Long-time behavior of an angiogenesis model with flux at the tumor boundary

arXiv:1202.4695v1 [math.AP] 21 Feb 2012 Long-time behaviour of an angiogenesis model with flux at the tumor boundary T. Cie´ slak and C. Morales-Rodrigo Institute of Mathematics, Polish Academy of Sciences, Warsaw, ´ Sniadeckich 8 E-mail address: [email protected] Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´aticas, Univ. de Sevilla, Calle Tarfia s/n, 41012-Sevilla, Spain E-mail address: [email protected] February 22, 2012 Abstract This paper deals with a nonlinear system of partial differential equations modeling a simplified tumor-induced angiogenesis taking into account only the interplay between tumor angiogenic factors and endothelial cells. Considered model assumes a nonlinear flux at the tumor boundary and a nonlinear chemotactic response. It is proved that the choice of some key parameters influences the long-time behaviour of the system. More precisely, we show the convergence of solutions to different semi-trivial stationary states for different range of parameters. AMS Classification. 35K45, 35K57, 92C17. Keywords. Chemotaxis, Asymptotic behaviour, Semigroup theory, Nonlinear elliptic eigenvalue problem, Angiogenesis, Nonlinear boundary condition. 1 Introduction Angiogenesis is a physiological process involving the new vessels sprout from a preexisting vasculature in response to a chemical stimuli. Angiogenesis is an important ingredient of a processes like development, growth and wound healing. However, angiogenesis is also induced by tumoral cells. In this paper we consider a model of tumor-induced angiogenesis that was proposed in [5]. Actually, in the above mentioned model some factors influencing angiogenesis are neglected to keep the model simple but sufficiently interesting from the analytical point of view. We refer the reader to [12] as a source of information about the progress in mathematical modelling and biological knowledge of angiogenesis process. We focus our attention on two key variables: the endothelial cells (ECs), denoted by u, and the tumor angiogenic factors (TAF), denoted by v. We assume that (ECs) that form the blood vessels wall are induced by the (TAF), factors that are generated by the tumor, to migrate chemotactically towards the tumor. We assume that the (ECs) and the (TAF) fill in a bounded and connected domain Ø ⊂IRdwith a regular boundary ∂Ø. In particular, neither the existence of extracellular matrix nor the activity of metalloproteinases is considered. But, what was new there, nonlinear flux of TAF on the tumor boundary was taken into account. The reason was that since ECs are supposed to react chemotactically to the TAFs, generating the large gradient of TAFs on the boundary February 22, 2012 T. Cie´slak and C. Morales-Rodrigo would probably make the tumour more dangerous. The aim of [5] was to study the interplay between the density of ECs and TAFs dependently on a parameter µmeasuring the strength of the flux on the tumor boundary and the nonlinearity Vmeasuring nonlinear response of ECs. In [5] the qualitative features of a model were studied in a local sense. We mean by that the local stability of steady states which were proven to exist in [5]. We complete the studies taken in [5] by analyzing the global stability of steady states. We shall prove the asymptotic convergence of solutions for different values of µ. To be more precise, we consider the case ∂Ø=Γ1∪Γ2, where Γ1∩Γ2=∅and Γiare closed and open sets in the relative topology of ∂Ø. We suppose that Γ2is the tumor boundary and Γ1is the blood vessel boundary. Our parabolic problem reads.                      