ANALYSIS OF A FREE BOUNDARY PROBLEM FOR AVASCULAR TUMOR GROWTH WITH A PERIODIC SUPPLY OF NUTRIENTS

. In this paper we study a free boundary problem for the growth of avascular tumors. The establishment of the model is based on the diﬀusion of nutrient and mass conservation for the two process proliferation and apopto-sis(cell death due to aging). It is assumed the supply of external nutrients is periodic. We mainly study the long time behavior of the solution, and prove that in the case c is suﬃciently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a positive periodic state.


1.
Introduction. Over the last forty years, a variety of partial differential equation models for tumor growth or therapy have been developed, cf. [1, 4-7, 13, 14, 20-25] and references therein. Most of those models are based on the reaction diffusion equations and mass conservation law. Analysis of such free boundary problems has drawn great interest, and many interesting results have been established, cf. [2, 3, 8-12, 16-19, 26-32] and references therein.
(5) where σ(r, t) represents nutrient concentration at radius r and time t; R(t) denote the external radius of tumor at time t; the term Γσ in (1) is the consumption rate of nutrient in a unit volume; φ(t) denotes the external concentration of nutrients, which is assumed to be a periodic function of a period ω. The two terms on the righthand side of (3) are explained as follows: The first term is the total volume increase in a unit time interval induced by cell proliferation, the proliferation rate is sσ; The second term is the total volume decrease in a unit time interval caused by natural death, and the natural death rate is sσ, hereσ is a constant, s is a scaling constant. c = T dif f usion /T growth is a positive constant which represents the ratio of the nutrient diffusion time scale to the tumor growth (e.g., tumor doubling) time scale, for details see (cf [16,19]). From [5,10] we know that T dif f usion ≈ 1min and T growth ≈ 1day, so that c 1. σ satisfies the compatibility condition σ 0 (R 0 ) = φ(0).
The model presented in this paper is similar to the model of Friedman and Reitich [19], but with one modification. The modification is as follows: In [19], the external concentration of nutrients is assumed to be a constant, so that instead of that Eq. (2) employed here. In this paper, as can be seen from (2), we assume that the external concentration of nutrients is a periodic function of a period ω. The idea of considering the periodic supply of external nutrients is motivated by [15]. In [15], through experiments, the authors observed that after an initial exponential growth phase leading to tumor expansion, growth saturation is observed even in the presence of a periodically applied nutrient supply. In this paper, we mainly discuss how the periodic supply of external nutrients influence the growth of avascular tumor growth. The results show the periodic supply of external nutrients have some influence on tumor growth. The main influence of the periodic supply of external nutrients on tumor growth is as follows: If c is sufficiently small and φ * >σ, we prove that the volume of the tumor will evolve to a positive periodic state. If the external concentration of nutrients is assumed to be a constantσ(>σ) and c is sufficiently small, Friedman and Reitich [19] have proved that the volume of the tumor will evolve to a positive steady state.
In the following of the paper, we assume that φ is a continuous differentiable function and always denotē and assume φ * > 0, then φ * ≤ φ(t) ≤ φ * for all t ≥ 0 and φ * ≤φ ≤ φ * follows. Moreover, since φ is a continuous differentiable function of a period ω, there exists a constant K such that |φ(t)| < K for all t ≥ 0. In [2], the authors have studied the limiting case where c = 0. By using a comparison method, the authors discussed the dynamical behavior of solutions to the model. In [2], Necessary and sufficient conditions for the global stability of tumor free equilibrium are given; the conditions under which there exists a unique periodic solution to the model are determined and they also show that the unique periodic solution is global attractor of all other positive solutions. In the limiting case where c = 0 Eq.(1), (2) can be solved exactly, and the exact expression of the evolution equation for R can be obtained. This is clearly not the case for present model and the method used in [2] can not be used to present model. Using Banach fixed point theorem, a compare method and some mathematical techniques, we mainly prove the existence and uniqueness of the global solution to the problem and asymptotic behavior of the solutions to the problem. The results show that in the case c is sufficiently small and φ * >σ, the volume of the tumor cannot expand unlimitedly and it will tend to a positive periodic state. We also show that in the case c is sufficiently small andφ <σ, the volume of the tumor also cannot expand unlimitedly and it will disappear as t → ∞..
The paper is arranged as follows: In Section 2 we prove the existence and uniqueness of the global solution to the system (1)- (5). Section 3 is devoted to the long time behavior of the solutions to the system (1)-(5).
Theorem 2.2. Assume that the condition (H) is satisfied and c is sufficient small (c < Γφ * K ). Then the system (1)-(5) has a unique solution σ(r, t), R(t) for all t ≥ 0.
Proof. For arbitrary T > 0, we introduce a metric space (S T , d) as follows: The set , and they satisfy the following conditions: It is clear that (S T , d) is a complete metric space.
We define a mapping F : (σ(r, t), R(t)) → (σ(r, t),R(t)) in the following way: And defineσ(r, t) = φ(t), for r >R(t), 0 ≤ t ≤ T. Using similar arguments as that in [10,11], we can prove F is a contraction for T > 0 is small. Therefore Banach fixed point theorem implies the local existence and uniqueness of a solution to the problem (1)-(5). Using Lemma 2.1, we can get the global existence and uniqueness of the solution.
3. Long time behavior of the solutions to (1)- (5). In this section, we study asymptotic behavior of the solutions to (1)-(5). First we consider the caseφ <σ.
Proof. By Lemma 2.1 (i) and the Eq.
From the left inequality above we can get . By the right inequality of (13), we have for ξ ∈ [0, ω] Next, we consider the caseφ >σ. Consider the corresponding quasi-stationary version of the problem (1)-(5) ∂v The solution to (14), (15) is Substituting (17) to (16), we have where Proof. Since the proof have been given in [2], we only sketch out it as follows for reader's convenience. Consider the following two equations For (I), firstly, they proved that above two equations have a unique positive constant solutions x 1 and x 2 respectively. Moreover, they also proved that if By using the fixed point theorem, they proved (I).
For (II), they used the method of reduction to absurdity. Assume that x(t) > x(t)(the proof when x(t) <x(t) is similar). Set By direct computation, one can get First, prove lim t→∞ y(t) exists, then prove lim t→∞ y(t) = 0. Thus, follows. This completes the proof of (II).

