Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions
aa r X i v : . [ m a t h . A P ] J a n ASYMPTOTIC STABILITY OF EXOGENOUS CHEMOTAXIS SYSTEMSWITH PHYSICAL BOUNDARY CONDITIONS
GUANGYI HONG ∗ AND ZHI-AN WANG
Abstract.
In this paper, we consider the exogenous chemotaxis system with physical mixedzero-flux and Dirichlet boundary conditions in one dimension. Since the Dirichlet boundarycondition can not contribute necessary estimates for the cross-diffusion structure in the system,the global-in-time existence and asymptotic behavior of solutions remain open up to date. Inthis paper, we overcome this difficulty by employing the technique of taking anti-derivative sothat the Dirichlet boundary condition can be fully used, and show that the system admits globalstrong solutions which exponentially stabilize to the unique stationary solution as time tendsto infinity against some suitable small perturbations. To the best of our knowledge, this is thefirst result obtained on the global well-posedness and asymptotic behavior of solutions to theexogenous chemotaxis system with physical boundary conditions.
Keywords.
Exogenous chemotaxis, steady state, asymptotic behavior, anti-derivative, energymethod Introduction
Chemotaxis, the directional movement of cells in response to a chemical stimulus gradient,is important for bacteria to find food (e.g., glucose) or to flee from poisons [33] and criticalto early development, normal function such as wound healing/inflammation and pathologicalprocess like cancer metastasis [34]. Mathematical models of chemotaxis were firstly developedby Keller-Segel in 1970s with two prototypes describing endogenous and exogenous chemotaxis,respectively. In the endogenous chemotaxis, cells respond to a chemical signal that is releasedfrom cells themselves. While in the exogenous chemotaxis, cells respond to an external chemicalsignal (such as oxygen, light or food). The typical example of endogenous chemotaxis is thespontaneous aggregation of
Dictyostelium discoideum (Dd) cells in response to the chemicalcyclic adenosine monophosphate (cAMP) secreted by Dd cells [3], which was first modeledmathematically by Keller and Segel in [17]. For such aggregation Keller-Segel models, thehomogeneous Neumann boundary conditions are usually prescribed to reproduce the aggregatingpatterns [11, 12]. The prominent example of exogenous chemotaxis was reported in [1] wheremotile
Escherichia coli placed at one end of a capillary tube containing an energy source andoxygen migrate out into the tube in the form of traveling bands clearly visible to the naked eye.The mathematical model was subsequently proposed by Keller and Segel in [18], which reads as ( u t = ∆ u − ∇ · ( u ∇ φ ( v )) in Ω ,v t = D ∆ v − uv m in Ω , (1.1)where u and v denote the bacterial density and oxygen concentration, respectively, at position x ∈ Ω and time t > D > m > φ ( v ) is called the chemotactic sensitivity function which typically hastwo prototypes: φ ( v ) = ln v (logarithmic sensitivity) and φ ( v ) = v (linear sensitivity). Thelogarithmic sensitivity was originally used in [18] based on the Weber-Fechner law (the sensoryresponse to a stimulus is logarithmic) which has various biological applications (cf. [10, 15, 21]). ∗ Corresponding author.
It was mentioned in [18, p.241] that the oxygen diffusion rate D is negligible (i.e. 0 < D ≪ D ≥ ≤ m ≤
1, while thestability of traveling wavefronts for the case m = 1 was obtained in [6, 8, 23–25, 30] and theinstability of pulsating wave for the case m = 0 was investigated in [9, 27].When considering the exogenous chemotaxis system (1.1) in a bounded domain Ω, the relevantphysical boundary conditions (for instance see the experiment in [1]) are ∂ ν u − u∂ ν v = 0 , v = v ∗ on ∂ Ω , (1.2)where ∂ ν = ∂∂ν is the normal derivative on the boundary with ν denoting the outward unitnormal vector of ∂ Ω, and the constant v ∗ > v . That is, thezero-flux boundary condition and Dirichlet boundary condition are imposed to cell density u andchemical concentration v , respectively. The Keller-Segel system (1.1) subject to the boundarycondition (1.2) has also been used in the chemotaxis-fluid model in [32] to describe the boundaryaccumulation layer of aerobic bacterial chemotaxis towards the drop edge (air-water interface)in a sessile drop mixed with Bacillus subtilis bacteria. The model in [32] reads u t + w · ∇ u = ∆ u − ∇ · ( u ∇ v ) ,v t + w · ∇ v = D ∆ v − uv,ρ ( w t + w · ∇ w ) = µ ∆ w + ∇ p − V b gu ( ρ b − ρ ) z , ∇ · w = 0 , (1.3)where u and v denote the bacterial and oxygen concentrations, respectively, and w is the fluidvelocity governed by the incompressible Navier-Stokes equations with the pure fluid density ρ andviscosity µ . p is a pressure function, V b gu ( ρ b − ρ ) z denotes the buoyant force along the upwardunit vector z where V b and ρ b are the bacterial volume and density, respectively, and g is thegravitational constant. With boundary conditions in (1.2) and the non-slip boundary conditionfor the fluid: w | ∂ Ω = 0, the works [7, 20, 32] have shown that the system (1.3) can numericallyreproduce the key features of experiment findings in [32] in two and three dimensions.Compared to a large number of results available to the endogenous chemotaxis models withNeumann boundary conditions (cf. [2, 11, 12]), the basic questions like the global well-posednessof the exogenous chemotaxis system (1.1) with physical boundary conditions in (1.2) still remainpoorly understood and only very limited analytical results are available so far. The primaryobstacle is that the estimate of ∇ v , which is needed for the global boundedness of solutions dueto the cross-diffusion structure in the first equation of (1.1), can not be achieved through thesecond equation of (1.1) with the Dirichlet boundary condition which gives no information on ∇ v . On the half line R + = (0 , ∞ ), the existence and stability of the unique stationary solution(¯ u, ¯ v ) of (1.1)-(1.2) with φ ( v ) = ln v was recently established in [5] for any m ≥
0, where (¯ u, ¯ v )is of a boundary (spike, layer) profile as D > φ ( v ) = v , the existence ofstationary solutions to (1.1)-(1.2) with m = 1 was proved in [19] for all dimensions and theexistence of global weak solutions was established in [36] in one dimension. The local existenceof weak solutions to (1.3) on the water-drop shaped domain as in [32] with (1.2) and w | ∂ Ω = 0was proved in [26]. These appear to be the only results in the literature for the Keller-Segelsystem (1.1) subject to the physical boundary conditions given (1.2). We also mention anotherresult in [4] where the existence of stationary solutions of (1.1) with φ ( v ) = v and m = 1 wasestablished for all dimensions when the Dirichlet condition for v in (1.2) was replaced by aboundary condition based on Henry’s law modeling the dissolution of gas in water. The purposeof this paper is to further make a progress in this direction for the Keller-Segel system (1.1) withlinear sensitivity and boundary conditions in (1.2) on a bounded interval I := (0 , XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 3 we consider the following problem u t = u xx − ( uv x ) x in I ,v t = Dv xx − uv in I , ( u, v )( x,
0) = ( u , v )( x ) in I (1.4)subject to the following boundary conditions: ( ( u x − uv x ) | x =0 , = 0 , v (0 , t ) = v (1 , t ) = v ∗ if D >
0, (1.5a)( u x − uv x ) | x =0 , = 0 if D = 0. (1.5b)By integrating the first equation of (1.4) along with the boundary condition (1.5a), one imme-diately finds that cell mass is conserved: Z I u ( x, t )d x = Z I u ( x )d x := M, where M > u, ¯ v ) of (1.4) with D > ¯ u xx − (¯ u ¯ v x ) x = 0 , x ∈ I ,D ¯ v xx − ¯ u ¯ v = 0 , x ∈ I , Z I ¯ u d x = M, (¯ u x − ¯ u ¯ v x ) | x =0 , = 0 , ¯ v | x =0 , = v ∗ . (1.6)The results of [19] assert that the stationary problem (1.6) with D > u, ¯ v ) which is of a boundary layer profile as D > D = 0, the system (1.4) with (1.5b) clearly has a unique constant solution ( M, u , v ) is a small perturbation ofthe stationary solution (¯ u, ¯ v ), then the system (1.4) with (1.5a)-(1.5b) admits a unique solution( u, v ) satisfying for any D ≥ k ( u, v ) − (¯ u, ¯ v ) k L ∞ → t → ∞ , where (¯ u, ¯ v ) = ( M,
0) if D = 0 and (¯ u, ¯ v ) is the non-constant stationary solution satisfying(1.6) if D >
0. As we know, this is the first result on the global well-posedness and asymptoticdynamics of the system (1.1)-(1.2). We note that it was shown in [19] that the unique non-constant stationary solution of (1.6) enjoys a boundary layer profile as
D > D = 0. With this fact and the boundary conditionsin (1.5), we may speculate that the solution of (1.4)-(1.5a) with D > D = 0 as D →
0, where the boundary layer will arise to correct thisdiscrepancy. Therefore the convergence of solutions to (1.4)-(1.5a) with
D > D → φ ( v ) = ln v and the Dirichlet boundary condition for u and Robin boundary condition for v are given, theconvergence of solutions for (1.1) with φ ( v ) = ln v and m = 1 as D → D → φ ( v ) and boundary conditions. Sketch of proof ideas . As mentioned previously, the boundary conditions for v in (1.5)refrain from deriving the estimates of v x which is, however, necessary to establish the globalwell-posedness of solutions of (1.4)–(1.5) due to the cross-diffusion structure in the first equation(1.4). To overcome this barrier, by observing that the first equation of (1.4) is conserved withzero-flux boundary condition on u , we develop an idea by considering the primitive function of u in space, say ̺ , and establish the equation of ̺ which no longer has cross-diffusion structureand the Dirichlet boundary condition of v can make essential contributions. As such, we derivethe boundedness and stability of ( ̺, v ) by the delicate (weighted) energy estimates first and then G.-Y. HONG AND Z.-A. WANG transfer the results to ( u, v ). This is our rough idea, and precise procedures are presented insection 3 for the case
D > D = 0. Indeed, the analysis for the case D = 0appears to be easier than D > v -equation is an ODE and lacks the diffusive dissipation,we need to make full use of the ODE structure along with the explicit formula of v to derive somedelicate higher-order estimates which, in turn, requires stronger smallness constraints upon theinitial datum compared to the case D >
D >
0. Finally, theasymptotic behavior of the solution for the case D = 0 will be proved in Sec. 4.2. Statement of main results
In this section, we introduce the results on the stationary problem (1.6) from [19] and stateour main results on the asymptotic stability of stationary solutions. Throughout the paper, wedenote by L ∞ , L , H and H k the standard function spaces L ∞ ( I ), L ( I ), H ( I ) and H k ( I ),respectively. We denote by ¯ I the closure of I and by C a generic time-independent constantwhich may take different values in different places. In the sequel, we often omit I withoutambiguity. Proposition 2.1 (Theorem 2.1 in [19]) . For any M ∈ (0 , ∞ ) , the problem (1.6) with D > admits a unique classical non-constant solution (¯ u, ¯ v ) ∈ C (¯ I ) ∩ C ∞ ( I ) such that ¯ u = M R I e ¯ v d x e ¯ v , ¯ u > , < ¯ v ≤ v ∗ for any x ∈ ¯ I . (2.1)Our first result is the asymptotic stability of stationary solutions obtained in Proposition 2.1for the initial-boundary value problem (1.4), (1.5a) as time goes to infinity. Theorem 2.1.