ut−∆u=−div(V(u)∇v) + lu −u2in Ø ×(0, T), vt−∆v=−v−cuv in Ø ×(0, T), ∂u ∂n =∂v ∂n = 0 on Γ1×(0, T), ∂u ∂n = 0,∂v ∂n =µv 1 + von Γ2×(0, T), u(x, 0) = u0(x), v(x, 0) = v0(x) in Ø, (1) where 0 < T ≤+∞,l, µ ∈IR, c > 0, V∈ C1(IR), V > 0 in (0,∞) with V(0) = 0; (2) and u0and v0are given non-negative and non-trivial functions. In [5, Theorem 3.1, Theorem 3.8] the existence and uniqueness of global-in-time bounded regular solutions, provided initial data are nonnegative and V∈L∞(0,+∞) is shown. Moreover in [5, Section 4] the existence of two semi-trivial steady-states (λ, 0), λ > 0 and (0, θµ) is shown provided µ > µ1(see also [14]), where µ1is the principal eigenvalue of the boundary eigenvalue problem            −∆v+v= 0 in Ω, ∂v ∂n = 0 on Γ1, ∂v ∂n =µv on Γ2. Furthermore, results concerning the linearized stability around the semi-trivial solutions to (1) are proven in [5]. First models of tumor induced angiogenesis that we are aware of are considered in [3] (see also [10] for a more elaborated model). A reduced model proposed in [10] is studied in [7]. The local stability of the homogeneous steady-states in one dimensional domains is shown there. In all the mentioned papers the boundary conditions are either zero Neumann or no-flux. In [6] the stationary problem of (1) with linear flux for vis studied. Finally let us mention [9] where the authors study the local solvability of a system of partial differential equations with a nonlinear boundary condition and a chemotaxis term. 2 Angiogenesis model with flux at the boundary February 22, 2012 The aim of this paper is to analyze the global stability for positive initial data. In particular we show global stability for some range of parameters (λ, µ) for which even the local stability is not known. It should be pointed out that the results of this paper could be extended even to more general forms of Vas soon as ku(t)k∞< C, for t≥0.(3) Observe that, if the above inequality holds, then the parabolic regularity asserts kv(t)k∞< C for any t > 0 and by [5, Theorem 3.1] the solution is global and regular. In particular, when Vis bounded in the L∞norm (see [5]) then (3) is satisfied. 2 Preliminaries For the reader’s convenience we collect here some results of interpolation theory and its applications to parabolic problems that will be used throughout the paper. a) Let E0,E1two normed spaces, we can define the real interpolation functor, denoted by (E0, E1)θ,p,0< θ < 1,1≤p≤+∞, (see for instance [13, Def. 22.1]). During the paper we will use the following property of the real interpolation functor (see [13, Lemma 25.2]): If (E0, E1)θ,p is a Banach space then ∃C > 0 such that kxk(E0,E1)θ,p ≤Ckxk1−θ E0kxkθ E1∀x∈E0∩E1. In the context of fractional Sobolev spaces this inequality reads, see [1, Theorem 7.2] ku(t)kWm,p ≤Cku(t)kθ Wk,p ku(t)k1−θ p(4) for m < kθ,θ∈(0,1). b) Let us consider a parabolic problem with a non-homogeneous boundary condition      zt+Az=f(t) in Ω ×(0, T), Bz=g(t) on ∂Ω×(0, T), z(x, 0) = z0(x),in Ω. (5) where Bz:= ∂z ∂n and Az:= −∆z+z. We define the space of functions Ws,p B:=      {z∈Ws,p(Ω) : Bz= 0}if 1 + 1/p < s ≤2, Ws,p(Ω) if −1 + 1/p < s < 1 + 1/p, (W−s,p′(Ω))′if −2 + 1/p < s ≤ −1 + 1/p. 3 February 22, 2012 T. Cie´slak and C. Morales-Rodrigo It is known that (A,B), as being in separated divergence form (see [1, pg. 21]), is normally elliptic. We denote by Aα−1the W2α−2,p B-realization of (A,B) (see [1, pg. 39] for the precise definition). Since (A,B) is normally elliptic then Aα−1generates an analytic semigroup [1, Theorem 8.5]. Moreover, if (f, g)∈ C((0, T); W2α−2,p B(Ω) ×W2α−1−1/p,p B(∂Ω)) for some T > 0 and 2α∈(1/p, 1 + 1/p) then for any t < T we rewrite (5) by the generalized variation of constants formula z(t) = e−tAα−1z0+Zt 0 e−(t−τ)Aα−1(f(τ) + Aα−1Bc αg(τ))dτ, where Bc αis the continuous extension of (B|Ker(A))−1to W2α−1−1/p,p(∂Ω). Since [0,+∞)⊂ρ(−Aα−1) (ρis the resolvent set) then by [1, Remark 8.6 c)] there exists a constant C≥1 such that kzkW2α,p B≤CkAα−1zkW2α−2,p B.(6) c) Let a, b, c ∈L∞(Ω), the eigenvalue problem            −∆z+a(x)z=λz in Ω, ∂z ∂n +b(x)z= 0 on Γ1, ∂z ∂n +c(x)z= 0 on Γ2. has a unique principal eigenvalue (i.e. an eigenvalue whose associated eigenfunction can be chosen positive in Ω) and it will be denoted by λ1(−∆ + a;N+b;N+c). 3 Convergence to the semi-trivial solution (l, 0) In the present section we deal with the convergence to the semi-trivial steady-state (l, 0). Throughout this section we assume (3). A sufficient condition guaranteeing (3) is the boundedness of V(see [5]). We will use the generalized variation of constants formula to estimate v, which is stated in the next lemma. Lemma 3.1. Let γ∈(1,+∞)and β∈(1,2α). Then, for every τ∈(0, t)there exists a constant δ∈(0, Re σ(Aα−1)) (σdenotes the spectrum) and θ=θ(β)∈(0,1) such that ke−(t−τ)Aα−1zkWβ,γ ≤C(t−τ)−θe−δ(t−τ)kzkW2α−2,γ B for every z∈W2α,γ B. Proof. By the choice of βwe have Wβ,γ B=Wβ,γ(Ω). As a consequence if we apply [1, Theorem 7.2] we get ke−(t−τ)Aα−1zkWβ,γ ≤Cke−(t−τ)Aα−1zkW2α,γ Bkθke−(t−τ)Aα−1zk1−θ W2α−2,γ B 4 Angiogenesis model with flux at the boundary February 22, 2012 for some θ∈(0,1). Next we apply (6) to the first norm on the right hand side and [8, Theorem 1.3.4] to deduce ke−(t−τ)Aα−1zkWβ,γ ≤C(t−τ)−θe−δ(t−τ)θe−δ(t−τ)(1−θ)kzkW2α−2,γ B, where δ∈(0, Re σ(Aα−1)). Lemma 3.2. Let γ∈(1,+∞),β∈(1,1 + 1/γ),µ∈[0, µ1)and 0< δ < ρ < α(µ)where α(µ)is defined as α(µ) := λ1(−∆ + 1; N,N − µ). Then, there exists C > 0such that, for t > 0, the v-solution to (1) satisfies v(x, t)≤Ce−ρt ∀(x, t)∈Ω×(0,+∞), kv(t)kWβ,γ ≤C(1 + t−θ)e−δtkv0kγ, where θ=θ(β)∈(0,1). Proof. A solution to the problem                wt−∆w+w= 0 in Ω ×(0, Tmax), ∂w ∂n = 0 on Γ1×(0, Tmax), ∂w ∂n =µw on Γ2×(0, Tmax), w(x, 0) = v0(x) in Ω, (7) is a supersolution to the v-equation of (1), therefore v(x, t)≤w(x, t). Since, for sufficiently large M,w=Me−ρtϕ1, with ϕ1a positive eigenfunction associated to α(µ), is a supersolution to (7), the pointwise estimate in the claim of the lemma follows. For the second one we pick f(t) := −cu(t)v(t), g(t) :=    0 on Γ1, µv(t) 1 + v(t)on Γ2. Taking the Wβ,γ-norm in a generalized variation of constants formula for vand using Lemma 3.1 we obtain kv(t)kWβ,γ ≤ ke−tAα−1v0kWβ,γ +Zt 0 ke−(t−τ)Aα−1(f(τ) + Aα−1Bc αg(τ))kWβ,γ ≤Ce−δtt−θkv0kW2α−2,γ B+Zt 0 (t−τ)−θe−δ(t−τ)kf(τ) + Aα−1Bc αg(τ)kW2α−2,γ Bdτ. Next, we estimate the last term in the above inequality using the fact that Aα−1Bc α∈ L(W2α−1−1/γ,γ(∂Ω), W 2α−2,γ B(Ω)) and the continuous embeddings Lγ(Ω) ֒→W2α−2,γ B, Lγ(∂Ω) ֒→W2α−1−1/γ,γ(∂Ω). 