Molecular

Contraction in L1 and large time behavior for a system arising in chemical reactions and molecular motors Michel Chipot 1 , Danielle Hilhorst 2 , David Kinderlehrer 3 , MichaÃl Olech 4 Abstract We prove a contraction in L1 property for the solutions of a nonlinear reaction–diffusion system whose special cases include intercellular transport as well as reversible chemical reactions. Assuming the existence of stationary solutions we show that the solutions stabilize as t tends to infinity. Moreover, in the special case of linear reaction terms, we prove the existence and the uniqueness (up to a multiplicative constant) of the stationary solution. Key words: weakly coupled system, molecular motor, transport, parabolic systems, contraction property. AMS subject classification: 34D23, 35K45, 35K50, 35K55, 35K57, 92C37, 92C45. 1 1 Introduction We start with two specific reaction-diffusion systems. The first one describes a reversible reaction and the other one a molecular motor. We first consider the reversible chemical reaction (see also Bothe [4], Bothe and Hilhorst [5], Desvillettes and Fellner [10] and Érdi and Tóth [11]). It involves a reaction-diffusion system of the form ¡ ¢ ut = d1 ∆u − αk rA (u) − rB (v) in Ω × (0, T ), Ω ⊂ Rd , (1.1) ¡ ¢ vt = d2 ∆v + βk rA (u) − rB (v) in Ω × (0, T ), Ω ⊂ Rd , Angewandte Mathematik, Universitaet Zurich, CH-8057 Zurich E-mail: [email protected] 2 Laboratoire de Mathématiques, CNRS and Université de Paris-Sud, 91405 Orsay Cédex, France; E-mail: [email protected] 3 Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213; E-mail: [email protected] 4 Instytut Matematyczny Uniwersytetu WrocÃlawskiego, pl. Grunwaldzki 2/4, 50-384 WrocÃlaw, Polska; Laboratoire de Mathématiques, CNRS Université de Paris-Sud, 91405 Orsay Cédex, France; E-mail: [email protected]. The preparation of this article has been partially supported by a KBN/MNiI grant 1 P03A 008 30 and a Marie Curie Transfer of Knowledge Fellowship of the European Community’s Sixth Framework Programme under the contracts MTKD-CT-2004-013389, DMS 0305794, DMS 0405343, and DMS 0806703. We would also like to mention the support of the Swiss National Science Foundation under the contract # 20-113287/1. 1 together with the homogeneous Neumann boundary conditions, where d1 , d2 , α, β, k and T are positive constants and where Ω is a bounded subset of Rd with smooth boundary. Such systems describe, with a suitable choice of the functions rA and rB , chemical reactions for two mobile species. For example, functions rA (u) = uk , rB (v) = v m correspond to a reversible reaction kA ⇋ mB. Reactions of the type q1 A1 + . . . qk Ak ⇋ q1 B1 + . . . qm Bm can also be described by similar systems with more complicated reactions terms. Another model problem is a system in d = 1 space dimension and n unknown variables u1 , . . . , un , n > 1, for intercellular transport, namely ¶ µ ∂ui ∂ ∂ui ′ = + ui ψi σ ∂t ∂x ∂x n X aij uj in QT = [0, 1] × (0, T ) + j=1 σ ∂ui + ui ψi′ = 0 on ∂QT = {0, 1} × (0, T ), ∂x where aii ≤ 0, aij ≥ 0 for all i ∈ {1, . . . , n}, i 6= j, n X aij = 0 for all i, j ∈ {1, . . . , n}. (1.2) i=1 It models transport via motor proteins in the eukaryotic cell where chemical energy is transduced into directed motion. A derivation of the system from a mass transport viewpoint is given in [7]. For an analysis of the steady state solutions and for further references we refer to [6], [12], [13], and [20]. In this paper we study the corresponding system in higher space dimension, namely ¡ ¢ ∂ui = div σi ∇ui + ui ∇ψi ∂t µX ¶ n ¡ ¢ + αi λij rj uj (x, t), x (1.3a) in QT , j=1 where i ∈ {1, . . . , n}, and ui (x, t) : QT → R+ , with QT = Ω × (0, T ), Ω an open bounded subset of Rd with smooth boundary, and T some positive constant. We supplement this system with