Asymptotic Behavior of a Tumor Angiogenesis Model with Haptotaxis

: This paper considers the existence and asymptotic behavior of solutions to the angiogenesis system p t = ∆ p − ρ ∇ · ( p ∇ w ) + λ p ( 1 − p ) , w t = − γ pw β in a bounded smooth domain Ω ⊂ R N ( N = 1, 2 ) , where ρ , λ , γ > 0 and β ≥ 1. More precisely, it is shown that the corresponding solution ( p , w ) converges to ( 1, 0 ) with an explicit exponential rate if β = 1, and polynomial rate if β > 1 as t → ∞ , respectively, in L ∞ -norm.


Introduction
As a physiological process, angiogenesis involves the formation of a new capillary network sprouting from a pre-existing vascular network. It has been recognized that the capillary growth through angiogenesis leads to vascularization of tumor, providing it with its own blood supply. During this process, the endothelial cells are induced by the fibronectin, which are bounded in Extracellular Matrix (ECM) and gathered under the effect of fibronection to form new vessels. In [1], Anderson et al. proposed a reaction-diffusion model to describe those procedures. In [2], Stevens and Othmer developed the so-called reinforced random walk to gain the understanding of the mechanism that causes the aggregation of myxobacteria. By the methodology established in [2], Levine et al. [3,4] derived models of the angiogenesis based on analysis of the relevant biochemical processes. We refer to [5,6] for the related research.
In this paper, we are concerned with the initial-boundary problem Here Ω ⊂ R N (N = 1, 2) is a bounded domain with smooth boundary ∂Ω and ν is unit outward normal vector. p denotes the density of endothelial cells and w represents the density of concentration of chemical substance such as fibronectin; ρ and γ are positive constant; β ≥ 1. f (p, w) denotes proliferation of endothelial cells; and χ(w) is called chemosensitivity. In addition to random motion, endothelial cells have haptotaxis migration, which responds to the gradient of attractant such as the fibronectin that is non-diffusible in the extracellular matrix.
Mathematical models with haptotaxis have attracted lots of attention [7][8][9][10][11][12]. For example, Corrias et al. [7] considered the system in Equation (1) with f ≡ 0 and β = 1 in the bounded domain and whole space, respectively. In fact, they proved that there exists a global-in-time L 1 -bounded weak solution when the domain is bounded and the sensitivity function χ(w) > 0, inf w≥0 wχ (w) χ(w) > −1. Moreover, the solution p converges to 1 |Ω| Ω p 0 in L 1 (Ω) weakly and c decays to zero in L p (Ω) (p < ∞) in strong topology. For the unbounded domain and the initial data satisfying p 0 log(1 + |x| N+1 ) ∈ L 1 (R N ), they obtained the existence of weak solution and the self-similar solutions thereof. Furthermore, they extended their previous result to the case of β ≥ 1 in [8]. Guarguaglini et al. [9] considered the variant of Equation (1) with χ(w) = 1 a+bw , f ≡ 0, and w-equation in whole line. Under some suitable assumption, it is shown that the corresponding system admits a global weak solution and local classic solution for sufficiently regular initial data. On the other hand, Rascle [12] showed the local existence and uniqueness of classic solutions of the system in Equation (1) with boundary condition with some L > 0 for all p ≥ 0, w ≥ 0. Liţcanu and Morales-Rodrigo [10] studied the system with χ(w) ≡ 1, and standard logistic growing source. Indeed, they proved the system in Equation (1) admits a globally L ∞ -bounded classic solution which converges to a constant with polynomial rate when β > 1 and exponential rate if β = 1 in L 2 -norm, respectively, when the initial data have a positive lower bound. It is noted that modeling approaches indicate that, in situations of significantly heterogeneous environment, adequate macroscopic limits of random walk rather lead to certain non-Fickian diffusion operator. We refer to [13,14] for the fractional diffusion and refer to [15,16] for the nonlinear diffusion. It is should be mentioned that Winkler [17] considered the related haptotaxis system of Equation (1), which describes the glioma spread in heterogeneous tissue, and proved the system has global weak solution with few initial data and the solution component p stabilizes towards a state involving infinite densities and other component w tends to zero. Finally, we would like mention some papers [18,19] where w satisfies d dt w = g(p, w).
and assumptions on g are much more restricted. Our aim is to consider the system in Equation (1) with χ(w) ≡ 1 and a standard logistic growing source, namely We assume the initial data satisfy The main result can be stated as follows: Theorem 1. Let γ λ, ρ be positive parameter and β ≥ 1. Then the problem in Equations (2) and (3) admits a globally L ∞ -bounded positive classic solution which satisfies Moreover, (1) If β = 1, then for any ∈ (0, min{λ 1 , 1, γ, λ}), there exists C( ) > 0 such that: where λ 1 is the first nonzero eigenvalue of −∆ in Ω with the homogeneous Neumann boundary condition.
(2) If β > 1, then for any ∈ (0, 1), there exist a constant C > 0 and t such that for all t > t .