5 February 22, 2012 T. Cie´slak and C. Morales-Rodrigo Therefore, we get kv(t)kWβ,γ ≤e−δtt−θkv0kγ+Ce−δt Zt 0 eδτ (t−τ)−θkf(τ)kLγ(Ω) +kg(τ)kLγ(∂Ω)dτ. (8) Observe that by (3) and the first part of the Lemma we have kf(τ)kLγ(Ω) ≤CkvkL∞(Ω) ≤Ce−ρτ , kg(τ)kLγ(∂Ω) ≤ kvkL∞(∂Ω) ≤Ce−ρτ . In view of the above bounds, (8) yields kv(t)kWβ,γ ≤Ce−δtt−θkv0kγ+Ce−δtkv0kγZt 0 e(δ−ρ)τ(t−τ)−θdτ. Next, by the choice of δand ρ,Z∞ 0 e(δ−ρ)τ(t−τ)−θdτ =C < +∞and the Lemma follows. Our purpose is to show that uconverges to steady states. To this end we treat separately the cases λ= 0, λ > 0. 3.1 Case λ= 0. Lemma 3.3. Let τ > 0and y∈C1(τ, +∞)∩L1(τ, +∞),y′∈L1(τ, +∞). Then lim t→+∞|y(t)|= 0. Proof. By the assumptions of the lemma we observe that for any k > 0 lim t→+∞Zt+k t|y(s)|+|y′(s)|ds = 0.(9) Let us assume that lim t→+∞|y(t)| 6= 0, then there exists a sequence {tn}n∈IN,tn→+∞, such that |y(tn)|> C > 0, for all n≥n0. We pick θ∈(0, k], then for any ε > 0 |y(tn+θ)| − |y(tn)|≤ |y(tn+θ)−y(tn)| ≤ Ztn+θ tn |y′(s)|ds ≤Ztn+k tn |y′(s)|ds < ε by (9). Therefore |y(s)|> C/2 for all s∈[tn, tn+k], n≥n0. The last assertion contradicts the fact that lim n→+∞Ztn+k tn |y(s)|ds = 0. In the following lemmata (u, v) is a solution to (1). Lemma 3.4. Let λ= 0 and t > τ > 0, then it holds µZt τZΓ2 V(u)v 1 + v+Zt τZΩ u2=ZΩ u(τ)−ZΩ u(t).(10) 6 Angiogenesis model with flux at the boundary February 22, 2012 Proof. Integrating the u-equation of (1) yields ZΩ ut=Z∂Ω∂u ∂n −V(u)∂v ∂n−ZΩ u2 =−µZΓ2 V(u)v 1 + v−ZΩ u2. So, integrating the last expression in time between τand twe get the result. Remark 3.5. By Lemma 3.4 we see that for any t > τ Zt τZΩ u2≤ ku(τ)k1. Theorem 3.6. Assume that 0≤µ < µ1and λ= 0, then lim t→+∞ku(t)kWm,p = 0, for any m < 1and p≥2. Proof. On multiplying the u-equation of (1) by uand integrating in space we obtain d 2dt ZΩ u2=ZΩ−|∇u|2+V(u)∇v· ∇u−u3−µZΓ2 V(u)uv 1 + v ≤(ǫ−1) ZΩ |∇u|2+C(ǫ)ZΩ |∇v|2−µZΓ2 V(u)uv 1 + v−ZΩ u3. (11) Therefore, we infer d 2dt ZΩ u2+ (1 −ǫ)ZΩ |∇u|2≤C(ǫ)kvk2 W1,2, and after integrating in time, thanks to Lemma 3.2 we arrive at ZΩ u(t)2−ZΩ u(τ)2+ (1 −ǫ)Zt τZΩ |∇u|2≤C(ǫ)Zt τ (1 + s−θ)2e−2δskv0k2 p. In particular we deduce that for t > τ Zt τZΩ |∇u|2≤C. By [5, Lemma 3.8] we find a bound ku(t)kC(Ω) ≤C, therefore,  d 2dt ZΩ u2≤CZΩ |∇u|2+C(ǫ)kvk2 W1,2+Cµ ZΓ2 V(u)v 1 + v+CZΩ u2. Thanks to (10), for t > τ Zt τ d 2dt ZΩ u2≤C. (12) Finally, Remark 3.5 and (12) together with Lemma 3.3 entail lim t→+∞ku(t)k2= 0. 7 February 22, 2012 T. Cie´slak and C. Morales-Rodrigo Also thanks to ku(t)kC(Ω) ≤Cfor all t > 0 we obtain lim t→+∞ku(t)kp= 0 for any p > 2. Next we recall that by [5, Lemma 3.7] for any 2β∈(k, 1) we find a bound on the Xβnorm of u, where Xβis a usual fractional space connected to a semigroup approach to parabolic equations, see [8]. Next, due to the fact that 2β∈(k, 1), we infer from the embedding Xβ֒→Wk,p (see for instance [8, Theorem 1.6.1]) that for all k < 1 and p≥2 ku(t)kWk,p ≤C. Next, (4) entails ku(t)kWm,p ≤Cku(t)kθ Wk,p ku(t)k1−θ p. Therefore, it holds lim t→+∞ku(t)kWm,p ≤Clim t→+∞ku(t)k1−θ p= 0.