Suppose that u ∈ H and v ∈ H with u ≥ , v ≥ such that R I u d x = M .Let (¯ u, ¯ v ) be the stationary solution given in Proposition 2.1 with R I ¯ u d x = M and define ϕ ( x ) = Z x ( u ( y ) − ¯ u ( y )) d y. Then there exists a constant δ > such that if k ϕ k H + k v − ¯ v k L ≤ δ , then the initial-boundary value problem (1.4) , (1.5a) admits a unique global solution ( u, v ) sat-isfying u ∈ C ([0 , ∞ ); H ) ∩ L (0 , ∞ ; H ) , v ∈ C ([0 , ∞ ); H ) ∩ L (0 , ∞ ; H ) , and the following asymptotic decay: k ( u − ¯ u, v − ¯ v )( · , t ) k L ∞ ≤ C e − αt for any t ≥ , (2.2) where α and C are positive constants independent of t . When D = 0, (1.4) becomes a PDE-ODE system which has a unique constant steady state( M,
0) with M = R I u d x satisfying the boundary condition (1.5b). Then we have our secondresult below. XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 5
Theorem 2.2.
Let ( u , v ) ∈ H × H with u ≥ , v ≥ such that R I u d x = M and define w ( x ) = Z x ( u ( y ) − M )d y. Then there exists a constant δ > such that if k w k H + k v k H ≤ δ , then the initial-boundary value problem (1.4) , (1.5b) admits a unique solution ( u, v ) in I × (0 , ∞ ) satisfying u ∈ C ([0 , ∞ ); H ) ∩ L (0 , ∞ ; H ) , v ∈ C ([0 , ∞ ); H ) . Furthermore, we have the following decay estimates: k ( u − M, v )( · , t ) k L ∞ ≤ C e − α t for any t > , (2.3) where α and C > are positive constants independent of t . Asymptotic stability for the case
D >
D >
Lemma 3.1 (cf. [28]) . For any f ∈ H ( I ) , there exists a constant c > such that k f k L ∞ ≤ c (cid:16) k f k L k f x k L + k f k L (cid:17) . (3.1) Furthermore, if f ∈ H ( I ) , then it holds that k f k L ∞ ≤ c k f k L k f x k L and k f k L ∞ ≤ c k f x k L (3.2) for some constants c , c > . A priori estimates.
First of all, integrating the first equation in (1.6), we see that thestationary solution (¯ u, ¯ v ) satisfies ¯ u x − ¯ u ¯ v x = 0 ,D ¯ v xx − ¯ u ¯ v = 0 , ¯ v (0) = ¯ v (1) = v ∗ , (3.3)with R I ¯ u d x = M . In view of the zero-flux boundary condition in (1.5a) for u , we know that themass of the bacteria is conserved for all time. This along with the fact R I u d x = R I ¯ u d x = M implies that Z I ( u ( x, t ) − ¯ u ( x ))d x = 0for any t ≥
0. Define ϕ ( x, t ) = Z x ( u ( y, t ) − ¯ u ( y ))d y, ψ = v − ¯ v, that is u = ϕ x + ¯ u, v = ψ + ¯ v. (3.4) G.-Y. HONG AND Z.-A. WANG
Substituting (3.4) into (1.4), integrating the first equation with respect to x and using (3.3), weobtain the following perturbation equations: ( ϕ t = ϕ xx − ϕ x ¯ v x − ¯ uψ x − ϕ x ψ x ,ψ t = Dψ xx − ¯ uψ − ¯ vϕ x − ϕ x ψ, (3.5)with the initial datum( ϕ, ψ )( x,
0) = ( ϕ , ψ ) = (cid:18)Z x ( u ( y ) − ¯ u ( y ))d y, v − ¯ v (cid:19) (3.6)and the boundary conditions ( ϕ, ψ )(0 , t ) = ( ϕ, ψ )(1 , t ) = 0 . (3.7)By the standard fixed point theorems (cf. [22, 29]), one can prove the local existence of solutionsto the initial-boundary value problem (3.5)–(3.7). Precisely, for any T >
0, if we define X (0 , T ) := { ( ϕ, ψ ) | ϕ ∈ C ([0 , T ]; H ∩ H ) ∩ L (0 , T ; H ) ,ψ ∈ C ([0 , T ]; H ∩ H ) ∩ ∈ L (0 , T ; H ) } and denote N ( T ) := sup ≤ t ≤ T (cid:0) k ϕ k H + k ψ k H (cid:1) , then we have the following local existence result. Proposition 3.1 (Local existence) . Let ϕ ∈ H ∩ H and ψ ∈ H ∩ H such that ϕ x + ¯ u ≥ , ψ + ¯ v ≥ for any x ∈ I . Then there exists a positive constant T depending on N (0) , ¯ u and ¯ v such that theinitial-boundary value problem (3.5) – (3.7) admits a unique solution ( ϕ, ψ ) ∈ X (0 , T ) satisfying N ( T ) ≤ N (0) and ϕ x + ¯ u ≥ , ψ + ¯ v ≥ for any ( x, t ) ∈ I × [0 , T ) . In order to study the asymptotic behavior of solutions to the problem (1.4), (1.5a), we firstestablish the global existence result for the initial-boundary value problem (3.5)–(3.7).
Proposition 3.2.
Assume the conditions of Proposition 3.1 hold. Then there exists a positiveconstant δ , such that if k ϕ k H + k ψ k L ≤ δ , then the problem (3.5) – (3.7) admits a uniqueglobal solution ( ϕ, ψ ) ∈ X (0 , ∞ ) which satisfies that for all t ≥ , k ϕ ( · , t ) k H + k ψ ( · , t ) k L ≤ C e − α t , k ϕ xx ( · , t ) k L + k ψ x ( · , t ) k H ≤ C (3.8) for some constants α > and C > independent of t . To ensure the global existence of solutions to the problem (3.5)–(3.7), by the local existenceresult and the standard continuation argument, it suffices to prove the following a priori esti-mates.