the Robin (noflux) boundary conditions σi ∂ui ∂ψi + ui = 0, ∂ν ∂ν i ∈ {1, . . . , n}, 2 on ∂Ω × (0, T ), (1.3b) where ν is the outward normal vector to ∂Ω, and the initial conditions u1 (x, 0) = u0,1 (x), . . . , un (x, 0) = u0,n (x), x ∈ Ω. (1.3c) We assume that the following hypotheses hold 1. The constants σi and αi ∈ R, where i ∈ {1, . . . , n}, are strictly positive; P 2. For i, j ∈ {1, . . . , n}, λii ≤ 0, λij ≥ 0 if i 6= j, nk=1 λkj = 0; 3. for all i ∈ {1, . . . , n}, the smooth functions ri are nondecreasing with respect to the first variable; ri (0, x) = 0 and we assume that the functions ψi are smooth as well; 4. ui (., 0) = u0i ∈ C(Ω), u0i > 0. In the linear case of the molecular motors, it amounts to choosing ri (s, x) = s, λij = aij and αi = 1 for all i, j ∈ {1, . . . , n}. (1.4) We denote by Problem (P) the system (1.3a) together with the boundary and initial conditions (1.3b), (1.3c), and admit without proof that Problem (P) possesses a unique smooth and bounded solution on each time interval (0, T ]. An essential idea for proving the existence of a solution would be to apply the Comparison principle Theorem 2.2 below to deduce that any solution of Problem (P) has to be nonnegative and bounded from above by a stationary solution. Finally, we note that because of the boundary conditions (1.3b) the quantity Z n X 1 ui (x, t) dx (1.5) αi i=1 Ω is conserved in time. The organization of this paper is as follows. In Section 2 we prove a comparison principle for Problem (P). The main idea, which permits to show that Problem (P) is cooperative, is a change of functions which transforms the Robin boundary conditions into the homogeneous Neumann boundary conditions. In Section 3 we establish a contraction in L1 property for the corresponding semigroup solution. Let us point out the similarity with an old result due to Crandall and Tartar [8] where they proved in a scalar case that in the presence of a conservation of the integral property such as (1.5), a comparison principle such as Theorem 2.2 is equivalent to a contraction in L1 property such as the inequality (3.4) below. As far as we know such an abstract result is not known in the case of systems. Section 4 deals with the large time behavior of the solutions. Supposing the existence of a stationary solution, we construct a continuum of 3 stationary solutions and prove that the solutions stabilize as t tends to infinity. Let us mention a result by Perthame [19] who proved the stabilization in the case of the two component one-dimensional molecular motor problem. Finally in Section 5, show the existence and uniqueness (up to a multiplicative constant) of the stationary solution of the molecular motor problem. Acknowledgment: The authors acknowledge the preliminary master thesis work of Aude Brisset about the corresponding two component system. They are grateful to the professors Piotr Biler, Stuart Hastings, Annick Lesne and Hiroshi Matano for very fruitful discussions. 2 Comparison principle First, we remark that the system of equations (1.3a) is cooperative. ∂ψi However, since nothing is known about the sign of the coefficients ∂ν in the Robin boundary conditions (1.3b), we cannot decide whether the Problem (P) is cooperative. This leads us to perform a change of variables which transforms the Robin boundary conditions into the homogeneous Neumann boundary conditions. 