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

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.


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 ⊂ IR d with 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 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 µ measuring the strength of the flux on the tumor boundary and the nonlinearity V measuring 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 where Γ 1 ∩ Γ 2 = ∅ and Γ i are closed and open sets in the relative topology of ∂Ø. We suppose that Γ 2 is the tumor boundary and Γ 1 is the blood vessel boundary. Our parabolic problem reads.
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 v is 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.
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 V as soon as Observe that, if the above inequality holds, then the parabolic regularity asserts v(t) ∞ < C for any t > 0 and by [5, Theorem 3.1] the solution is global and regular. In particular, when V is bounded in the L ∞ norm (see [5]) then (3) is satisfied.

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 E 0 , E 1 two normed spaces, we can define the real interpolation functor, denoted by (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 (E 0 , E 1 ) θ,p is a Banach space then In the context of fractional Sobolev spaces this inequality reads, see [1, Theorem 7.2] for m < kθ, θ ∈ (0, 1). b) Let us consider a parabolic problem with a non-homogeneous boundary condition where Bz := ∂z ∂n and Az := −∆z + z.
We define the space of functions -realization of (A, B) (see [1, pg. 39] for the precise definition). Since (A, B) is normally elliptic then A α−1 generates an analytic semigroup [1,Theorem 8.5]. Moreover, if 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 has a unique principal eigenvalue (i.e. an eigenvalue whose associated eigenfunction can be chosen positive in Ω) and it will be denoted by 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.
is a supersolution to the v-equation of (1), therefore v(x, t) ≤ w(x, t). Since, for sufficiently large M , w = M e −ρt ϕ 1 , with ϕ 1 a 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 Taking the W β,γ -norm in a generalized variation of constants formula for v and using Next, we estimate the last term in the above inequality using the fact that (Ω)) and the continuous embeddings Observe that by (3) and the first part of the Lemma we have In view of the above bounds, (8) yields Next, by the choice of δ and ρ, Our purpose is to show that u converges to steady states. To this end we treat separately the cases λ = 0, λ > 0.

Case
Proof. By the assumptions of the lemma we observe that for any k > 0 lim t→+∞ t+k t |y(s)| + |y ′ (s)| ds = 0.
Lemma 3.4. Let λ = 0 and t > τ > 0, then it holds Proof. Integrating the u-equation of (1) yields So, integrating the last expression in time between τ and t we get the result. Proof. On multiplying the u-equation of (1) by u and integrating in space we obtain Therefore, we infer and after integrating in time, thanks to Lemma 3.2 we arrive at In particular we deduce that for t > τ t τ Ω |∇u| 2 ≤ C.
Thanks to (10), for t > τ Finally, Remark 3.5 and (12)  Also thanks to u(t) C(Ω) ≤ C for all t > 0 we obtain lim t→+∞ u(t) p = 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 X β norm of u, where X β is 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 X β ֒→ W k,p (see for instance [8, Theorem 1.6.1]) that for all k < 1 and p ≥ 2 u(t) W k,p ≤ C.
Next, (4) entails Remark 3.7. Let us point out that if we pick m such that m − d/p > 0 then W m,p (Ω) is embedded in C(Ω).

