Asymptotic dynamics of an anti-angiogenic system in tumour growth

This paper deals with the Neumann initial boundary problem for anti-angiogenic system in tumour growth. The known results show that the problem possesses a unique global-in-time bounded classical solution for some sufficiently smooth initial data. For the large time behaviour of the global solution, by establishing some estimates based on semigroup theory, we prove that the solution approaches to the homogeneous steady state as , where is the spatial mean of the initial data for the endothelial cell tip density.


Introduction
Angiogenesis is the development of new blood vessels from any nearby preexisting vasculature, and it is widely recognized to play a crucial role in cancer metastatic cascade. In the process of metastasis, the endothelial cells from the primary tumour migrate to grow giving rise to secondary tumour. Moreover, it has been observed experimentally that when the primary tumours remain, the secondary tumours cannot be discovered, whereas upon removal of the primary tumours often leads to the rapid growth of many large secondary tumours (Anderson et al., 2000;Folkman, 1995;O'Reilly et al., 1997O'Reilly et al., , 1994. In order to describe explicitly the phenomenon, Anderson et al. (2000) proposed the following partial differential equation system: where n(x, t), c(x, t) and a(x, t) are the endothelial cell tip density, the concentrations of tumour angiogenic factors and the concentrations of angiostatin in the one-dimension domain [0, L], respectively. D n > 0 is the endothelial cell random motility coefficient, D c > 0 and D a > 0 respectively represent the diffusion coefficients of CONTACT Qingshan Zhang qingshan11@yeah.net tumour angiogenic factors and angiostatin. The parameters χ, κ, α 0 , λ 1 and λ 2 are nonnegative constants. It is important and interesting to qualitatively investigate the model (1). The authors in Anderson et al. (2000) studied the steady solution with the boundary condition Moreover, under this framework, Wei and Cui (2008) proved the corresponding initial boundary value problem has a unique global classical solution. When the blood vessel is located at x 1 and the secondary tumour located at x 2 (0 < x 1 < x 2 < L) and the influence of the both ends is neglected, the authors in Yang and Lu (2020) consider (1) with the special no-flux boundary conditions (2) and the initial conditions for all x ∈ (0, L). They obtain the boundedness of the global classical solution for system (1)-(3) with some sufficiently smooth initial data. Compared with the global solvability results (Wei & Cui, 2008;Yang & Lu, 2020;Zhang & Tao, 2019), very few information seems to know about the qualitative asymptotic dynamics on the problem (1)-(3). In the present paper, we investigate the large time behaviour of the global solution obtained in Yang and Lu (2020) and we prove that the solution (n, c, a) converges to (n 0 , 0, 0) uniformly in the large time, wherē The rest of the paper is organized as follows. In Section 2, we derive some useful lemmas and prove the main result.
In Section 3, we present a numerical example to illustrate the effectiveness of the theoretical analysis result. We then give the conclusion and discussion in Section 4. Throughout the paper, we set We denote by C various positive constants which may vary from step to step. (·) represents Gamma function. We define L p ( ) to be Lebesgue space.

Preliminaries and main results
As a preparation of the proof, let first give an explicit bounds for the global classical solution to the problem (1)-(3).

Proof:
The proof of global solvability in problem (1)-(3) can be found in Yang and Lu (2020). Following the same steps as in the proof of Zhang (2016, Lemma 3.2), we get which yields (4) from a direct calculation. Since n, c and a are nonnegative functions, we get the differential inequalities . Therefore, the maximum principle gives the estimates (5) and (6).
In order to study of the large time behaviour of solutions for (1), we first establish a time-space estimate for ∂n ∂x .
Proof: Multiplying the second equation in (1) by c and integrating over [0, L], we obtain for all T > 0. Similarly, we test the third equation in (1) by a and integrating over [0, L] to get for all T > 0. Multiplying the first equation in (1) with n, integrating by parts and using Young's inequality and (6), Integrating over (0, T) and using (4), (6), (8) and (9), we get (7).
We now establish the L p -estimate for ∂c ∂x and ∂a ∂x .

