Global Existence and Uniqueness of Solutions for a Free Boundary Problem Modeling the Growth of Tumors with a Necrotic Core and a Time Delay in Process of Proliferation

Abstract

We study a mathematical model for the growth of necrotic tumors with time delays in proliferation. By transforming this problem into an initial-boundary value problem in fixed domain of a coupled system of a parabolic equation and one integrodifferential equation with time delays, in which all equations involve discontinuous terms, and using the approximation method combined with Schauder fixed point theorem, we prove that this problem has a unique global solution in any time interval [0,𝑇].

1. Introduction

Tumor is a major threat in public health; it can cause serious problems for people at all ages. However, tumor progression is a complex process. The understanding of its dynamics is one of the great challenges of modern medical science. To describe the growth of solid tumors, an increasing number of mathematical models in forms of free boundary problems of partial differential equations have been proposed and studied during the past several decades (see, e.g., [1–12] and the references cited therein). Most of those models are based on the reaction-diffusion equations and mass conservation law. Analysis of such mathematical models has drawn great interest, and many results have been established; compare [13–22] and references therein. Many numerical results have been performed; compare [1, 2, 13, 23, 24] and references therein. Analysis of such models can help us to examine and distinguish different functions of different mechanisms involved in tumor-growth process and may also assist us in assessing effects of various drug treatments and chemotherapy.

The first equation of the model which we will study in this paper is a reaction-diffusion equation for the nutrient concentration 𝜎(𝑟,𝑡), which we present in dimensionless form: 𝑐𝜕𝜎=1𝜕𝑡𝑟²𝜕𝑟𝜕𝑟²𝜕𝜎(𝜎)𝐻𝜕𝑟−𝐹𝜎−𝜎_𝑁,0<𝑟<𝑅(𝑡),𝑡>0, (1) where 𝐻 is the Heaviside function and 𝑐 represents the time scale of diffusion of nutrient compared to the time scale of the tumor doubling within the tumor. 𝐹(𝜎) in (1) is the consumption rate of nutrient in a unit volume in the region 𝜌(𝑡)<𝑟<𝑅(𝑡), 𝑡>0; (1) means dead cells do not consume the nutrient. 𝜎_𝑁 is the minimal nutrient concentration needed for proliferation; that is, 𝜎_𝑁 is a positive constant representing a threshold value for necrosis. In the region where 𝜎>𝜎_𝑁, nutrient is enough to sustain (at least a portion of) tumor cells alive, whereas in the region where 𝜎≤𝜎_𝑁, nutrient is insufficient to sustain any tumor cells alive and all cells there are dead cells.