Case λ > 0.
Assume that there exists δ 0 and t 0 such that for t > t 0 > 0. Next, we examine the long time behavior for u under the hypothesis (14).
In the sequel we shall give sufficient conditions on V (u) implying (14).
Proof. On multiplying the u-equation by u − λ we have (16) Having in mind that (1 + v) 2 ≥ 1, the hypothesis (14) and the Sobolev trace embedding By Lemma 3.2 we can deduce u(t) − λ 2 2 ≤ Ce −θ 1 t for 0 < θ 1 < min{2δ 0 , β}. At this point we can argue exactly as in the end of the proof of Theorem 3.6. Namely, by the bound on u in L ∞ we infer the bound on the L p norm of u, p > 2. Next, we use the estimate of u in W k,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 V implying (14). Actually, only the behavior of V around zero matters. Roughly speaking we require a superlinear growth of V in the neighbourhood of zero. From now on we assume that there exist C, δ > 0, k 0 > 1 + d/2, j > d/2 such that for all s ∈ (0, δ).
Remark 3.9. The condition (18) is satisfied, for example, for functions 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: Lemma 3.10. Assume that (18) holds. Moreover, for some D, µ > 0, η > 1,ǫ and C(ǫ) are given byǫ Then, if δ > 0 is small enough, the following conditons are satisfied simultaneously for s ∈ (0, δ), C( ǫ)g(δ) < 1/2 (20) and Proof. Thanks to (18), we have Hence, for η < k 0 and δ sufficiently small (21) is satisfied. Next, owing to (18), we observe that Thus, (19) can be assured for η < k 0 and δ small enough. Moreover, it is straightforward to see that (20) is also satisfied for 1 < η < min{k 0 , 1 + j}.
Proof. Let δ > 0 be a fixed constant defined in (18). Given a function f , we define the negative part of f as a nonpositive function as follows Our purpose is to show that (u − δ) − (t) ∞ ≤ δ/2 for every t > t 0 which implies (14). In order to obtain the previous estimate we multiply the u-equation by (u − δ) − and we integrate in space to obtain where Previous inequality can be rewritten in terms of f (δ) defined before Lemma 3.10 as Thanks to the Sobolev trace embedding W 1,2 (Ω) ֒→ L 2 (∂Ω) and having in mind that (v + 1) 2 ≥ 1, we arrive at Therefore, we obtain In view of the nonnegativity of u we have Owing to (23), from (22) we see that (g(δ) was defined before Lemma 3.10) Due to the nonpositivity of (u − δ) − and (20) we have By the Hopf lemma and zero Neumann data on the boundary for u we see that there exists δ 1 such that u(t 0 ) > δ 1 . Hence choosing δ < δ 1 and using (19), (25) and Lemma 3.2 we infer from (24) (u − δ) − (t) 2 2 ≤ (2C(ǫ)f (δ) + 2µ ǫ)C(β), for t > t 0 > 0. We shall show that for some η > 1. To this end notice that choosing ǫ = C(ǫ) = 1/2, we are in a position to apply Lemma 3.10 with D = C(β). As a consequence, forǫ as it is chosen in Lemma 3.10, (20),(19), (21) and 2µǫC(β) ≤ δ 2η 2 are satisfied simultaneously. Hence (26) is shown.
In the following Theorem, we eliminate the restriction on µ of Theorem 3.6. However, we require the additional condition (28) on V . Proof. On the one hand, we multiply the u-equation of (1) by u and we integrate in the space variable to obtain with On the other hand, we multiply the v-equation of (1) by ϕ(u). Integrating in space, we obtain Inserting the above equality into (29) we have (30) Next we estimate v t . Multiplying the v-equation by v t and integrating over Ω we see that Therefore, by the uniform bound of v in C(Ω) we deduce After integrating over the interval (τ, t) we find, by Lemma 3.4, that for t ≥ τ Next, by (28) we obtain from (30) that By Lemma 3.4 and (28) we get for t ≥ τ . According to (33) and (31) we find upon integration of (32) over the time interval (τ, t) that for t ≥ τ t τ Ω |∇u| 2 ≤ C.
From the last estimate, a similar argument to the one used previously yields Finally, we can infer the result arguing as in the end of the proof of Theorem 3.6.
Next we prove a lemma which we will use in the proof of Theorem 4.4. As a by-product of the following lemma we learn a qualitative information that v is bounded away from 0 for times large enough. We shall obtain a lower bound on v by considering a subsolution to an elliptic problem which is also a subsolution to a second equation in (1). By Theorem 4.2 there exists t 0 > 0 such that 0 ≤ u(t) < ǫ 0 for all t ≥ t 0 > 0. We claim that there exists δ > 0 such that w = δϕ 1 is a subsolution to                w t − ∆w + (1 + cu)w = 0 in Ω × (t 0 , +∞), ∂w ∂n = 0 on Γ 1 × (t 0 , +∞), ∂w ∂n = µ w 1 + w on Γ 2 × (t 0 , +∞).
Therefore v(x, t) ≥ δϕ 1 ≥ c 1 . It remains to prove the claim. By the strong maximum principle v(x, t 0 ) > c > 0. Thus there exists δ > 0 such that δϕ 1 < v(x, t 0 ). Moreover, choosing δ > 0 such that k(1 + δ) < µ we make sure that ∂w ∂n ≤ µ w 1 + w on Γ 2 × (t 0 , +∞). Hence the claim is shown and the lemma follows. Now we are in a position to prove the main result of this section. To this end we make use of the theorem by Amann and López-Gómez, see [2], stating the equivalence between positivity of principal eigenvalue and existence of stricly positive supersolution of some elliptic problems (the previous version of this theorem for the Dirichlet problem was shown in [11]). Proof. Let z(t) = v(t) − θ µ . Then z solves the following parabolic problem We multiply (35) by z to obtain In order to estimate the right-hand side of (36) for t ≥ t 0 , we pick γ > 1 such that where c 1 is given in (34). For each t ≥ t 0 we consider the eigenvalue problem            −∆w + w = λw in Ø, ∂w ∂n = 0 on Γ 1 , ∂w ∂n = µγw (1 + v(t))(1 + θ µ ) on Γ 2 . (38) Next, we see that θ µ is a strict supersolution of            −∆w + w = 0 in Ø, ∂w ∂n = 0 on Γ 1 , ∂w ∂n = µγw (1 + v(t))(1 + θ µ ) on Γ 2 .