t ) − m ∞ k C γ (¯Ω) ≤ C and k m ( · , t ) − m ∞ k C γ (¯Ω) ≤ C (5.17)with certain C > t > . Observing from C γ ( ¯Ω) ֒ → L ∞ (Ω) ֒ → L (Ω) and C γ ( ¯Ω) ֒ → W , ∞ (Ω) ֒ → L (Ω) that the first embedding of each is compact, we thus apply an Ehrling lemmato obtain some C > η > k w k L ∞ (Ω) ≤ η C k w k C γ (¯Ω) + C k w k L (Ω) for each w ∈ C γ ( ¯Ω) (5.18)and k w k W , ∞ (Ω) ≤ η C k w k C γ (¯Ω) + C k w k L (Ω) for each w ∈ C γ ( ¯Ω) . (5.19)Since Lemmas 5.3–5.4 allow for a choice of t > k n ( · , t ) − n ∞ k L (Ω) ≤ η C for any t > t , k c ( · , t ) − m ∞ k L (Ω) ≤ η C for any t > t and k m ( · , t ) − m ∞ k L (Ω) ≤ η C for any t > t as well as k u ( · , t ) k L (Ω) ≤ η C for any t > t , (1.10) thereby follows from (5.13), (5.14) and (5.17) in conjunction with applications of (5.18) toboth w = n − n ∞ and w = u as well as of (5.19) to both w = c − m ∞ and w = m − m ∞ . cknowledgments The author is supported by the National Natural Science Foundation of China (Grant No. 11901298),the Fundamental Research Funds for the Central Universities (Grant No. KJQN202052), and theBasic Research Program of Jiangsu Province (Grant No. BK20190504).
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