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 behavior 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 pre-existing 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 modeling 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) form the blood vessels wall are induced by the (TAF), factors that are generated by the tumor, to migrate chemotactically toward the tumor. We assume that the (ECs) and the (TAF) fill in a bounded and connected domain Ω ⊂ R d with a regular boundary ∂Ω. In particular, neither the existence of extracellular matrix nor the activity of metalloproteinases is considered. But, following [5], nonlinear flux of TAF on the tumor boundary is taken into account. This way we include the fact that ECs are supposed to react chemotactically to the TAF, thus generating the large gradient of TAF on the boundary. This, in turn, is supposed to make the tumor more dangerous. The aim of [5] was to study the interplay between the density of ECs and TAF 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 ∂Ω. We suppose that Γ 2 is the tumor boundary and Γ 1 is the blood vessel boundary. Our parabolic problem reads.
where 0 < T ≤ +∞, λ, μ ∈ R, c > 0, and u 0 and v 0 are given nonnegative and nontrivial 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, θ μ ) is shown provided μ > μ 1 (see also [14]), where μ 1 is the principal eigenvalue of the boundary eigenvalue problem 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 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 in our investigations, we assume In particular, (3) is satisfied when V is bounded (see [5]). However, most of the results of this paper could be extended even to more general forms of V , for example, V (u) = u or V (u) = u p , p > 1, as soon as we know that (3) holds. By the regularity of V , we have just to apply the following estimate Observe that the parabolic regularity asserts v(t) ∞ < C for any t > 0 once we know (3) and by [5,Theorem 3.1], the solution is global and regular.

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.
Vol. 64 (2013) Long-time behavior of an angiogenesis model 1627 a) Let E 0 , E 1 be two normed spaces embedded in a common topological Hausdorff space E, we can define the real interpolation function, 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]): In the context of fractional Sobolev spaces, this inequality reads cf. [1, (5.20 for m < kθ, θ ∈ (0, 1). b) Let us consider a parabolic problem with a nonhomogeneous boundary condition where Bz := ∂z ∂n and We define the space of functions It is known that (A, B), as being in separated divergence form (see [1, pg. 21]), is normally elliptic. We denote by A α−1 the W 2α−2,p B -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 1628 Tomasz Cieślak and Cristian Morales-Rodrigo ZAMP has a unique principal eigenvalue (i.e., an eigenvalue whose associated eigenfunction can be chosen positive in Ω) and it will be denoted by

Convergence to the semi-trivial solution (λ, 0)
In the present section, we deal with the convergence to the semi-trivial steady state (λ, 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.
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 is a supersolution to the v-equation of (1); therefore, v(x, t) ≤ w(x, t). Since, for sufficiently large M, w = Me −ρt ϕ 1 , with ϕ 1 a positive eigenfunction associated with α(μ), is a supersolution to (7), the pointwise estimate in the claim of the lemma follows. For the second one, we pick Vol. 64 (2013) Long-time behavior of an angiogenesis model 1629 Taking the W β,γ -norm in a generalized variation of constants formula for v and using Lemma 3.1, we obtain Next, we estimate the last term in the above inequality using the fact that 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, ∞ 0 e (δ−ρ)τ (t − τ ) −θ dτ = C < +∞ and the Lemma follows.

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 d 2dt Therefore, we infer d 2dt Vol. 64 (2013) Long-time behavior of an angiogenesis model 1631 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.
By [5, Theorem 3.8], we find a bound u(t) C(Ω) ≤ C, therefore, d 2dt 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 Next, (4) entails

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 d 2dt 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 −θ1t 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 neighborhood of zero. From now on, we assume that there exist C, δ > 0, k for all s ∈ (0, δ). Next, we introduce some notation that will be used 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:
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 f − := min{f, 0}.

Tomasz Cieślak and Cristian Morales-Rodrigo ZAMP
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 d 2dt Previous inequality can be rewritten in terms of f (δ) defined before Lemma 3.10 as d 2dt Vol. 64 (2013) Long-time behavior of an angiogenesis model 1635 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 d 2dt 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) d 2dt 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) for some η > 1. To this end notice that choosing = C( ) = 1/2, we are in a position to apply Lemma 3.10 with D = C(β). As a consequence, for˜ as it is chosen in Lemma 3.10, (19), (20), (21) and 2μ˜ C(β) ≤ δ 2η 2 are satisfied simultaneously. Hence, (26) is shown.

Convergence to the semi-trivial solution (0, θ µ )
Throughout this Section, additionally to the boundedness of u, we assume that there exist constants 0 < c m < C M and α ≥ 1 such that Remark 4.1. Let us observe that when V (0) = 0 and (2) holds, then (28) is true for α = 1. Moreover, if V ∈ C k for k ≥ 1 with V k (0) = 0 and V j (0) = 0 for j < k, then (28) holds true for α = k.
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 d 2dt On the other hand, we multiply the v-equation of (1) by ϕ(u). Integrating in space, we obtain