Proposition 3.3 ( A priori estimates) . For any
T > and any solution ( ϕ, ψ ) ∈ X (0 , T ) to theproblem (3.5) – (3.7) with ( ϕ , ψ ) ∈ H , there exists a suitably small C > independent of T such that if k ϕ k H + k ψ k L ≤ C , then we have k ϕ ( · , t ) k H + k ψ ( · , t ) k L ≤ C e − α t , k ϕ xx ( · , t ) k L + k ψ x ( · , t ) k H ≤ C in [0 , T ] and Z t (cid:0) k ϕ k H + k ψ k H + k ϕ τ k H + k ψ τ k H (cid:1) d τ ≤ C (cid:0) k ϕ k H + k ψ k H (cid:1) in [0 , T ] , where α and C are positive constants independent of T . XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 7
We shall prove Proposition 3.3 by the argument of a priori assumption. That is, we firstassume that the solution ( ϕ, ψ ) to the problem (3.5)–(3.7) satisfy for any t ∈ [0 , T ], k ϕ ( · , t ) k H + k ψ ( · , t ) k L ≤ δ, k ϕ xx ( · , t ) k L + k ψ x ( · , t ) k H ≤ σ in [0 , T ] , (3.9)where 0 < δ < M are positive constants to be determined later, and then derive the apriori estimates with (4.6) to ensure the global existence of solutions. Finally, we show thatthe solution exactly satisfies the a priori assumption (4.6) and close the argument. Beforeproceeding, we note that by (3.9) along with (3.1), (3.2) and (3.7), we get k ϕ k L ∞ ≤ Cδ, k ψ k L ∞ ≤ Cδ σ , k ϕ x k L ∞ ≤ Cδ + Cδ σ (3.10)for some constant C > δ , σ and T .The following simple properties on the stationary solution are of importance in studying theasymptotic behavior of solutions. Lemma 3.2.
Let (¯ u, ¯ v ) be the stationary solution of (1.4) , (1.5a) stated in Proposition 2.1.Then it holds that < C − ≤ ¯ u, ¯ v ≤ C (3.11) for some constant C > , and that D ¯ v x ≤ ¯ v ¯ u. (3.12) Proof.
According to Proposition 2.1, the proof of (3.11) is trivial and hence we prove (3.12)only. Since 0 < ¯ v ( x ) ≤ v ∗ for any x ∈ ¯ I , then there exists an x ∈ (0 ,
1) such that0 < ¯ v ( x ) = min x ∈ ¯ I ¯ v ( x ) and ¯ v x ( x ) = 0 . Multiplying the second equation in (1.6) by ¯ v x followed by an integration from x to x , we have D v x = Z xx ¯ u ¯ vv y d y = λ Z ¯ v ( x )¯ v ( x ) s e s d s ≤ λ Z ¯ v s e s d s, with λ = M R I e ¯ v d x , where we have used the following identity¯ u = M R I e ¯ v d x e ¯ v =: λ e ¯ v (3.13)from (2.1). Hence, we get, thanks to (3.13) and integration by parts, that D ¯ v x ≤ λ ¯ v e ¯ v − λ Z ¯ v s e s d s = ¯ v ¯ u − λ Z ¯ v s e s d s ≤ ¯ v ¯ u . The proof is completed. (cid:3)
Now let us turn to estimates on the solution ( ϕ, ψ ). We begin with the following weighted L estimate. Lemma 3.3.
For any solution ( ϕ, ψ ) ∈ X (0 , T ) to the problem (3.5) – (3.7) satisfying (3.9) , itholds that Z I (cid:18) ϕ ¯ u + ψ ¯ v (cid:19) d x + Z t (cid:0) k ϕ x k L + k ψ x k L (cid:1) d τ ≤ C (cid:0) k ϕ k L + k ψ k L (cid:1) (3.14) for any t ∈ [0 , T ] , provided that δ is suitably small, where C > is a constant independent of t .Proof. Multiplying the first equation in (3.5) by ϕ ¯ u followed by an integration over I , we getafter using integration by parts that12 dd t Z I ϕ ¯ u d x + Z I ϕ x ¯ u d x = − Z I ϕϕ x (cid:20) ¯ v x + (cid:18) u (cid:19) x (cid:21) d x − Z I ψ x ϕ d x − Z I ψ x ϕϕ x ¯ u d x. (3.15) G.-Y. HONG AND Z.-A. WANG
By the first equation in (3.3), we have (cid:18) u (cid:19) x + ¯ v x ¯ u = − u (¯ u x − ¯ u ¯ v x ) = 0 , and thus − Z I ϕϕ x (cid:20) ¯ v x + (cid:18) u (cid:19) x (cid:21) d x = 0 . (3.16)Thanks to (3.10) and the Cauchy-Schwarz inequality, we get Z I ψ x ϕϕ x ¯ u d x ≤ k ϕ k L ∞ k ψ x k L k ϕ x k L ≤ Cδ (cid:0) k ϕ x k L + k ψ x k L (cid:1) . (3.17)Inserting (3.16) and (3.17) into (3.15) gives12 dd t Z I ϕ ¯ u d x + Z I ϕ x ¯ u d x ≤ − Z I ψ x ϕ d x + Cδ (cid:0) k ϕ x k L + k ψ x k L (cid:1) . (3.18)To proceed, multiplying the second equation in (3.5) by ψ ¯ v and then integrating the resultingequation over I , we have12 dd t Z I ψ ¯ v d x + D Z I ψ x ¯ v d x = − Z I (cid:20) ¯ u ¯ v ψ + Dψψ x (cid:16) v (cid:17) x (cid:21) d x − Z I ψϕ x d x − Z I ψ ϕ x ¯ v d x, (3.19)where, by virtue of (1.6) and (3.12), it holds that − Z I (cid:20) ¯ u ¯ v ψ + Dψψ x (cid:16) v (cid:17) x (cid:21) d x = − Z I ψ (cid:18) ¯ u ¯ v + D ¯ v xx v − D ¯ v x ¯ v (cid:19) d x = − Z I ψ (cid:18)
32 ¯ u ¯ v − D ¯ v x ¯ v (cid:19) d x ≤ − Z I ¯ u ¯ v ψ d x + Z I ¯ u ¯ v ψ d x ≤ − Z I ¯ u ¯ v ψ d x. For the last term on the right hand side of (3.19), by (3.2), (3.9), (3.11) and the H¨older inequality,we get − Z I ψ ϕ x ¯ v d x ≤ C k ψ k L ∞ k ψ k L k ϕ x k L ≤ C k ϕ x k L k ψ x k L ≤ Cδ k ψ x k L . We thus have from (3.19) that12 dd t Z I ψ ¯ v d x + D Z I ψ x ¯ v d x + 12 Z I ¯ u ¯ v ψ d x ≤ − Z I ψϕ x d x + Cδ k ψ x k L . (3.20)Adding (3.20) with (3.18), we then arrive at12 dd t Z I (cid:18) ϕ ¯ u + ψ ¯ v (cid:19) d x + Z I (cid:18) ϕ x ¯ u + D ψ x ¯ v + ¯ uψ v (cid:19) d x ≤ Cδ (cid:0) k ϕ x k L + k ψ x k L (cid:1) . This along with (3.11) implies that12 dd t Z I (cid:18) ϕ ¯ u + ψ ¯ v (cid:19) d x + 1 C min { , D } (cid:0) k ϕ x k L + k ψ x k L (cid:1) ≤ Cδ (cid:0) k ϕ x k L + k ψ x k L (cid:1) . Therefore it holds that dd t Z I (cid:18) ϕ ¯ u + ψ ¯ v (cid:19) d x + β (cid:0) k ϕ x k L + k ψ x k L (cid:1) ≤ , (3.21)provided that Cδ ≤ C min { , D } =: β . (3.22)Integrating (3.21) over (0 , t ), we then get (3.14). The proof of Lemma 3.3 is complete. (cid:3) In the next lemma, we are going to derive the estimate on ϕ x . XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 9
Lemma 3.4.