Remark 1.
The system under consideration is very similar to the problem in [10]. However, we remove the condition that the initial data must have a positive lower bound to reach the L ∞ -convergence of solution.
The crucial idea towards to the proof of Theorem 1 in our approach is to derive a bound for for some C > 0. Furthermore, with the help of estimate of q = pe −ρw and Gagliardo-Nirenberg interpolation theorem, we can show the component p(x, t) converges to 1 in L ∞ (Ω) as t → ∞ with exponential rate if β = 1 and polynomial rate in L ∞ -norm if β > 1, respectively. This paper is organized as follows. In Section 2, we prove the system exists a globally L ∞ -bounded classic solution for any β ≥ 1 by the iterated method. In Section 3, we show the solution converge to stationary solution in L ∞ -norm when N = 2 and establish the explicit decay rate of solution in case of β > 1 and β = 1, respectively. In Section 4, we prove the same result in one-dimensional setting.

Local Existence
In this section, we state the local existence and uniqueness of solution to problem in Equations (2) and (3).

Global Existence
This section is devoted to the global existence of classic solution to problem in Equations (2) and (3). To this end, we derive some prior estimates of (p, w) and henceforth fix τ = min{1, T max 6 }.
Lemma 3. We have following estimates for all t ≤ T max .
In the following, we use induction to prove Equation (19). For m = 1 and any given t > τ, we can find some In the above inequality, the fact t t−τ b(s)ds ≤ C has been used. Now, we assume is valid for m = k − 1 ≥ 1. Due to the Gagliardo-Nirenberg inequality, It follows that With the help of induction, one can see that Equation (19) is valid for all m ≥ 1 . The argument above is still valid in one-dimension setting with some adaption.
Proof. Now, we define q k = (q − k) + , where (·) + denotes the positive part of function and Ω k (t) = {x ∈ Ω : q(x, t) > k}. Then, it is observed that where the constant C = C(λ, ρ, ||w 0 || L ∞ (Ω) , β). Here, we add the term Ω q 2 k to both sides to get Then, the Gagliardo-Nirenberg inequality and Young inequality yield By using the Gagliardo-Nirenberg inequality again, we have: For the term ||q k || L 1 (Ω) , we apply the Hölder inequality and Sobolev inequality to obtain Due to one-dimension setting, we use the embedding W 1,2 (Ω) → L ∞ (Ω) to get same inequality above. Therefore, Equation (24) can be rewritten as: Now, we set and choose = 1. By following proof of Lemma 3.13 in [20], we get where C is independent of k and t. We solve this inequality to get We choose positive constant η which is sufficiently small and K large enough such that when From previous lemma, we know Because ϕ is a non-negative and non-increasing function, Lemma B.1 in Appendix B of [21] implies that there exist k 0 ≤ ∞ such that ϕ(k 0 ) = 0. Therefore, q k 0 ≡ 0. This completes the proof.
for all t > 0.

Steady States
The positive steady states of Equation (2) can be defined by: By the argument in [20], we have wherew is any function in W 2,s (Ω).

L ∞ -Convergence of Solution in Two Dimensions
In this section, we discuss the asymptotic behavior of solution in the spatially two-dimension of problem in Equations (2) and (3). First, we define the Lyapunov functional Proof. The simple computation shows that Combining above two equalities, we complete the proof.
Then, we define 3.1. Asymptotic Behavior When β > 1 Lemma 8. There exists constant C > 0 such that Proof. By solving the w-equation, we obtain and thus w −1 > ||w 0 || −1 L ∞ (Ω) by the positivity of p. Now, combining the above inequalities with the fact that s(1 − s) log s < 0 for all s > 0, we arrive at Equation (34).
Then, we use the well-known L p -L q estimates of heat semigroup and the Gagliardo-Nirenberg inequality to show the L ∞ -convergence of solution.
Now, we define h(q, w) = λq(1 − e ρw q) + ργq 2 e ρw w β . Applying variation-of-constant formula to q-equation yields q = e t∆ q(·, 0) + t 0 e (t−s)∆ (∇w · ∇q + h(q, w)). (50) Then, the well-known L p -L q estimate of heat semigroup shows that Then, the Hölder inequality with the boundedness of w and p show that where C ! and C 2 are independent of t. Then, Gronwall inequality yields where Γ represents the gamma function. Proof. The L 2 -convergence of p can be proved by the argument in Lemmas 5.9, 5.10, and 5.11 in [20]. Furthermore, the Gagliardo-Nirenberg inequality implies that Then, Equation (53) follows by the boundedness of p and Lemma 12.
Now, we establish the explicit decay rate of p(x, t) − 1 and w(x, t) with respect to L ∞ -norm.