2.1 The change of unknown functions Performing the change of variables wi (x, t) = ui (x, t) e ψi (x)/σ i , i ∈ {1, . . . , n}, (2.1) we deduce from (1.3) that w ~ := (w1 , . . . , wn ) satisfies the parabolic problem ¡ −ψ (x) ¢ ∂wi ψ (x) = σi e i /σi div e i /σi ∇wi ∂t µX ¶ n ¡ ¢ ψi (x)/σ −ψj (x)/σ j i + αi e λij rj wj (x, t) e ,x (2.2) in QT , j=1 together with the homogeneous Neumann boundary conditions ∂wi = 0, ∂ν i ∈ {1, . . . , n}, on ∂Ω, (2.3) and the initial conditions wi (x, 0) = u0,i (x) e ψi (x)/σ i , i ∈ {1, . . . , n}, 4 x ∈ Ω. (2.4) In the following, we denote by Problem PN — the problem (2.2), (2.3), (2.4). To begin with we define the operators ¡ −ψ (x) ¢ ∂wi ψ (x) − σi e i /σi div e i /σi ∇wi ∂t µX ¶ n ¡ ¢ ψ (x) −ψ (x) − αi e i /σi λij rj wj (x, t) e j /σj , x Li (wi ) = (2.5) in QT . j=1 We say that (w1 , . . . , wn ) is a subsolution of Problem PN if Li (wi ) 6 0 in QT , ∂wi 6 0 on ∂Ω × (0, T ), ∂ν wi (x, 0) 6 wi (x, 0), x ∈ Ω (2.6) for all i ∈ {1, . . . , n}. We define similarly a supersolution (u1 , . . . , un ) of Problem PN by the inequalities Li (wi ) > 0 in QT , ∂wi > 0 on ∂Ω × (0, T ), ∂ν wi (x, 0) > wi (x, 0), x ∈ Ω. (2.7) The following comparison theorem holds ([2], [21]). Theorem 2.1. Let (w1 , . . . , wn ) and (w1 , . . . , wn ), be a sub - and a super - solution, respectively, for the operators Lj defined by (2.5) with j ∈ {1, . . . , n}, which means that (2.6) and (2.7) hold for i ∈ {1, . . . , n}. Then wi 6 wi in QT . Moreover, for all i ∈ {1, . . . , n} such that wi 6 wi and wi 6≡ wi on {t = 0} × Ω then wi < wi in QT . ¥ This comparison theorem immediately translates into a comparison theorem for solutions of the original Problem (P). For all i ∈ {1, . . . , n}, we define the operators ¡ ¢ Li (ui ) = (ui )t − div σi ∇ui + ui ∇ψi ¶ µX n (2.8) − αi λij rj (uj , x) in QT . j=1 The following result holds. Theorem 2.2. Let (u1 , . . . , un ) and (u1 , . . . , un ), be a sub - and a super - solution, respectively, for the operators Lj , defined by (2.8) with j ∈ {1, . . . , n}. Then ui 6 ui in QT . Moreover, for all i ∈ {1, . . . , n} such that ui 6 ui and ui 6≡ ui on {t = 0} × Ω then ui < ui in QT . ¥ 5 Next we state two immediate corollaries of Theorem 2.2. Corollary 2.3. (uniqueness) If (u11 , . . . , u1n ) and (u21 , . . . , u2n ) are solucondition (u0,1 , . . . , u0,n ) ∈ ¡tions ofnProblem (P) with the same initial 1 C(Ω)) , then for all i ∈ {1, . . . , n}, ui = u2i . ¥ Corollary 2.4. (positivity) If (u1 , . . . , un ) is the solution of¡Problem ¢n (P) with the nonnegative initial condition (u0,1 , . . . , u0,n ) ∈ C(Ω) , then for all i ∈ {1, . . . , n}, ui > 0. Moreover, for all i ∈ {1, . . . , n}, such that u0,i > 0 and u0,i 6≡ 0, ui > 0 in Ω. ¥ 3 Contraction property ¡ ¢n The purpose of this section is to show a contraction in L1 (Ω) property for the ¡ solutions ¢n of Problem (P) with the initial conditions belonging to L∞ (Ω) . The main steps of the proof rely upon arguments due to [3] and [18]. We first introduce some notation. We suppose that the functions (u11 , . . . , u1n ) and (u21 , . . . , u2n ) are the solutions of Problem (P) with the initial conditions (u10,1 , . . . , u10,n ) and (u20,1 , . . . , u20,n ), respectively. Define (3.1) (U1 , . . . , Un ) := (u11 − u21 , . . . , u1n − u2n ). Then ¡ ¢ (Ui )t = div σi ∇Ui + Ui ∇ψi n X ¡ ¢ λij rj (u1j (x, t), x) − rj (u2j (x, t), x) in QT , + αi j=1 (3.2) ∂Ui ∂ψi σi + Ui = 0 on ∂Ω × (0, T ), ∂ν ∂ν Ui (x, 0) = U0,i (x) for x ∈ Ω, together with U0,i = u10,i − u20,i , (3.3) for each i ∈ {1, . . . , n}. Next we prove the following contraction in L1 property. Theorem 3.1. For all t > 0, 1 1 kU1 (·, t)kL1 (Ω) + . . . + kUn (·, t)kL1 (Ω) α1 αn 1 1 6 kU0,1 (·)kL1 (Ω) + . . . + kU0,n (·)kL1 (Ω) , (3.4) α1 αn 6 where Ui and U0,i , i ∈ {1, . . . , n}, are defined by (3.1) and (3.3), respectively. Proof Dividing each partial differential equation of (3.2) by αi and summing them up, we obtain µ n ¶ X n d X 1 1 Ui = div (σi ∇Ui + Ui ∇ψi ) dt αi αi i=1 i=1 + n X n X i=1 j=1 ³ ´ λij rj (u1j (x, t), x) − rj (u2j (x, t), x) n X 1 = div (σi ∇Ui + Ui ∇ψi ) α i=1 i ¾ n n ½³ ´X X 1 2 λij + rj (uj (x, t), x) − rj (uj (x, t), x) = i=1 j=1 n X i=1 1 div (σi ∇Ui + Ui ∇ψi ) , αi where we have