Under the conditions of Lemma 3.3, let ( ϕ, ψ ) ∈ X (0 , T ) be a solution to theinitial-boundary value problem (3.5) – (3.7) satisfying (3.9) . Then for any given σ > , it holdsfor any t ∈ [0 , T ] that k ϕ k H + k ψ k L ≤ C (cid:0) k ϕ k H + k ψ k L (cid:1) e − α t , (3.23) provided δ is suitably small, where α is determined by (3.32) .Proof. Multiplying the first equation in (3.5) by ϕ t and integrating the resulting equation over I , we get by integration by parts that12 dd t Z I ϕ x d x + Z I ϕ t d x = − Z I ϕ x ¯ v x ϕ t d x − Z I ¯ uψ x ϕ t d x − Z I ϕ t ϕ x ψ x d x. (3.24)Next we estimate the terms on the right hand side of (3.24). By (3.11), (3.12) and the Cauchy-Schwarz inequality, we deduce that − Z I ϕ x ¯ v x ϕ t d x ≤ k ¯ v x k L ∞ k ϕ x k L k ϕ t k L ≤ ε k ϕ t k L + C ε k ϕ x k L , (3.25) − Z I ¯ uψ x ϕ t d x ≤ k ¯ u k L ∞ k ϕ t k L k ψ x k L ≤ ε k ϕ t k L + C ε k ψ x k L for any ε >
0. For the last term on the right hand side of (3.24), it follows from (3.10) and theCauchy-Schwarz inequality that − Z I ϕ t ϕ x ψ x d x ≤ k ϕ x k L ∞ k ϕ t k L k ψ x k L ≤ C (cid:16) δ + Cδ σ (cid:17) (cid:0) k ϕ t k L + k ψ x k L (cid:1) . (3.26)Substituting (3.25)–(3.26) into (3.24) and choosing δ small enough such that C (cid:16) δ + Cδ σ (cid:17) ≤ , (3.27)we get after taking ε suitably small that12 dd t Z I ϕ x d x + 12 Z I ϕ t d x ≤ C k ψ x k L . (3.28)Adding (3.28) with (3.21) multiplied by a sufficiently large constant K > Kβ > C ,it then follows thatdd t Z I (cid:18) ϕ ¯ u + ψ ¯ v + ϕ x (cid:19) d x + β (cid:0) k ϕ x k L + k ψ x k L + k ϕ t k L (cid:1) ≤ β >
0. Multiplying (3.29) by e α t with α being a constant to be determined later,we havedd t (cid:26) e α t Z I (cid:16) ϕ ¯ u + ψ ¯ v + ϕ x (cid:17) d x (cid:27) + e α t (cid:20) β ( k ϕ x k L + k ψ x k L ) − Z I α (cid:16) ϕ ¯ u + ψ ¯ v + ϕ x (cid:17) d x (cid:21) ≤ , (3.30)where we have ignored k ϕ t k L on the left hand side of (3.29) due to β >
0. By (3.2), (3.7),(3.11) and the Sobolev inequality k f k L ≤ C k f x k L for any f ∈ H asserting Z I (cid:18) ϕ ¯ u + ψ ¯ v (cid:19) d x ≤ C (cid:0) k ϕ x k L + k ψ x k L (cid:1) for some constant C >
0, we get from (3.30) thatdd t (cid:26) e α t Z I (cid:18) ϕ ¯ u + ψ ¯ v + ϕ x (cid:19) d x (cid:27) ≤ , (3.31)provided that α ≤
12 min (cid:26) β C , β (cid:27) . (3.32) This along with (3.11) gives rise to k ϕ k H + k ψ k L ≤ C (cid:0) k ϕ k H + k ψ k L (cid:1) e − α t . (3.33)We thus finish the proof of Lemma 3.4. (cid:3) In what follows, we derive some higher-order estimates for the solution.
Lemma 3.5.
Let ( ϕ, ψ ) ∈ X (0 , T ) be a solution to the initial-boundary value problem (3.5) – (3.7) satisfying (3.9) and assume the conditions of Lemma 3.3 hold. Then it holds for any t ∈ [0 , T ] that k ϕ xx k L + k ψ x k H ≤ σ, (3.34) and that Z t (cid:0) k ϕ x k H + k ψ x k H + k ϕ τ k H + k ψ τ k H (cid:1) d τ ≤ C (cid:0) k ϕ k H + k ψ k H (cid:1) , (3.35) provided that δ and k ϕ k H + k ψ k L are suitably small, where σ is given by (3.50) .Proof. Let us begin with the estimate on ψ x . Multiplying the second equation in (3.5) by ψ t followed by an integration with respect to x , we get12 dd t Z I (cid:0) Dψ x + ¯ uψ (cid:1) d x + Z I ψ t d x = − Z I ψ t ϕ x ψ d x − Z I ¯ vϕ x ψ t d x, (3.36)where, thanks to (3.2), (3.9) and the Cauchy-Schwarz inequality, the terms on the right handside can be estimated as follows: − Z I ψ t ϕ x ψ d x ≤ C k ψ k L ∞ k ψ t k L k ϕ x k L ≤ C k ψ x k L k ψ t k L k ϕ x k L ≤ Cδ (cid:0) k ψ t k L + k ψ x k L (cid:1) , − Z I ¯ vϕ x ψ t d x ≤ k ¯ v k L ∞ k ϕ x k L k ψ t k L ≤ ε k ψ t k L + C ε k ϕ x k L for any ε >
0. We thus update (3.36), after taking ε and δ suitably small, asdd t Z I (cid:0) Dψ x + ¯ uψ (cid:1) d x + Z I ψ t d x ≤ C (cid:0) k ϕ x k L + k ψ x k L (cid:1) . This along with (3.11) and (3.14) implies for any t ∈ [0 , T ] that k ψ ( · , t ) k H + Z t k ψ τ k L d τ ≤ C k ψ x k L + C (cid:0) k ϕ k L + k ψ k L (cid:1) . (3.37)To complete the proof, it now remains to derive the H -estimates. Differentiating (3.5) withrespect to t , we get ( ϕ tt = ϕ xxt − ¯ v x ϕ xt − ¯ uψ xt − ϕ xt ψ x − ϕ x ψ xt ,ψ tt = Dψ xxt − ¯ uψ t − ¯ vϕ xt − ϕ xt ψ − ϕ x ψ t . (3.38)Multiplying the first equation in (3.38) by ϕ t and the second one by ψ t , integrating the resultingequation over I , we get12 dd t Z I (cid:0) ϕ t + ψ t (cid:1) d x + Z I (cid:0) ϕ xt + Dψ xt + ¯ uψ t (cid:1) d x = − Z I ¯ v x ϕ xt ϕ t d x − Z I ¯ uψ xt ϕ t d x − Z I ¯ vϕ xt ψ t d x − Z I ϕ x ψ xt ϕ t d x − Z I ϕ xt ψψ t d x − Z I ϕ x ψ t d x − Z I ϕ xt ψ x ϕ t d x. (3.39) XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 11
We now estimate the terms on the right hand side of (3.39). By (3.11), (3.12) and the Cauchy-Schwarz inequality, we have − Z I ¯ v x ϕ xt ϕ t d x − Z I ¯ uψ xt ϕ t d x − Z I ¯ vϕ xt ψ t d x ≤ ε Z I (cid:0) ψ xt + ϕ xt (cid:1) d x + C ε Z I (cid:0) ϕ t + ψ t (cid:1) d x (3.40)for any ε >
0. Thanks to (3.9), (3.10) and the Cauchy-Schwarz inequality, we derive for 0 < δ ≤ − Z I ϕ x ψ xt ϕ t d x − Z I ϕ xt ψψ t d x − Z I ϕ x ψ t d x ≤ C k ϕ x k L ∞ k ψ xt k L k ϕ t k L + k ψ k L ∞ k ϕ xt k L k ψ t k L + C k ϕ x k L k ψ t k L ≤ C (cid:16) δ + δ σ (cid:17) k ψ xt k L k ϕ t k L + Cδ σ k ϕ xt k L k ψ t k L + Cδ k ψ t k L ≤ Cδ (cid:0) k ψ xt k L + k ϕ xt k L (cid:1) + C (cid:16) δ + δ σ (cid:17) (cid:0) k ϕ t k L + k ψ t k L (cid:1) . (3.41)For the last term on the right hand side of (3.39), we get from (3.2) and Young’s inequality that − Z I ϕ xt ψ x ϕ t d x ≤ C k ϕ xt k L k ϕ t k L k ψ x k L ≤ ε k ϕ xt k L + C ε k ϕ t k L k ψ x k L (3.42)for any ε >