(13) Remark 3.7. Let us point out that if we pick msuch that m−d/p > 0then Wm,p(Ω) is embedded in C(Ω). 3.2 Case λ > 0. Assume that there exists δ0and t0such that u(t)> δ0>0 (14) for t > t0>0. Next, we examine the long time behavior for uunder the hypothesis (14). In the sequel we shall give sufficient conditions on V(u) implying (14). Theorem 3.8. Let 0≤µ < µ1and assume the the hypothesis (14)) is satisfied, then there exists θ > 0such that ku(t)−λkWm,p ≤Ce−θt,(15) for all t≥t0and any m < 1,p≥2. Proof. On multiplying the u-equation by u−λwe have d 2dt ZΩ (u−λ)2=−ZΩ |∇u|2+ZΩ V(u)∇v· ∇u−µZΓ2 vV (u) 1 + v(u−λ)−ZΩ u(u−λ)2 ≤ −1 2ZΩ |∇u|2+kVk2 ∞ 2ZΩ |∇v|2+ +µkV(u)(u−λ)k2,Γ2ZΓ2 v2 (1 + v)21/2 −ZΩ u(u−λ)2. (16) Having in mind that (1 + v)2≥1, the hypothesis (14) and the Sobolev trace embedding W1,2(Ω) ֒→L2(∂Ω) we get d dt ZΩ (u−λ)2+ 2δ0ZΩ (u−λ)2≤Ckvk2 W1,2+µCkvkW1,2.(17) 8 Angiogenesis model with flux at the boundary February 22, 2012 By Lemma 3.2 we can deduce ku(t)−λk2 2≤Ce−θ1t for 0 < θ1<min{2δ0, β}. At this point we can argue exactly as in the end of the proof of Theorem 3.6. Namely, by the bound on uin L∞we infer the bound on the Lpnorm of u, p > 2. Next, we use the estimate of uin Wk,p,k < 1, p ≥2, coming from [5, Lemma 3.7], in order to conclude (15). In the rest of this section we give sufficient conditions on Vimplying (14). Actually, only the behavior of Varound zero matters. Roughly speaking we require a superlinear growth of Vin the neighbourhood of zero. From now on we assume that there exist C, δ > 0, k0>1 + d/2, j > d/2 such that 0< V (s)< Csk0,|V′(s)| ≤ Csj(18) for all s∈(0, δ). Remark 3.9. The condition (18) is satisfied, for example, for functions V(u) = uα 1 + uα with α > 1 + d/2. Next we introduce some notation that will be of importance in the proof of (14). Moreover we formulate a lemma which we need in the main part of the proof of (14). Let f(δ), g(δ) be defined in a following way: f(δ) := sup s∈(0,δ) V2(s), g(δ) := sup s∈(0,δ) (2(s−δ)2 −V′(s)2+ 2V2(s)). Lemma 3.10. Assume that (18) holds. Moreover, for some D, µ > 0,η > 1,˜ǫand C(˜ǫ) are given by ˜ǫ=δ2η 2µD, C(˜ǫ) = µD 2δ2η. Then, if δ > 0is small enough, the following conditons are satisfied simultaneously C(eǫ)V2(s) sδ≤λ−δ(19) for s∈(0, δ), C(eǫ)g(δ)<1/2 (20) and f(δ)D≤δ2η 2.(21) 9 February 22, 2012 T. Cie´slak and C. Morales-Rodrigo Indeed, −∆θµ+θµ= 0 in Ω, ∂θµ ∂n = 0 on Γ1, Finally by the choice of γ(see (37)) and Lemma 4.3 we have ∂θµ ∂n =µθµ 1 + θµ >µγθµ (1 + v(t))(1 + θµ)on Γ2. Therefore, by [2, Theorem 2.4] we get λ1>0, the principal eigenvalue of (38). Next, the variational characterization of the principal eigenvalue entails λ1= inf ϕ∈H1(Ω) ZΩ |∇ϕ|2+ZΩ ϕ2−µγ ZΓ2 ϕ2 (1 + v(t))(1 + θµ) ZΩ ϕ2 . Thus, for all ϕ∈H1(Ω) we have λ1γ−1ZΩ ϕ2≤γ−1ZΩ |∇ϕ|2+γ−1ZΩ ϕ2−µZΩ ϕ2 (1 + v(t))(1 + θµ). In particular, we can apply it in (36) to obtain the following inequality d 2dt ZΩ z2+ (1 −γ−1)ZΩ |∇z|2+ (1 −γ−1+λ1γ−1)ZΩ z2≤ZΩ cuvz . Therefore, there exists M > 0 such that d 2dt ZΩ z2+ (1 −γ−1)ZΩ |∇z|2+ZΩ z2≤MZΩ u2. Integrating the above estimate on the time interval (τ, t) we obtain for t≥τ, Zt τZΩ |∇z|2+ZΩ z2≤C. (39) In view of (39) one infers Zt τ d dt ZΩ z2≤C for t≥τ. Finally, the result follows by Lemma 3.3. Acknowledgement.T. 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