used Hypothesis 2. This, together with the boundary conditions (1.3b), implies the conservation in time of the quantity Z n d X 1 Ui (x, t) dx = 0. (3.5) dt αi i=1 Ω Let us look closer at the nonlinear term in (3.2). We can write, for fixed index i n X j=1 ¢ ¡ λij rj (u1j (x, t), x) − rj (u2j (x, t), x) = n X j=1 λij Uj Z1 0 n X ∂ rj (θu1j + (1 − θ)u2j , x)dθ = Aij Uj . ∂u j=1 Freezing the functions uki for i ∈ {1, . . . , n}, k ∈ {1, 2}, we deduce that the functions U1 , . . . , Un satisfy a system of the form n ´ X ³ Aij Uj in QT , (3.6) (Ui )t = div σi ∇Ui + Ui ∇ψi + j=1 with the boundary and initial conditions ∂Ui ∂ψi + Ui = 0 on ∂Ω × (0, T ), ∂ν ∂ν Ui (x, 0) = U0,i (x), x ∈ Ω. σi 7 (3.7) for i ∈ {1, . . . , n}, where Aij are functions of space and time. In order to make the notation more concise, we write ¢ ¡ ~ 0 = U0,1 , . . . , U0,n , U ¡ ¢ ~ = U1 , . . . , Un , U ¢ ¡ ~ ± = U± , . . . , U± , U 0 0,1 0,n ¡ ¢ ~ ± = U ± , . . . , Un± , U 1 where s+ = max{s, 0}, s− = max{−s, 0}. By (3.6), (3.7) and Corol~ in the form lary 2.3 we can write U ¡ ¢ ~ )(x, t) = S(t) U ~ 0 (x) = S1 (t)U ~ 0 , . . . , Sn (t)U ~ 0 (x) (U with some operator S(t). We set ¢ ¡ ¢ ¡ ψ (x) ψ (x) W1 , . . . , Wn = − U1 e 1 /σ1 , . . . , Un e n /σn , eij = Aij eψi (x)/σi e−ψj (x)/σj . Then, the system of equations (3.6) and A can be expressed in the form ¡ Wi ¢ t = σi e ψi (x)/σ i n ³ ´ X ψ (x) eij Wj div e− i /σi ∇Wi + A in QT , (3.8) j=1 with the boundary and initial conditions ∂Wi = 0 on ∂Ω × (0, T ), ∂ν ψ (x) Wi (x, 0) = −U0,i e i /σi , x ∈ Ω, (3.9) (3.10) for i ∈ {1, . . . , n}. Next we show that the solutions Wi of the problem (3.8) – (3.10) with nonpositive initial conditions are nonpositive in Ω for all t ∈ (0, T ). To that purpose we consider the auxiliary problem ¡ Wi ¢ n ´ X ³ γij Wj 6 0 in QT , − − ϑ (x)div ζ (x)∇W i i i t (3.11) j=1 ∂Wi 6 0 on ∂Ω × (0, T ), ∂ν Wi (x, 0) = W0,i (x) 6 0 x ∈ Ω, (3.12) (3.13) for i ∈ {1, . . . , n}. We assume that ϑi (x) and ζi (x) are nonnegative in Ω and that the coefficients γij satisfy the same assumptions as the coefficients λij in Problem (P). The following result holds. 8 Lemma 3.2. Let (W1 , . . . , Wn ) be a smooth and bounded solution of the problem (3.11) – (3.13) with nonpositive initial conditions W0,i on a time interval [0, T ]. Then Wi (x, t) 6 0 in Ω × (0, T ]. Moreover, for each i ∈ {1, . . . , n} such that W0,i 6 0 and W0,i 6≡ 0, Wi < 0 in Ω × (0, T ]. Proof The result of Lemma 3.2 follows from the fact that the system (3.11), (3.12), (3.13), with the inequalities {6} replaced by the equalities {=}, is a cooperative system. However, for the sake of completeness, we present a proof below. We first remark that, in view of [21, Remark (i), p. 191], one can always satisfy the condition n X γij 6 0 for all i ∈ {1, . . . , n}, (3.14) j=1 ¡ ¢n for the matrix of coefficients γij i,j=1 by performing the change of variables W i = Wi e−ct for all i ∈ {1, . . . , n} and c > 0 large enough. Thanks to the regularity of each Wi , we can apply Theorem 15, p. 191 from [21] to conclude that Wi − M 6 0 in Ω × [0, T ] for some M 0 and all i ∈ {1, . . . , n}. In fact, we can deduce that Wi − M < 0 in Ω × (0, T ). Indeed, if for some k ∈ {1, . . . , n}, Wk = M in an interior point (x̃, t̃) ∈ Ω × (0, T ), then Theorem 15, p. 191 in [21] implies that Wk ≡ M for all 0 6 t < t̃, which is impossible since Wk (x, 0) 6 0. If the maximum M of Wk is attained at a boundary point P ∈ ∂Ω × (0, T ) then either there exists an open ball K ⊂ Ω × (0, T ) such that P ∈ ∂K and Wk − M < 0 in K, and the last part of Theorem 15, p. 191 in [21] contradicts the boundary inequality (3.12), or for all open balls K ⊂ Ω × (0, T ) such that P ∈ ∂K there exists a point (x̃, t̃) ∈ K such that Wi (x̃, t̃) = M , and we proceed as in the case before. f > 0, such that