0, where we have used the fact ϕ t (0 , t ) = ϕ t (1 , t ) = 0 due to (3.7). Substituting(3.40)–(3.42) into (3.39) and then taking ε and δ small enough, we get that12 dd t Z I (cid:0) ϕ t + ψ t (cid:1) d x + 12 Z I (cid:0) ϕ xt + Dψ xt + ¯ uψ t (cid:1) d x ≤ C (cid:16) δ σ (cid:17) Z I (cid:0) ϕ t + ψ t (cid:1) d x + k ϕ t k L k ψ x k L . (3.43)Recalling (3.14) and (3.28), we have Z I ϕ x d x + Z t Z I ϕ τ d x d τ ≤ k ϕ x k L + C Z t k ψ x k L d τ ≤ C (cid:0) k ϕ k H + k ψ k L (cid:1) , (3.44)provided δ is small enough and (3.27) holds. Combining (3.37) with (3.43) and (3.44), we thenget Z I (cid:0) ϕ t + ψ t (cid:1) d x + Z t Z I (cid:0) ϕ xτ + ψ xτ + ¯ uψ τ (cid:1) d x d τ ≤ C Z I (cid:0) ϕ xx + ϕ x + ψ xx + ψ x (cid:1) d x + (cid:16) sup ≤ τ ≤ t k ψ x k L (cid:17) Z t Z I ϕ τ d x d τ + C (cid:16) δ σ (cid:17) Z t Z I (cid:0) ϕ τ + ψ τ (cid:1) d x d τ ≤ C (cid:0) k ϕ xx k L + k ψ x k H + k ϕ x k L (cid:1) + C (cid:16) δ σ (cid:17) (cid:0) k ϕ k L + k ψ k L (cid:1) + Cδ σ k ψ x k L + C (cid:0) k ϕ k H + k ψ k L (cid:1) (cid:0) k ϕ k L + k ψ k H (cid:1) (3.45)for any t ∈ [0 , T ], where ϕ t | t =0 = ϕ xx − ϕ x ¯ v x − ¯ uψ x − ϕ x ψ x and ψ t | t =0 = Dψ xx − ¯ uψ − ¯ vϕ x − ϕ x ψ from (3.5) have been used. From (3.1), (3.11), (3.12), the first equation in (3.5)and the Cauchy-Schwarz inequality, we have Z I ϕ xx d x ≤ C Z I (cid:0) ϕ t + ϕ x + ψ x (cid:1) d x + k ϕ x k L ∞ Z I ψ x d x ≤ C Z I (cid:0) ϕ t + ϕ x + ψ x (cid:1) d x + C (cid:0) k ϕ x k L k ϕ xx k L + k ϕ x k L (cid:1) Z I ψ x d x ≤ C Z I (cid:0) ϕ t + ϕ x + ψ x (cid:1) d x + 12 k ϕ xx k L + C k ϕ x k L (cid:0) k ψ x k L + k ψ x k L (cid:1) , and thus Z I ϕ xx d x ≤ C Z I (cid:0) ϕ t + ϕ x + ψ x (cid:1) d x + C k ϕ x k L (cid:0) k ψ x k L + k ψ x k L (cid:1) . This together with (3.14), (3.37), (3.44) and (3.45) yields that Z I ϕ xx d x ≤ C (cid:0) k ϕ xx k L + k ψ x k H + k ϕ x k L (cid:1) + C (cid:16) δ σ (cid:17) (cid:0) k ϕ k L + k ψ k L (cid:1) + Cδ σ k ψ x k L + C (cid:0) k ϕ k L + k ψ k L (cid:1) (cid:0) k ϕ k L + k ψ k H (cid:1) , (3.46)and that Z t Z I ϕ xx d x d τ ≤ C Z t Z I ( ϕ τ + ϕ x + ψ x )d x d τ + sup ≤ τ ≤ t (cid:0) k ψ x k L + k ψ x k L (cid:1) Z t k ϕ x k L d τ ≤ C (cid:0) k ϕ k H + k ψ k L (cid:1) + (cid:0) k ϕ k L + k ψ k L (cid:1) (cid:0) k ψ k H + k ϕ k L (cid:1) ≤ C (cid:0) k ϕ k H + k ψ k L (cid:1) (cid:0) k ϕ k L + k ψ k H (cid:1) . (3.47)where (3.27) has been used. Similarly, recalling (3.2), (3.7), (3.33), (3.37), (3.44), (3.45) and thesecond equation in (3.5), we have Z I ψ xx d x ≤ C Z I (cid:0) ψ t + ψ + ϕ x (cid:1) d x + k ψ k L ∞ Z I ϕ x d x ≤ C Z I (cid:0) ψ t + ψ + ϕ x (cid:1) d x + k ψ k L k ψ x k L Z I ϕ x d x ≤ C (cid:0) k ϕ xx k L + k ψ x k H + k ϕ x k L (cid:1) + C (cid:16) δ σ (cid:17) (cid:0) k ϕ k L + k ψ k L (cid:1) + Cδ σ k ψ x k L + C (cid:0) k ϕ k H + k ψ k L (cid:1) (cid:0) k ϕ k L + k ψ k H (cid:1) (3.48)and Z t Z I ψ xx d x d τ ≤ C Z t Z I ( ψ τ + ψ + ϕ x )d x d τ + (cid:16) sup τ ∈ [0 ,t ] k ψ ( · , τ ) k H (cid:17) Z t Z I ϕ x d x d τ ≤ C (cid:0) k ψ k H + k ϕ k L (cid:1) + C (cid:0) k ϕ k L + k ψ k L (cid:1) (cid:0) k ψ k H + k ϕ k L (cid:1) ≤ C (cid:0) k ϕ k L + k ψ k L (cid:1) (cid:0) k ϕ k L + k ψ k H (cid:1) . (3.49)Combining (3.37), (3.46) and (3.48), we arrive at k ϕ xx ( · , t ) k L + k ψ x ( · , t ) k H + Z t Z I (cid:0) ϕ xτ + ψ xτ + ¯ uψ τ (cid:1) d x d τ ≤ C (cid:0) k ϕ xx k L + k ψ x k H (cid:1) + C (cid:16) δ σ (cid:17) (cid:0) k ϕ k H + k ψ k L (cid:1) + Cδ σ k ψ x k L + C (cid:0) k ϕ k H + k ψ k L (cid:1) (cid:0) k ϕ k L + k ψ k H (cid:1) . Consequently, if we take σ = max (cid:8) C (cid:0) k ϕ xx k L + k ψ x k H (cid:1) , (cid:9) (3.50)and set both δ and k ϕ k H + k ψ k L small enough such that (3.27) and C (cid:16) δ σ (cid:17) (cid:0) k ϕ k H + k ψ k L (cid:1) + Cδ σ k ψ x k L + C (cid:0) k ϕ k H + k ψ k L (cid:1) (cid:0) k ϕ k L + k ψ k H (cid:1) ≤ σ (3.51) XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 13 are satisfied, then it holds that k ϕ xx ( · , t ) k L + k ψ x ( · , t ) k H + Z t Z I (cid:0) ϕ xτ + ψ xτ + ¯ uψ τ (cid:1) d x d τ ≤ σ . (3.52)This gives (3.34). Differentiating the first equation in (3.5) with respect to x leads to ϕ xxx = ϕ xt + ¯ v xx ϕ x − ¯ v x ϕ x − ¯ u x ψ x − ¯ vψ xx − ϕ xx ψ x − ϕ x ψ xx , which in combination with (1.6), (3.11), (3.12), (3.14), (3.47), (3.49), (3.52) and the Sobolevinequality (3.1) yields that Z t Z I ϕ xxx d x d τ ≤ C (cid:0) k ϕ k H + k ψ k H (cid:1) , (3.53)provided k ϕ k H + k ψ k L is suitably small. Similarly, we utilize (3.3), (3.11), (3.12), (3.14),(3.46) and (3.52) to get Z t Z I ψ xxx d x d τ ≤ C (cid:0) k ϕ k H + k ψ k H (cid:1) , (3.54)provided k ϕ k H + k ψ k L is suitably small. Combining (3.47), (3.49), (3.53) and (3.54), wethen obtain (3.35) and finish the proof of Lemma 3.5. (cid:3) Proof of Proposition 3.3 . According to Lemmas 3.2–3.5, to finish the proof of Proposition3.3, it suffices to close the a priori assumptions (3.9). To this end, we first fix σ by (3.50) andchoose δ and k ϕ k H + k ψ k L suitably small such that (3.22), (3.27) and (3.51) hold. Then inview of (3.23) and (3.34), the a priori assumption (3.9) is closed provided that k ϕ k H + k ψ k L is small enough. The proof is completed. (cid:3) Proof of Theorem 2.1.
With the unique solution ( ϕ, ψ ) ∈ X (0 , ∞ ) obtained in Proposi-tion 3.2 to the reformulated problem (3.5)–(3.7), in view of (3.4), we conclude that the initial-boundary value problem (1.4), (1.5a) admits a unique global solution ( u, v ) satisfying, u ∈ C ([0 , ∞ ); H ) ∩ L (0 , ∞ ; H ) , v ∈ C ([0 , ∞ ); H ) ∩ L (0 , ∞ ; H ) . Furthermore, according to (3.8), we have k ( u − ¯ u, v − ¯ v )( · , t ) k L ≤ C e − α t , k ( u x − ¯ u x , ϕ x − ¯ v x )( · , t ) k L ≤ C for any t ≥ , where C > t . This along with the Sobolev inequality (3.1) implies that k ( u − ¯ u, v − ¯ v )( · , t ) k L ∞ ≤ C k ( u − ¯ u, v − ¯ v )( · , t ) k L k ( u x − ¯ u x , v x − ¯ v x )( · , t ) k L + C k ( u − ¯ u, v − ¯ v )( · , t ) k L ≤ C e − α t for any t ≥ . This gives (2.2) with α = α . We thus finish the proof of Theorem 2.1. (cid:3) Asymptotic stability for the case D = 0In this section, we are devoted to studying the large time behavior of solutions to the problem(1.4), (1.5b) with D = 0. As in the case D >
0, the heart of the matter is to derive someuniform-in-time estimates on the solution.
A priori estimates.
Now we consider the system (1.4) with D = 0: ( u t = u xx − ( uv x ) x in I ,v t = − uv in I , (4.1)subject to the following initial and boundary conditions ( ( u, v ) | t =0 = ( u , v )( x ) , u ≥ , ( u x − uv x ) | x =0 , = 0 . (4.2)We shall show that ( u, v ) → ( M,
0) in L ∞ as t → + ∞ , where M = R I u d x . To this end, we first reformulate the problem by defining w ( x, t ) = Z x ( u ( y, t ) − M )d y with w | t =0 = Z x ( u − M )d y =: w ( x ) , which leads to the following problem in terms of ( w, v ): w t = w xx − M v x − w x v x ,v t = − M v − w x v,w (0 , t ) = w (1 , t ) = 0 , ( w, v ) | t =0 = ( w , v )( x ) . (4.3)Similar to the case D > X (0 , T ) := (cid:8) ( w, v ) | w ∈ C ([0 , T ]; H ∩ H ) ∩ L (0 , T ; H ) , v ∈ C ([0 , T ]; H ) (cid:9) . Precisely, we have the following local existence result.