Lemma 3.3. Let (σ(r, t), R(t))
is the solution to (1)- (5). Assume that the condition (H) is satisfied and for some 0 < T ≤ ∞ and ε > 0 Assume further that 0 ≤ r ≤ R 0 , Then there exist positive constants c 0 and C independent of c, T, L, M, K and R 0 but depend of ε, L 0 , M 0 , K 0 such that for arbitrary 0 ≤ r ≤ R(t), 0 ≤ t < T and 0 < c ≤ c 0 .
Proof. By direct computation, we have Hypothesis (20) implies that for 0 < r < R(t), t ≥ 0, where C depends only on φ * , Γ and ε. Since the function is monotone increasing for r > 0 and q(R) = 1, one can get By (15) and (20), we have Then by comparison principle we obtain Similarly arguments can prove that σ − (r, t) ≤ σ(r, t) for 0 ≤ r ≤ R(t), 0 ≤ t < T.
Proof. The proof of (1) can be found in [19] and the proof of (3),(4) can be found in [12]. Next we prove (2), from Lemma 3.3 [10] we know that is strictly monotone decreasing for all x > 0. By simple computation, it follows that From [28] we know that lim x→0 xp (x) p (x) = 1, then we have −2 < xp (x) p (x) < 1. This completes the proof.
Proof. Assume that (23) is not valid for some t. It follows that there exists T > 0 such that for 0 ≤ t < T, By (13) and the fact that for 0 ≤ t < T, 1 2 we have |R (t)| ≤ L 0 , L 0 is a positive constant independent of c and T . Obviously |σ(r, 0) − v(r, 0)| ≤ φ * . By Lemma 3.3, it follows for arbitrary 0 ≤ r ≤ R(t), 0 ≤ t < T and 0 < c ≤ c 0 . Then we have for t > 0 It follows that for T > 0 From Lemma 3.4 (1) we known function p(x) is monotone decreasing for any x > 0, ) − sσ < 0 by choosing ε sufficiently small, then if c 0 is sufficiently small and 0 < c ≤ c 0 it follows that R (T ) < 0 which contracts to the fact R (T ) ≥ 0.
If R(T ) = 1 2 min(R(T ), ε), similar arguments can prove the desired assertion. This completes the proof.
Assume further that for some ε > 0 and all t ≥ 0, Then there exist positive constants c 0 , T 0 and C independent of c such that the following assertions holds: If 0 < c ≤ c 0 , for any α ∈ (0, α 0 ], if the inequalities hold for all 0 ≤ r ≤ R(t), t ≥ 0, then also the inequalities . By the mean value theorem and the inequality (26), we have for all 0 ≤ r ≤ R(t), t ≥ 0. Here and hereafter we use the same notation C to denote various different costive constants independent c and α. It follows that Since |R (t)| ≤ α for all t ≥ 0, by Lemma 3.3, there exists positive constants c 0 such that |σ(r, t) − v(r, t)| ≤ Cα(c + e − Γt c ) (30) for arbitrary 0 ≤ r ≤ R(t), t ≥ 0 and 0 < c ≤ c 0 .
By the fact that te −t < 1(⇔ e −t < 1 t ) for t > 0, one can get that then there exists T 0 > 1, when t > T 0 , there holds ≤ sCαcR(t).
Consider initial value problems Similar arguments as that in [2] Theorem 3.1, one can prove the following assertions: Let φ(t) be a positive periodic function of period ω. Assume that φ * >σ holds. Then there exist positive constants c 0 independent of c such that if 0 < c < c 0 (I) there exists a unique ω−priodic positive solutionR ± (t) to Eq.(32).
(II) for any other positive solutions R ± (t) to (32) and (33), the following assertion holds: lim Consider the following initial problems and Denote the solution to (37), (38) by R 1 (t) and the solution to (35), (36) by R 2 (t).