Wi 6 M f < M in Ω × [0, T ] for Hence, there exists M all i ∈ {1, . . . , n}. Then we can repeat the reasoning for all M > 0 until M = 0. Indeed, if this would not be the case, we find the least f in Ω × [0, T ], which leads real number M > 0, with Wi 6 M 6 M c < M with the same again to the existence of a real number 0 6 M property. This contradicts the fact that M was defined as the least such real number. ¥ Since the functions u1i , u2i are bounded on Ω × [0, T ], it follows that the functions Wi are bounded on Ω × [0, T ] for all i ∈ {1, . . . , n}. Then we are in a position to apply Lemma 3.2 with ϑi (x) = eψi/σi , eij for i, j ∈ {1, . . . , n}. We deduce that ζi (x) = σi e−ψi/σi and γij = A the solutions Wi of the problem (3.8) – (3.10) with nonpositive initial conditions are nonpositive in Ω for all t ∈ (0, T ). 9 Next we remark that the above reasoning can be applied either ~ 0 replaced by U + or with U ~ 0 replaced by U − . This permits to with U 0 0 ~ + , Si (t)U ~ − > 0 and that show that Si (t)U 0 0 ~ ± > 0 if U ~ ± 6≡ 0. Si (t)U 0 0 (3.15) We easily compute n n X X ° ° 1° 1° °Ui (·, t)° 1 − °U0,i (·)° 1 L (Ω) L (Ω) αi αi i=1 i=1 n n X X ° ° 1° 1° ~ + − Si (t)U ~ −° 1 − °Si (t)U °U0,i (·)° 1 = 0 0 L (Ω) L (Ω) α α i=1 i i=1 i n Z X ª © 1n ~ + , Si (t)U ~− = max Si (t)U 0 0 αi i=1 Ω (3.16) Z n X ª ªo © + © 1 1 − + − ~ ~ dx dx − Ui,0 + Ui,0 min Si (t)U0 , Si (t)U0 − αi αi i=1 = n Z X ¢ 1¡ ~ + + Si (t)U ~ − dx − Si (t)U 0 0 αi i=1 Ω n Z X −2 i=1 Ω n Z X = −2 i=1 Ω n X i=1 Ω 1 αi Z Ω ª © + − dx Ui,0 + Ui,0 ª © 1 ~ + , Si (t)U ~ − dx min Si (t)U 0 0 αi © ª 1 ~ + , Si (t)U ~ − dx 6 0, min Si (t)U 0 0 αi (3.17) which completes the proof of (3.4). ¥ ¡ ¢n Corollary 3.3. Let (u10,1 , . . . , u10,n ), (u20,1 , . . . , u20,n ) ∈ C(Ω) be as in Theorem 3.1. Moreover, let us assume that for at least one index k ∈ {1, . . . , n} the difference u10,k − u20,k changes the sign. Then, the inequality (3.4) is strict for all t > 0, so that solution satisfies a strict contraction property. 10 4 Large time behavior of solutions In this section we assume the ¡ existence and ¢n uniqueness of a positive solution ~v = (v1 , . . . , vn ) ∈ C(Ω) ∩ C 2 (Ω) of the elliptic problem µX ¶ n ¡ ¢ ¡ ¢ div σi ∇vi + vi ∇ψi + αi λij rj vj (x), x = 0 in Ω, (4.1) j=1 ∂ψi ∂vi + vi = 0 on ∂Ω, (4.2) ∂ν ∂ν Z n X 1 vi (x) dx = 1, (4.3) αi σi i=1 Ω for i ∈ {1, . . . , n}. Definition 4.1. We say that a vector function ~v = (v1 , . . . , vn ) ∈ ¡ ¢n C(Ω) is nonnegative (resp. positive) if vi (x) > 0 (resp. vi (x) > 0) for all x ∈ Ω and all i ∈ {1, . . . , n}. Next we introduce the semigroup notation for the unique solution of Problem (P), namely ´ ³ ~u(t) = T (t) ~u0 = T1 (t) ~u0 , . . . , Tn (t) ~u0 , ¢n ¡ with the initial data ~u0 ∈ C(Ω) . The method of the proof is based upon an idea of Osher and Ralston [18]. It mainly exploits the contraction properties for the nonlinear semigroup T (t) given by Theorem 3.1 and Corollary 3.3. A similar reasoning was developed in other contexts by Bertsch and Hilhorst [3], Hilhorst and Hulshof [14] and Hilhorst and Peletier [15]. ¡ ¢n We suppose there exists a set H ⊂ C(Ω) ∩ C 2 (Ω) of positive stationary solutions with the following property which we denote by S: ¢n ¡ For each f~ = (f1 , . . . , fn ) ∈ C(Ω) ∩ C 2 (Ω) either f~ ∈ H or there exists (ξ1 , . . . , ξn ) ∈ H , such that fi − ξi changes the sign for at least one index i ∈ {1, . . . , n}. One can prove that a set H satisfying Property S exists in at least two cases: i) In the case of the system (1.1) where the Robin boundary conditions reduce to the homogeneous Neumann boundary conditions, 11 the set H is given by n −1 H = (a, b) : a > 0, b = rB (rA (a)) Z ³ a u b v´ o and + = + dx . α β α β Ω For more details we refer to [5]. ii) In the case of the molecular motor with a linear n-component