Proof:
The proof is based on semigroup arguments. Differentiating the variation-of-constants formula with respect to x shows that where A 1 is the realization of the operator −D c (·) xx in L p ( ) equipped with homogeneous Neumann boundary conditions. According to the smoothing estimates for the Neumann heat semigroup (Winkler, 2010), we can find C 1 > 0 and C 2 > 0 such that e −tA 1 ϕ x L p ( ) ≤ C 1 e −λt ϕ x L p ( ) for all ϕ ∈ W 1,2 ( ) and e −tA 1 ϕ x L p ( ) ≤ C 2 1 + t − 1 2 e −λt ϕ L p ( ) (12) for all ϕ ∈ L p ( ) with 1 ≤ p ≤ ∞. Then, we can estimate Let A 2 be the realization of the operator −D a (·) xx in L p ( ) equipped with homogeneous Neumann boundary conditions. Using similar arguments to the variation-ofconstants formula of the third equation we obtain the boundedness of a x (·, t) L p ( ) for all t ∈ (0, T max ).
In the proof of convergence for the first component of solution n, we shall need the following lemma.

Proof:
The proof is based on compactness arguments, see Hirata et al. (2017, Lemma 4.6) for details.

Proof:
We first prove that there exist θ > 0 and C > 0 such that for all t ≥ 1. In fact, note that n is the solution of the Neumann problem by the Cauchy inequality and Therefore, by the use of (10) and (11), according to the standard parabolic regularity arguments (see Li et al., 2015, Lemma 4.3 andPorzio &Vespri, 1993, Theorem 1.3), we have (14). Applying the Poincaré's inequality with ϕ = n and = (0, L), it follows from (7) that with some C 4 > 0. As a direct consequence of Lemma 2.4, we have n(·, t) →n 0 in C 0 (¯ ) as t → ∞, which completes the proof of the lemma.
With the aid of asymptotic property of n, we can now acquire the decay results concerning c and a. Lemma 2.6: For the initial data n 0 ∈ C 0 (¯ ), c 0 ∈ W 1,∞ ( ) and a 0 ∈ W 1,∞ ( ), we have the second and third components of solution for the system (1) Then the comparison principle gives ∞). Upon directly solving the initial value problem (15), we have for all t ∈ [T, ∞). Then, according to (5), we get  Therefore, exponentially as t → ∞. This completes the proof.
With the aid of the above lemmas, we can now get the main result of the present paper.

Simulations
In order to illustrate the asymptotic result of the present paper, we present a numerical simulation with Python 3.7.4. Since the two equations in (1) governing the concentrations of tumour angiogenic factors and angiostatin are of the similar structure, we fix a = 0 for simplicity. Other system parameters are D n = D c = D a = χ = 1, λ 1 = λ 2 = κ = 0, L = 3. We choose the initial data as From Figures 1(a and b) we find that the solution component n(x, t) approaches to 2 (the mean of n 0 (x) on the interval [0, L]) and c(x, t) converges to 0 uniformly as time t goes on, which is consistent with our analytical result in Theorem 2.7.

Conclusion
It is importance and interesting to understand the system (1) from a mathematical point of view. In the present paper, we study the large time behaviour of the solution. We derive the solution stabilizes to the spatially uniform equilibrium (n 0 , 0, 0) uniformly on [0, L] as t → ∞. The result can provide deep insight into the process of suppression of secondary tumours by the primary tumour. Time delay always accompanies with anti-angiogenic system in tumour growth, which may lead to oscillation and instability Qian, Li, Zhao, et al., 2020;Suriyon & Piyapong, 2020). But these may require different method, it needs to be discussed in the future work.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work is supported by Youth Talent Promotion Project of Henan Province (No. 2019HYTP035) and Project funded by China Postdoctoral Science Foundation (No. 2018M630824).