Proposition 4.1.
Assume w ∈ H ∩ H and v ∈ H such that w x + M ≥ and v ≥ . Thenthere exists a positive constant T ∗ depending on the initial data such that there exists a uniquesolution ( w, v ) ∈ X (0 , T ∗ ) to the problem (4.3) with k w k H + k v k H ≤ (cid:0) k w k H + k v k H (cid:1) and w x + M ≥ , v ≥ for any ( x, t ) ∈ I × [0 , T ∗ ) . Next, we state the global existence result for the initial-boundary value problem (4.3), fromwhich we can obtain the global existence and large time behavior of the solution ( u, v ) to theproblem (4.1), (4.2).
Proposition 4.2.
Let w ∈ H ∩ H and v ∈ H such that w x + M ≥ and v ≥ . Thenthere exists a positive constant δ ∗ such that if k w k H + k v k H ≤ δ ∗ , the unique solution of theproblem (4.3) obtained in Proposition 4.1 exists globally in time. Furthermore, it holds that k w xx ( · , t ) k L + k v xx ( · , t ) k L + k w ( · , t ) k H + k v ( · , t ) k H ≤ C e − α t t ≥ , (4.4) where C > is a constant independent of t . To prove Proposition 4.2, by the local existence result and the standard continuation argu-ment, we just need to establish some a priori estimates as stated in the following proposition.
XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 15
Proposition 4.3.
For any
T > , let ( w, v ) ∈ X (0 , T ) be a solution to the initial-boundaryvalue problem (4.3) . Then there exists a constant ˆ C > independent of T such that if k w k H + k v k H ≤ ˆ C , then the solution ( w, v ) possesses the following estimates: k w ( · , t ) k H + k v ( · , t ) k H ≤ C e − α t , (4.5a) k w xx k L + k v xx ( · , t ) k L + Z t ( k w k H + k w τ k H + k v τ k H )d τ ≤ C (4.5b) for any t ∈ [0 , T ] , where α is as in (4.8) , the constant C > is independent of T .Proof. The proof of Proposition 4.3 consists of Lemmas 4.1–4.3 below. (cid:3)
Before proceeding, we assume that the solution ( w, v ) to the problem (4.3) satisfy the following a priori assumptions: k w ( · , t ) k H + k v ( · , t ) k H ≤ δ, k w xx ( · , t ) k L ≤ σ for any t ∈ [0 , T ] , (4.6)where 0 < ˜ δ < σ ≥ H -norm of ( w, v ) with (4.6). Lemma 4.1.
For any
T > , let ( w, v ) ∈ X (0 , T ) be a solution to the initial-boundary valueproblem (4.3) satisfying (4.6) . Then it holds for ˜ σ ≥ that k w k H + k v k H ≤ C (˜ σ k v k H + k w k H ) for all t ∈ [0 , T ] , (4.7) provided ˜ δ is suitably small. Furthermore, we have the following decay estimate k w k H + k v k H ≤ C (˜ σ k v k H + k w k H )e − α t for all t ∈ [0 , T ] , (4.8) where α and C are positive constants independent of t and ˜ σ .Proof. We divide the proof into three steps.
Step 1:
Estimates on w . Multiplying the first equation in (4.3) followed by an integration over I , we have 12 dd t Z I w d x + Z I w x d x = − Z I wM v x d x − Z I w x wv x d x. (4.9)With integration by parts and the Cauchy-Schwarz inequality, we get − Z I wM v x d x = Z I M vw x d x ≤ η Z I w x d x + C η Z I v d x (4.10)for any η >
0. In view of (4.6), the Cauchy-Schwarz inequality and the Sobolev inequality (3.2),we derive − Z I w x wv x d x ≤ C k w k L ∞ k w x k L k v x k L ≤ C ˜ δ (cid:0) k w x k L + k v x k L (cid:1) . (4.11)Inserting (4.10) and (4.11) into (4.9), for suitably small ˜ δ and η , it holds thatdd t Z I w d x + Z I w x d x ≤ C Z I (cid:0) v + v x (cid:1) d x. (4.12)To proceed, multiplying the first equation in (4.3) by w t and then integrating the resultingequation over I , we get12 dd t Z I w x d x + Z I w t d x = − M Z I v x w t d x − Z I w x v x w t d x. (4.13)For the first term on the right hand side of (4.13), we utilize the Cauchy-Schwarz inequality toget − M Z I v x w t d x ≤ η k w t k + C η k v x k L for any η >
0. For the last term, from (3.1) and (4.6), we have k w x k L ∞ ≤ C ˜ σ ˜ δ + C ˜ δ, (4.14)which along with the Cauchy-Schwarz inequality implies that Z I w x v x w t d x ≤ k w x k L ∞ k w t k L k v x k L ≤ C (˜ σ ˜ δ + ˜ δ ) (cid:0) k w t k L + k v x k L (cid:1) . Therefore, after taking η suitably small (e.g., η < ) and choosing ˜ δ small enough such that C (cid:16) ˜ σ ˜ δ + ˜ δ (cid:17) ≤ , (4.15)we update (4.13) as dd t Z I w x d x + Z I w t d x ≤ C k v x k L . This together with (4.12) givesdd t Z I ( w + w x )d x + Z I ( w x + w t )d x ≤ C Z I (cid:0) v + v x (cid:1) d x. (4.16) Step 2:
Estimates on v . In view of (4.14), it holds that M ≤ w x + M ≤ M , (4.17)provided C (˜ σ ˜ δ + ˜ δ ) ≤ M . (4.18)Therefore we test the second equation in (4.3) against v to get12 dd t Z I v d x + M Z I v d x ≤
12 dd t Z I v d x + Z I ( w x + M ) v d x = 0 . That is, dd t Z I v d x + M Z I v d x ≤ . (4.19)Differentiating the second equation in (4.3) with respect to x gives v xt = − ( M + w x ) v x − w xx v. (4.20)Multiplying (4.20) by v x and then integrating the resulting equation over I , it follows that12 dd t Z I v x d x + Z I ( M + w x ) v x d x = − Z I w xx vv x d x (4.21)with − Z I w xx vv x d x ≤ M k v x k L + C Z I w xx v d x ≤ M k v x k L + CM k v k L ∞ Z I w xx d x ≤ M k v x k L + C ˜ σ M (cid:0) k v x k L k v k L + k v k L (cid:1) ≤ M k v x k L + C (cid:18) ˜ σ M + ˜ σ M (cid:19) k v k L , where we have used (3.1), (4.6) and the Cauchy-Schwarz inequality. Furthermore, thanks to(4.17) and the fact ˜ σ ≥
1, we have from (4.21) that12 dd t Z I v x d x + M Z I v x d x ≤ C ˜ σ k v k L . (4.22) XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 17
Combining (4.22) with (4.19), we then havedd t Z I (˜ σ v + v x )d x + ˆ c Z I ( v + v x )d x ≤ , (4.23)where ˆ c > M but independent of ˜ σ . Step 3:
Decay estimates. Combining (4.23) with (4.16) yields thatdd t Z I (˜ σ v + v x + w + w x )d x + ˆ c (cid:0) k v k H + k w x k L + k w t k L (cid:1) ≤ , where ˆ c is a constant depending on M but independent of ˜ σ . Consequently, we have k v k H + k w k H + Z t (cid:0) k v k H + k w x k L + k w τ k L (cid:1) d τ ≤ C (˜ σ k v k H + k w k H ) (4.24)for any t ∈ [0 , T ], where we have used the fact ˜ σ ≥
1. The estimate (4.7) is proved. To showthe decay estimate (4.8), multiplying (4.23) by e ˆ α t with ˜ σ ˆ α ≤ ˆ c , we deduce thatdd t (cid:26) e ˆ α t Z I (˜ σ v + v x )d x (cid:27) ≤ , (4.25)which immediately yields that Z I (˜ σ v + v x )d x ≤ e − ˆ α t Z I (˜ σ v + v x )d x. (4.26)Since w (0 , t ) = w (1 , t ) = 0, we have k w k L ≤ ˜ C k w x k L for some constant ˜ C >
0. Multiplying(4.16) by e ˆ α t with ˆ α < min { ˜ C , ˆ α } , it follows thatdd t (cid:26) e ˆ α t Z I ( w + w x )d x (cid:27) ≤ C e ˆ α t Z I (cid:0) v + v x (cid:1) d x, (4.27)and thus Z I ( w + w x )d x ≤ C (cid:0) k w k H + ˜ σ k v k H (cid:1) e − ˆ α t , where we have used (4.26) and ˜ σ ≥
1. The proof is completed. (cid:3)
In the next lemma, we establish estimate for w xx . Lemma 4.2.