system the set H is given by © ª H = c~v : c ∈ R+ , where ~v is a unique solution of the elliptic problem (4.1) – (4.3). Proposition 4.2. The continuum H is such that for each ¡ ¢n f~ = (f1 , . . . , fn ) ∈ C(Ω) ∩ C 2 (Ω) either f~ ∈ H , or there exists (ξ1 , . . . , ξn ) ∈ H such that fi − ξi changes the sign for at least one index i ∈ {1, . . . , n}. Proof i) In the case of system (1.1) the proof is rather obvious since the continuum H is composed of constant pairs. ii) In the case of the molecular motor, let us assume that f~ 6∈ H . Then there does not exist any positive constant c such that c ~v = f~. In particular, there exists an index i ∈ {1, . . . , n} such that vi is not proportional to fi , or in other words cvi 6= fi for all c > 0. Without loss of generality we can assume that the first coordinate has this property. Let x0 ∈ Ω be arbitrary. Since v1 is strictly positive in Ω, we can define c0 = so that ¡ f1 (x0 ) , v1 (x0 ) ¢ f1 − c0 v1 (x0 ) = 0. ¡ ¢ ª © Let Z = x ∈ Ω : f1 − c0 v1 (x) = 0 . From the continuity of f1 and v1 , Z is ¡closed as ¢a subset of¡ Ω. If there ¢ exist x1 , x2 ∈ Z c , such that f1 − c0 v1 (x1 ) and f1 − c0 v1 (x2 ) are of Now suppose that ¡ different¢ signs, then the proof is complete. f1 − c0 v1 (x) is positive for all x ∈ Z c . In particular ¡ ¢ f1 − c0 v1 (x̃) = d > 0 12 for some fixed x̃ ∈ Z c . Then choosing ε = ¡ However d we see that 2v1 (x̃) ¢ d f1 − (c0 + ε)v1 (x̃) = > 0. 2 ¢ f1 − (c0 + ε)v1 (x0 ) < 0. ¢ ¡ We proceed similarly when f1 − c0 v1 (x) is negative for all x ∈ Z c. ¥ ¡ In¡the sequel ¢n we suppose that the initial data ~u0 = (u0,1 , . . . , u0,n ) from C(Ω) also satisfy the following property: There exists ~h ∈ H such that 0 6 ~u0 6 ~h in Ω, (4.4) and remark that this property is satisfied in both the cases (i) and (ii). ¡ ¢n Proposition 4.3. Let ~u0 = (u0,1 , . . . , u0,n ) ∈ C(Ω) satisfy the property (4.4). Then the solution (u1 , . . . , un ) of Problem (P) is such that 0 6 ~u(t) 6 ~h for all t > 0. Proof We remark that ~0 is a subsolution of Problem (P) and that ~h is a supersolution, and apply Theorem 2.2. ¥ Next we prove the°main ° result of this section. To that purpose we first define the norm ° · °1 by n X ° ° ° 1° ° f~ ° = °fi ° 1 . 1 L (Ω) αi i=1 Note that this ¡ ¢n norm is equivalent to the usual product norm in the space L1 (Ω) . ¡ ¢n Theorem 4.4. For all nonnegative ~u0 = (u0,1 , . . . , u0,n ) ∈ C(Ω) there exists f~ = (f1 , . . . , fn ) ∈ H , such that ° ° lim ° T (t) ~u − f~ °1 = 0. t→∞ Proof The proof consists of several steps. To begin with we define the ω-limit set n ¡ ¢n ω(~u0 ) = ~g ∈ L1 (Ω) : there exists a sequence tk → ∞ o ° ° as k → ∞, such that lim ° T (tk ) ~u0 − ~g °1 = 0 , (4.5) k→∞ 13 The organization of the proof is as follows. First we show that ω(~u0 ) is not empty. In the second step we define the Lyapunov functional ° ° ~ =° °~ V(ξ) ξ−w ~ °1 , where w ~ is a stationary solution and check that it is constant on ω(~u0 ). We then deduce that ω(~u0 ) ⊂ H , and finally prove that ω(~u0 ) consists of exactly one function. Step 1. ω(~u0 ) is not empty. Let ε > 0 be arbitrary. Suppose that Ω′ ⊂⊂ Ω satisfy ¯ ¯ ¯Ω \ Ω′ ¯ 6 ε . 2K and set K= n X 2 khi kC(Ω) , αi (4.6) i=1 where ~h has been introduced in (4.4). We ¡ ∞have¢nalready proved in Proposition 4.3 that T (t) ~u¡0 is bounded in L (Ω) . Therefore there exist a vector function ~g ∈ L∞ (Ω))n and a sequence {~u(tk )} such that ~u(tk ) ⇀ ~g weakly in (L2 (Ω))n , (4.7) as tk → ∞. Next we deduce from [16, Chap. III, Theorem 10.1] that there exists a positive constant C such that ¯ ¯ ¯ui (x1 , t) − ui (x2 , t)¯ 6 C|x1 − x2 |α for all x1 , x2 ∈ Ω′ and all t > 0. Therefore, it follows from the Ascoli-Arzelà Theorem (see, e.g., [1, Theorem 1.33]) that ~u(tk ) → ~g ′ as tk → ∞, uniformly in Ω . We choose t0 large enough