Lemma 14.
For any ∈ (0, 1), there exist t and constant C( ) such that for all t > t Proof. The explicit expression of w(x, t) shows that Then, we can choose positive ∈ (0, 1) and t such that for all x ∈ Ω whenever t > t . Now, we multiply the term γ( which leads to Equation (54).
Proof. We choose ∈ (0, 1) and t such that p(x, t) > 1 − for all x ∈ R n . By multiplying the p-equation with p − 1 and integrating over Ω, we have This implies that which leads to Equation (56).

Lemma 16.
For any ∈ (0, 1), there exist constant C( ) and t such that for all t > t for all t > t .
Proof. By the Gagliardo-Nirenberg inequality in the two-dimensional setting, we can obtain The Gagliardo-Nirenberg inequality still has a similar form in the one-dimensional setting. Then, Lemmas 5, 12, and 15 imply Equation (57).

Lemma 17.
For any ∈ (0, 1), there exist constant C > 0 and t such that for all t > t .
Proof. Now, we choose the same in Lemma 14. Then, the explicit expression of w implies that Then, we obtain Equation (58).
Collecting all the lemmas above, we infer that Theorem 3. If the initial data satisfy Equation (3), then the solution satisfies Moreover, for some ∈ (0, 1), there exists t > 0 such that for any t > t and w(·, t) W 1,4 ≤ C ( ) w 0

Asymptotic Behavior When β = 1
By checking the proof of Lemmas 10 and 11, a similar result is also valid for β = 1 which can be stated as follows and lim t→∞ ||w(·, t)|| L ∞ (Ω) = 0.
At this position, we focus on the explicit decay rate of solutions.

Lemma 18.
For any ∈ (0, 1), there exists C > 0 such that Proof. It is observed that Now, Equation (63) implies that there exist > 0 and t such that p(x, t) > 1 − whenever t > t for all x ∈ Ω. By integrating Equation (66) over Ω, we get which implies Equation (65).
By checking the proof of Lemmas 13, 15 and 16, we can establish the following explicit decay rate.

L ∞ -Convergence in One Dimensions
In this section, we establish the explicit decay rate of p(x, t) − 1 and w(x, t) for β = 1 and β > 1 in one-dimensional setting respectively. In should be mentioned here that the results in previous subsection are still valid in the one-dimensional case.
The following lemma plays a crucial role in establishing the uniformly convergence of p(x, t) − 1 when β ≥ 1.

Lemma 20.
There exists a constant C > 0 such that: where q = pe −ρw .
Then, we integrate it over Ω × (0, ∞) to obtain We notice that Thus, the inequality in Equation (72) can be deduced by Equations (73) and (74) and Lemma 5 by choosing = 1 2 .

Lemma 22. If the initial data satisfy Equation (3), then we have
Proof. By integrating the w-equation in time variable, we have for arbitrary t > 0, which leads to Equation (75).

Asymptotic Behavior When β > 1
Now, we focus our attention on the decay property of solutions for β > 1.
Lemma 23. If the initial data satisfy Equation (3), then we have: Proof. By the Poincaré-Wirtinger inequality, we have This indicates that the term t 0 ||p(·, s) − p(s)|| 2 L ∞ (Ω) is bounded. Now, we choose t 0 such that overlinep(t) > 1 2 for all t > t 0 We notice Then, for any t > max{ Then, the assertion now follows from Equation (78).
The explicit polynomial decay rate can be established by the argument in two-dimensional setting which can be stated as Lemma 25. If the initial data satisfy Equation (3), then, for any ∈ (0, 1), there exist t > 0 such that for all t > t > 0 and Now, from all the above lemmas, we get Theorem 6. If the initial data satisfy Equation (3) and β > 1, then the solution satisfies lim t→∞ p(·, t) L ∞ (Ω) + w(·, t) L ∞ (Ω) = 0.
Now, we choose some > 0 and t such that p(t) > 1 − whenever t > t . By applying the Gronwall inequality, we have and w(·, t) L ∞ (Ω) ≤ C( )e −γ(1− )t , where λ 1 is the first nonzero eigenvalue of −∆ in Ω with the homogeneous Neumann boundary condition.
Author Contributions: All the authors contributed to each part of this study equally. All authors have read and agreed to the published version of the manuscript.