Assume the conditions of Lemma 4.1 hold. Then the solution ( w, v ) ∈ X (0 , T ) to the problem (4.3) satisfies k w xx k L ≤
32 ˜ σ for any t ∈ [0 , T ] , provided that ˜ δ and k v k H + k w k H are suitably small, where ˜ σ ≥ is determined by (4.40) .Proof. Differentiating the first equation in (4.3) with respect to t , we have w tt = w xxt − M v xt − w xt v x − w x v xt . (4.28)Multiplying (4.28) by w t followed by an integration over I , we get12 dd t Z I w t d x + Z I w xt d x = − M Z I v xt w t d x − Z I w xt v x w t d x − Z I w x v xt w t d x. (4.29)We next estimate the terms on the right hand side of (4.29). Using integration by parts and theCauchy-Schwarz inequality, one has − M Z I v xt w t d x = M Z I v t w tx d x ≤ Z I w xt d x + C Z I v t d x. Furthermore, recalling (4.17) and the second equation in (4.3), we have Z I v t d x ≤ C Z I v ( w x + M ) d x ≤ C k v k L . (4.30) It thus holds that − M Z I v xt w t d x ≤ Z I w xt d x + C Z I v d x, (4.31)provided C (˜ σ ˜ δ + ˜ δ ) ≤ M . It follows from (3.2) and the Cauchy-Schwarz inequality that − Z I w xt v x w t d x ≤ Z I w xt d x + C Z I w t v x d x ≤ Z I w xt d x + C k w t k L ∞ k v x k L ≤ Z I w xt d x + C k w t k L k w xt k L k v x k L ≤ Z I w xt d x + C k w t k L k v x k L . (4.32)For the last term on the right hand side of (4.29), integration by parts leads to Z I w x v xt w t d x = − Z I w xx v t w t d x − Z I w x v t w xt d x. (4.33)Recalling the first equation in (4.3), we have Z I w xx d x ≤ Z I w t d x + M Z I v x d x + Z I w x v x d x, where, thanks to the Sobolev inequality (3.1) and the Cauchy-Schwarz inequality, we deducethat Z I w x v x d x ≤ k w x k L ∞ Z I v x d x ≤ C (cid:0) k w x k L k w xx k L + k w x k L (cid:1) k v x k L ≤ k w xx k L + C k w x k L k v x k L + C k w x k L k v x k L . It then follows that Z I w xx d x ≤ C k w t k L + C k w x k L k v x k L + C k w x k L k v x k L . (4.34)This, along with (3.2), (4.30) and the Cauchy-Schwarz inequality, yields − Z I w xx v t w t d x ≤ C Z I w xx d x + C Z I v t w t d x ≤ C Z I w xx d x + C k w t k L ∞ Z I v t d x ≤ C k w xx k L + C k w t k L k w tx k L k v k L ≤ k w xt k L + C k w xx k L + C k w t k L k v k L ≤ k w tx k L + C k w t k L + C k w t k L k v k L + C k w x k L k v x k L (1 + k v x k L ) . (4.35)It now remains to estimate the last term on the right hand side of (4.33). By (3.1), (4.30), (4.34)and the Cauchy-Schwarz inequality, we get − Z I w x v t w xt d x ≤ Z I w xt d x + C Z I v t w x d x ≤ Z I w xt d x + C k w x k L ∞ Z I v t d x ≤ Z I w xt d x + C (cid:0) k w x k L k w xx k L + k w x k L (cid:1) k v k L ≤ Z I w xt d x + C k w xx k L + C k w x k L k v k L + C k w x k L k v k L ≤ Z I w xt d x + C k w t k L + C k w x k L k v x k L + C k w x k L k v x k L + C k w x k L k v k L + C k w x k L k v k L . (4.36) XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 19
With (4.35) and (4.36), we then update (4.33) as Z I w x v xt w t d x ≤ Z I w xt d x + C k w t k L + C k w t k L k v k L + C k w x k L k v x k L + C k w x k L k v x k L + C k w x k L k v k L + C k w x k L k v k L (4.37)for any η >
0. Combining (4.29), (4.31), (4.32) and (4.37), we arrive atdd t Z I w t d x + Z I w xt d x ≤ C k v t k L + C k w t k L (1 + k v k H ) + C k w x k L k v k H + C k w x k L k v k H . Integrating the above inequality over [0 , t ] for any t ∈ [0 , T ], thanks to (4.7), (4.30) and w t | t =0 = w xx − M v x − w x v x from (4.3) , one can show that Z I w t d x + Z t Z I w xτ d x d τ ≤ C k w xx k L + k ( w x + M ) v x k L + C Z t k v τ k L d τ + sup τ ∈ [0 ,t ] k v k H Z t (cid:0) k w τ k L + k w x k L (cid:1) d τ + sup τ ∈ [0 ,t ] k v k H Z t k w x k L d τ ≤ C k w xx k L + C (cid:16) ˜ σ k v k H + k w k H (cid:17)(cid:16) σ k v k H + k w k H (cid:17) . (4.38)Recalling (4.7) and (4.34), we then get Z I w xx d x ≤ C k w t k L + C k w x k L k v x k L + C k w x k L k v x k L ≤ C k w xx k L + C (cid:16) ˜ σ k v k H + k w k H (cid:17)(cid:16) σ k v k H + k w k H (cid:17) , (4.39)where we have used (4.7). Consequently, we obtain Z I w xx d x ≤
94 ˜ σ for any t ∈ [0 , T ] , provided C k w xx k L + C (cid:16) ˜ σ k v k H + k w k H (cid:17)(cid:16) σ k v k H + k w k H (cid:17) ≤
94 ˜ σ . (4.40)It should be pointed out the constraint (4.40) on ˜ σ is reachable. Indeed, if we fix the constant ˜ σ ∈ (max { , √ C k w xx k L } , + ∞ ), then (4.40) is automatically satisfied provided k v k H + k w k H is suitably small. The proof is complete. (cid:3) Remark 4.1.
According to (4.39) , we can fix the constant ˜ σ ∈ (max { , √ C k w xx k L } , + ∞ ) .Furthermore, let the constant ˜ δ suitably small such that (4.15) and (4.18) hold. Then by (4.7) and Lemma 4.2, if k v k H + k w k H is small enough, the a priori assumption (4.6) is closed. Now we have closed the a priori assumptions in (4.6) and proved most of the estimates in(4.5). To guarantee the global existence of the solution ( w, v ), we need to derive some morehigher-order estimates (i.e., the rest of the estimates in (4.5)), and ultimately end the proof ofProposition 4.3.
Lemma 4.3.
Under the conditions of Lemmas 4.1–4.2, we get for any t ∈ [0 , T ] that k v xx ( · , t ) k L + Z t (cid:0) k w xxx k L + k v τ k H (cid:1) d τ ≤ C, where C > is a constant independent of t . Proof.
Differentiating the first equation in (4.3) with respect to x , we get w xxx = w xt + M v xx + w xx v x + w x v xx . We thus derive, thanks to (3.1), (4.7), (4.38) and (4.39), that Z t k w xxx k L d τ ≤ Z t k w xτ k L d τ + Z t k v x k L ∞ k w xx k L d τ + M Z t k v xx k L d τ + Z t k w x k L ∞ k v xx k L d τ ≤ C + C Z t k v xx k L d τ (4.41)for any t ∈ [0 , T ], where the constant C > t . From the second equation in(4.3), we have v ( x, t ) = v e − R t ( w x + M )d τ , (4.42) v txx = − ( w x + M ) v xx − w xxx v − w xx v x . (4.43)Testing (4.43) against v xx , and then integrating the resulting equation over (0 , t ) for any t ∈ (0 , T ], we get Z I v xx ( · , t )d x + Z t Z I ( w x + M ) v xx d x d τ = − Z t Z I w xxx vv xx d x d τ − Z t Z I w xx v x v xx d x d τ, (4.44)where, due to (4.17), (4.41) and (4.42), it holds that Z t Z I w xxx vv xx d x d τ ≤ C Z t k v k L ∞ k w xxx k L k v xx k L d τ ≤ C Z t e − M τ k w xxx k L k v xx k L d τ ≤ M Z t Z I v xx d x d τ + Z t e − M τ k v xx k L d τ + C (4.45)for some constant C > t . For the last term on the right hand side of (4.44),by (3.1), (4.7) and the Cauchy-Schwarz inequality, we have − Z t Z I w xx v x v xx d x d τ ≤ M Z t Z I v xx d x d τ + Z t k v x k L ∞ Z I w xx d x d τ ≤ M Z t Z I v xx d x d τ + Z t ( k v x k L + k v xx k L ) k w xx k L d τ ≤ M Z t Z I v xx d x d τ + C Z t k w xx k L (1 + k v xx k L )d τ. (4.46)Inserting (4.45) and (4.46) into (4.44), by (4.17), it follows that Z I v xx ( · , t )d x + M Z t Z I v xx d x d τ ≤ C Z t (cid:16) e − M τ + k w xx k L (cid:17) k v xx k L d τ + C Z t k w xx k L d τ + C (4.47)for any t ∈ [0 , T ]. Furthermore, by (4.24) and (4.34), one can show for any t ∈ [0 , T ] that Z t k w xx k L d τ ≤ C Z t k w τ k L d τ + sup t ∈ [0 ,T ] (cid:0) k v x k L + k v x k L (cid:1) Z t k w x k L d τ ≤ C, (4.48)where the constant C > k v k H and k w k H . This along with (4.47) and theGronwall inequality implies that Z I v xx ( · , t )d x + M Z t Z I v xx d x d τ ≤ C. (4.49) XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 21
Combining (4.49) with (4.41) further yields that Z t k w xxx k L d τ ≤ C (4.50)for any t ∈ [0 , T ], where the constant C > t . Finally, recalling the secondequation in (4.3), we get by virtue of (3.1), (4.7), (4.48)–(4.50) that Z t k v τ ( · , τ ) k H d τ ≤ C for any t ∈ [0 , T ]. We thus finish the proof of Lemma 4.3. (cid:3) Proof of Theorem 2.2.