such that for all tk > t0 ° ° ° ~u(·, tk ) − ~g (·)°1 ,Ω′ 6 ε , (4.8) 2 ° ° where ° · °1 ,Ω′ corresponds to the L1 norm in Ω′ . We deduce that, in view of (4.6) and (4.7) that ¯ ° ¯ ° ° ~u(·, tk ) − ~g (·)°1,Ω\Ω′ 6 K ¯Ω \ Ω′ ¯ 6 ε , 2 which together with (4.8) yields ° ° ° ~u(·, tk ) − ~g (·)° 6 ε. 1 Step 2. ω(~u0 ) ⊂ H . Indeed, let ~g ∈ ω(~u0 ) and suppose ~g ∈ / H . According to Proposition 14 4.2 we can find a steady state solution w ~ ∈ H , such that at least one component of w ~ − ~g changes the sign. Without loss of generality we can assume that it happens for the first component, namely that f1 −w1 changes the sign. We remark that, by the contraction property in Theorem 3.1, the functional ° ° ~ =° °~ V(ξ) ξ−w ~° 1 ¡ ¢n is a Lyapunov functional for Problem (P), where ξ~ ∈ L1 (Ω) . Next we describe some of its properties. Property (a) The functional V is constant on ω(~u0 ). Since T (t) w ~ = w ~ and T (t) has the contraction property (3.4), the functional V is nonincreasing in time along the trajectory T (t) ~u0 , which yields ° ¡ ¢ ° V T (t) ~u0 = ° T (t) ~u0 − w ~ °1 ° ° ° ° ~ ° 6 ° ~u0 − w = ° T (t) ~u0 − T (t) w ~° < ∞ . 1 1 ¡ ¢ Thus there exists a finite limit V ∗ of V T (t) ~u0 as t → ∞. Let ~h1 , ~h2 ∈ ω(~u0 ). We can find a sequence tk → ∞ as k → ∞, such that ° ° ° ° ° T (t2k ) ~u0 − ~h1° → 0 and ° T (t2k+1 ) ~u0 − ~h2° → 0, 1 1 ¡ ¢ ¡ ¢ as k tends to ∞. It follows that V ~h1 = V ~h2 = V ∗ . Property (b) The ω-limit set ω(~u0 ) is invariant with respect to the semigroup T (t) , namely if ~h ∈ ω(~u0 ), then for all t > 0 also T (t) ~h ∈ ω(~u0 ). ° ° Let the sequence tk → ∞ as k → ∞ be such that ° T (tk ) ~u0 −~h°1 → 0. From the contraction property (3.4) ° ° ° ° ° T (tk + t) ~u0 − T (t) ~h° = ° T (t) T (tk ) ~u0 − T (t) ~h° 1 ° ° 1 6 ° T (tk ) ~u0 − ~h° . 1 Since the last term above tends to 0 as k tends to ∞ this shows that T (t) ~h ∈ ω(~u0 ). Now, remember that ~g ∈ ω(~u0 ) is such that ~g ∈ / H and w ~ ∈ H is such that the first component of w ~ − ~g changes the sign in Ω. Then, Corollary 3.3 yields ° ° V(T (t) ~g ) = ° T (t) ~g − w ~ °1 ° ° ~ k1 = V(~g ), = ° T (t) ~g − T (t) w ~ °1 < k~g − wk 15 for all t > 0, which contradicts Property (a). Therefore ~g ∈ H . Step 3. The set ω(~u0 ) contains only one element. Suppose that ~g1 , ~g2 ∈ ω(~u0 ). Then we can find ° two sequences ° °tk , sk tending to ∞ as k → ∞, such that sk 6 tk and ° T (tk ) ~u0 −~g1°1 , ° T (sk ) ~u0 − ° ~g2°1 → 0 as tk → ∞. Since ω(~u0 ) ⊂ H , it follows that ° ° ° ° k~g1 − ~g2k1 6 ° T (tk ) ~u0 − ~g1°1 + ° T (tk ) ~u0 − ~g2°1 ° ° ° ° = ° T (tk ) ~u0 − ~g1°1 + ° T (tk − sk ) T (sk ) ~u0 − T (tk − sk ) ~g2°1 ° ° ° ° 6 ° T (tk ) ~u0 − ~g1°1 + ° T (sk ) ~u0 − ~g2°1 , which tends to 0 as k → ∞. ¥ 5 Stationary solutions for the linear molecular motor problem In this section we show the existence and the uniqueness (up to a multiplicative constant) of the classical stationary solution of the problem for the molecular motor. We suppose that Ω is an open bounded subset of Rd with smooth boundary ∂Ω. We consider the linear system n ¡ ¢ X div σi ∇vi (x) + vi (x)∇ψi (x) + aij vj (x) = 0 in Ω, (5.1) j=1 where i ∈ {1, . . . , n}, n > 1. The system (5.1) is supplemented with the Robin boundary conditions σi ∂vi ∂ψi + vi = 0 on ∂Ω, ∂ν ∂ν (5.2) where i ∈ {1, . . . , n}. Thus, the problem can be written as A~v = 0, with a linear operator A in a suitable Banach space X of functions on Ω, to be made precise later. Moreover, we impose the integral constraint n Z X vi (x) dx = 1. (5.3) The adjoint problem i=1 Ω A∗ ϕ ~= 0 to (5.1), in a dual space X ∗ , is now σi ∆ϕi − ∇ψi · ∇ϕi + n X j=1 16 aji ϕj = 0, in Ω, (5.4) with the Neumann boundary conditions for each i = 1, . . . , n ∂ϕi = 0 on ∂Ω. (5.5) ∂ν P Since nj=1 aji = 0, the problem (5.4) has the obvious solution ϕ ~ = (ϕ1 , . . . , ϕn ) = (1, . . . , 1). (5.6) We are going to apply the Krein-Rutman theorem on the first eigenvalues and eigenvectors of positive operators, and this will permit us to conclude that the problem (5.1)–(5.2) has a one-dimensional space of solutions. Therefore, under the additional constraint (5.3), the original problem (5.1)–(5.2) has a unique solution. Perthame and Souganidis sketched this argument for n > 1 and d = 1 in [20]. Pn Theorem 5.1. Under the assumption j=1 aji = 0, there exists a unique smooth solution ~v of the system (5.1)–(5.3). Before proving Theorem 5.1 we recall some basic definitions as well as the Krein-Rutman theorem from [9, Ch. VIII, p. 188–191]. Definition 5.2 (Reproducing cone). We say that a closed set K in X is a cone, if it possesses the following properties: i) 0 ∈ K, ii) u, v ∈ K =⇒ αu + βv ∈ K, for all α, β > 0, iii) v ∈ K and −v ∈ K =⇒ v = 0. © A cone K ⊂ ª X is said to be reproducing if X = K − K ≡ k1 − k2 : k1 , k2 ∈ K . Definition 5.3 (Dual cone). If K is a cone in X , then the set K ∗ ⊂ X ∗ is said to be a dual cone if hf ∗ , vi > 0, for every v ∈ K. Definition 5.4 (Strict positivity). Let B be a linear operator on X . Then B is said to be strongly positive if Bv ∈ K o for all v ∈ K such that v 6= 0. Theorem 5.5. Let K be a reproducing cone in a Banach space X , with nonempty interior K o 6= ∅, and let B be a strongly positive compact operator on K in a sense of Definition 5.4. Then the spectral radius of B, r(B), is a simple eigenvalue of B and B∗ , and their associated eigenvectors belong to K o and (K ∗ )o . More precisely, there exists a unique associated eigenvector in K o (resp. (K ∗ )o ) of norm 1. Furthermore, all other eigenvalues are strictly less in absolute value than r(B). 17 ¡ ¢n We will apply Theorem 5.5 to the space X = C(Ω) ⊂ ¡Proof ¢ n L1 (Ω) endowed with the usual supremum norm, and the operators B = (λI − A)−1 : X → X , B ∗ = (λI − A∗ )−1 : X ∗ → X ∗ , where λ > 0 is a strictly positive real number to be fixed later. Let ª © K = ~u ∈ X : ui (x) > 0 for each x ∈ Ω, i = 1, . . . , n . We remark that K is a reproducing cone, with nonempty interior © ª K o = ~u ∈ X : inf ui (x) > 0, i = 1, . . . , n . x∈Ω From the standard theory [17, Theorem 2.1 and Theorem 3.1, Ch. 7] for elliptic partial differential linear systems, the boundary value problem σi ∆ϕi − ∇ψi · ∇ϕi + n X aji ϕj − λϕi = fi in Ω, (5.7) i=1 e> with the homogeneous Neumann conditions (5.5) on ∂Ω, for λ = λ 0 sufficiently large, has a solution ϕ ~ = (ϕ1 , . . . , ϕn ) ∈ X for each f~ = (f1 , . . . , fn ) ∈ X . Moreover, if fi (x) > 0 for each i = 1, . . . , n, and x ∈ Ω, then ϕi (x) > 0 (in fact, ϕi (x) > 0 in Ω), which is a consequence of the maximum principle (cf. also Example 3 on p. 196– ¡ ¢ e −A∗ −1 is a strongly positive 197 in [9]). Thus, the operator B∗ = λI and compact operator, and by Theorem 5.5, the largest eigenvalue µ of B and B∗ is simple. Since −σi ∆ϕi + ∇ψi · ∇ϕi − n X j=1 e i = λϕ e i aji ϕj + λϕ in Ω ∂ϕi = 0 on ∂Ω, ∂ν for all i ∈ {1, . . . , n}, with ϕ ~ = (ϕ1 , . . . , ϕn ) = (1, . . . , 1), and since ³¡ ¢ ´ 1 e − A∗ −1 is a simple (1, . . . , 1) ∈ (K ∗ )o , it follows that = r λI e ¡ ¢λ e − A∗ −1 . Applying again Theorem 5.5, eigenvalue of the operator λI ¡ ¢ 1 e − A −1 we deduce that is the largest eigenvalue of the operator λI e λ and that it is simple, and that there exists ~v ∈ K o ⊂ X such that ´−1 ³ 1 e −A ~v = ~v , λI e λ 18 which is equivalent to A~v = 0. This proves the existence of the solution of the problem (5.1)–(5.3). ¥ Bibliography 1. R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975. 2. N. D. Alikakos, P. Hess, H. Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Differential Equations, 1989, 2, 82, 322–341. 3. M. Bertsch and D. 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