With the global existence result on the initial-boundary valueproblem (4.3) and the decay estimates in (4.4) for ( w, v ) at hand, by the same process as in theanalysis for the case
D > (cid:3)
Acknowledgement
G. Hong is partially supported from the CAS AMSS-POLYU Joint Laboratory of AppliedMathematics postdoctoral fellowship scheme. Z.A. Wang was supported in part by the HongKong Research Grant Council General Research Fund No. PolyU 153031/17P (Q62H) andinternal grant No. ZZHY from HKPU.
References [1]
J. Adler , Chemotaxis in bacteria , Science, 153 (1966), pp. 708–716.[2]
N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler , Toward a mathematical theoryof Keller-Segel models of pattern formation in biological tissues , Math. Models MethodsAppl. Sci., 25 (2015), pp. 1663–1763.[3]
J. Bonner , The Cellular SIime Molds , 2nd ed, Princeton University Press, 1967.[4]
M. Braukhoff and J. Lankeit , Stationary solutions to a chemotaxis-consumption modelwith realistic boundary conditions for the oxygen , Math. Models Methods Appl. Sci., 29(2019), pp. 2033–2062.[5]
J. Carrillo, J. Li, and Z.-A. Wang , Boundary spike-layer solutions of thesingular keller–segel system: existence and stability , Proc. Lond. Math. Soc.,(Doi:10.1112/plms.12319, 2020).[6]
M. Chae and K. Choi , Nonlinear stability of planar traveling waves in a chemotaxismodel of tumor angiogenesis with chemical diffusion , J. Differential Equations, 268 (2020),pp. 3449–3496.[7]
A. Chertock, K. Fellner, A. Kurganov, A. Lorz, and P. A. Markowich , Sink-ing, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolutionnumerical approach , J. Fluid Mech., 694 (2012), pp. 155–190.[8]
K. Choi, M.-J. Kang, and A. F. Vasseur , Global well-posedness of large perturbations oftraveling waves in a hyperbolic-parabolic system arising from a chemotaxis model , J. Math.Pures Appl. (9), 142 (2020), pp. 266–297.[9]
P. Davis, P. van Heijster, and R. Marangell , Absolute instabilities of travelling wavesolutions in a keller–segel model , Nonlinearity, 30 (2017), p. 4029.[10]
S. Dehaene , The neural basis of the Weber–Fechner law: a logarithmic mental numberline , Trends Cogn. Sci., 7 (2003), pp. 145–147. [11]
T. Hillen and K. Painter , A user’s guide to PDE models for chemotaxis , J. Math. Biol.,58 (2009), pp. 183–217.[12]
D. Horstmann , From 1970 until present: the Keller-Segel model in chemotaxis and itsconsequences. I , Jahresber. Deutsch. Math.-Verein., 105 (2003), pp. 103–165.[13]
Q. Hou, C.-J. Liu, Y.-G. Wang, and Z. Wang , Stability of boundary layers for a viscoushyperbolic system arising from chemotaxis: one-dimensional case , SIAM J. Math. Anal., 50(2018), pp. 3058–3091.[14]
Q. Hou and Z. Wang , Convergence of boundary layers for the Keller-Segel system withsingular sensitivity in the half-plane , J. Math. Pures Appl. (9), 130 (2019), pp. 251–287.[15]
Y. Kalinin, L. Jiang, Y. Tu, and M. Wu , Logarithmic sensing in escherichia colibacterial chemotaxis , Biophys. J., 96 (2009), pp. 2439–2448.[16]
E. Keller and G. Odell , Necessary and sufficient conditions for chemotactic bands ,Math. Biosci., 27 (1975), pp. 309–317.[17]
E. F. Keller and L. A. Segel , Initiation of slime mold aggregation viewed as an insta-bility , J. Theor. Biol., 26 (1970), pp. 399–415.[18]
E. F. Keller and L. A. Segel , Traveling bands of chemotactic bacteria: a theoreticalanalysis , J. Theor. Biol., 30 (1971), pp. 235–248.[19]
C.-C. Lee, Z. Wang, and W. Yang , Boundary-layer profile of a singularly perturbed non-local semi-linear problem arising in chemotaxis , Nonlinearity, 33 (2020), pp. 5111–5141.[20]
H. G. Lee and J. Kim , Numerical investigation of falling bacterial plumes caused bybioconvection in a three-dimensional chamber , Eur. J. Mech. B Fluids, 52 (2015), pp. 120–130.[21]
H. A. Levine, B. D. Sleeman, and M. Nilsen-Hamilton , A mathematical model forthe roles of pericytes and macrophages in the initiation of angiogenesis. I. The role ofprotease inhibitors in preventing angiogenesis , Math. Biosci., 168 (2000), pp. 77–115.[22]
H. Li and K. Zhao , Initial-boundary value problems for a system of hyperbolic balancelaws arising from chemotaxis , J. Differential Equations, 258 (2015), pp. 302–338.[23]
J. Li, T. Li, and Z. Wang , Stability of traveling waves of the Keller-Segel system withlogarithmic sensitivity , Math. Models Methods Appl. Sci., 24 (2014), pp. 2819–2849.[24]
T. Li and Z. Wang , Nonlinear stability of traveling waves to a hyperbolic-parabolic systemmodeling chemotaxis , SIAM J. Appl. Math., 70 (2009), pp. 1522–1541.[25]
T. Li and Z. Wang , Asymptotic nonlinear stability of traveling waves to conservation lawsarising from chemotaxis , J. Differential Equations, 250 (2011), pp. 1310–1333.[26]
A. Lorz , Coupled chemotaxis fluid model , Math. Models Methods Appl. Sci., 20 (2010),pp. 987–1004.[27]
T. Nagai and T. Ikeda , Traveling waves in a chemotactic model , J. Math. Biol., 30(1991), pp. 169–184.[28]
L. Nirenberg , An extended interpolation inequality , Ann. Scuola Norm. Sup. Pisa Cl. Sci.(3), 20 (1966), pp. 733–737.[29]
T. Nishida , Nonlinear hyperbolic equations and related topics in fluid dynamics ,D´epartement de Math´ematique, Universit´e de Paris-Sud, Orsay, 1978. PublicationsMath´ematiques d’Orsay, No. 78-02.[30]
H. Peng and Z.-A. Wang , Nonlinear stability of strong traveling waves for the singularkeller–segel system with large perturbations , J. Differential Equations, 265 (2018), pp. 2577–2613.[31]
H. R. Schwetlick , Travelling fronts for multidimensional nonlinear transport equations ,Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 17 (2000), pp. 523–550.
XOGENOUS CHEMOTAXIS SYSTEMS WITH PHYSICAL BOUNDARY CONDITIONS 23 [32]
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler, andR. E. Goldstein , Bacterial swimming and oxygen transport near contact lines , Proc. Natl.Acad. Sci. USA, 102 (2005), pp. 2277–2282.[33]
G. Wadhams and J. Armitage , Making sense of it all: bacterial chemotaxis , NatureReviews Molecular Cell biology, 5 (2004), pp. 1024–1037.[34]
Y. Wang, C.-L. Chen, and M. Iijima , Signaling mechanisms for chemotaxis , Develop-ment, Growth & Differentiation, 53 (2011), pp. 495–502.[35]
Z.-A. Wang , Mathematics of traveling waves in chemotaxis–review paper , Discrete Contin.Dyn. Syst. Ser. B, 18 (2013), pp. 601–641.[36]
M. Winkler , Singular structure formation in a degenerate haptotaxis model involvingmyopic diffusion , J. Math. Pures Appl. (9), 112 (2018), pp. 118–169.
Guangyi HongDepartment of Applied MathematicsHong Kong Polytechnic UniversityHung Hom, Kowloon, Hong Kong, P. R. China
Email address : [email protected] Zhi-an WangDepartment of Applied MathematicsHong Kong Polytechnic UniversityHung Hom, Kowloon, Hong Kong, P. R. China
Email address ::