Let 𝑑=(1/𝑐) and 𝐹(𝜎)=𝑐𝑓(𝜎); we have 𝜕𝜎1𝜕𝑡=𝑑𝑟²𝜕𝑟𝜕𝑟²𝜕𝜎(𝜎)𝐻𝜕𝑟−𝑓𝜎−𝜎_𝑁,0<𝑟<𝑅(𝑡),𝑡>0. (2) The initial and boundary conditions for the necrotic phase of the tumor are 𝜕𝜎()(𝑅(𝑡))𝜕𝑟0,𝑡=0,𝜎,𝑡=𝜎_∞,𝑡>0,𝜕𝜎𝜌(𝑡)=𝜕𝑟−0,𝑡𝜕𝜎𝜌(𝑡)𝜎𝜌(𝑡)𝜌(𝑡)𝜕𝑟+0,𝑡,𝑡>0,−0,𝑡=𝜎+0,𝑡=𝜎_𝑁𝜎()()(𝑡)𝜌(𝑡)(𝑡)𝑅(𝑡)(𝑡),𝑡>0,𝑟,𝑡=𝜓𝑟,𝑡,0≤𝑟≤𝑅,−𝜏≤𝑡≤0,=𝜙,−𝜏≤𝑡≤0,=𝜑,−𝜏≤𝑡≤0. (3) The second equation of the model 𝑑𝑅=𝑠𝑑𝑡𝑅²_𝜌()^{𝑅(𝑡−𝜏)}_𝑡−𝜏𝜎(𝑟,𝑡−𝜏)𝑟²−𝑑𝑟_{𝜌(𝑡)𝑅(𝑡)}𝜎𝑟²𝑑𝑟−₀^𝜌(𝑡)𝜆𝑟²𝑑𝑟,𝑡>0, (4) describes the evolution of the tumor radius 𝑅(𝑡) and is obtained by applying mass balance equations with adequate constitutive laws, where 𝑅(𝑡) and 𝜌(𝑡) denote the external and necrotic radius of tumor at time 𝑡, respectively, and 𝜎_∞ denotes the external concentration of nutrients, which is assumed to be a constant. The three terms on the right-hand side of (4) have the following explanation. The first term is the total tumor volume increase in a unit time interval induced by cell proliferation; the rate of cell proliferation in a unit volume is assumed to be proportional to 𝜎. The second term is the total volume shrinkage in a unit time interval caused by cell apoptosis; the rate of cell apoptosis in a unit volume is assumed to be 𝑠𝜎. The third term is the total volume shrinkage induced by dead cell dissolution; the rate of cell dissolution in a unit volume is assumed to be 𝑠𝜆. 𝑠 is a scaling constant.

Equations (1) and (3) are from Bueno et al. [15]. And (4) is from [1]. The process of tumor growth can be divided into several different stages, starting from the very early stage of solid tumor without necrotic core inside (cf. [19, 20, 22, 25, 26]). In the present paper we focus on the next stage, that is, the process of necrotic core formation. In this stage there are three main cellular processes: proliferation, apoptosis, and necrosis (cf. [1, 2, 8, 13, 14, 16, 27, 28]). For process of necrotic core formation, the study by using the method of mathematical models was initiated by Byrne in [1]. Recently this study has drawn the attention of some other researchers; compare Bueno et al. [15], Cui [16], Foryś and Mokwa-Borkowska [13], and references cited therein. But all of them are related to quasistationary version; that is, 𝑐=0, and Cui [16] and Foryś and Mokwa-Borkowska [13] studied the particular cases in which 𝐹(𝜎)=Γ𝜎 and 𝐹(𝜎)=𝑎, respectively, where Γ and 𝑎 are two constants.

The model we studied in this paper is established by modifying the model studied in [15] by introducing a time delay in proliferation as that in Byrne [1]. In their model, the cell proliferation rate in unit volume is 𝑆(𝜎), where 𝑆 is a continuous function which satisfies some conditions; that is, (4) is as follows (𝜏=0): 𝑑𝑅=𝑠𝑑𝑡𝑅²_𝜌(𝑡)^𝑅(𝑡)𝑆(𝜎(𝑟,𝑡))𝑟²−𝑑𝑟_{𝑅(𝑡)𝜌(𝑡)}𝜎𝑟²𝑑𝑟−₀^𝜌(𝑡)𝜆𝑟²𝑑𝑟,𝑡>0. (5) But for simplicity we assume that the cell proliferation rate in unit volume is proportional to its concentration; that is, 𝑆(𝜎)=𝜇𝜎. The methods presented in this paper can be extended to the case that the cell proliferation rate in unit volume is 𝑆(𝜎), where 𝑆 is a continuous function that satisfies some conditions. It should be pointed out that, in [15], the authors only study the existence and uniqueness of stationary solutions to the model (2)–(4), but in our study we mainly discuss the existence and uniqueness of the solution to the model (2)–(4).

2. Main Results

In this paper, we always use the following notation 𝐵₁(0)=𝑦∈𝑅³,𝑦≤1,𝑄_𝑇=𝐵₁(0)×[],𝑊0,𝑇_𝑝2,1𝑄_𝑇=𝑢∈𝐿^𝑝𝑄_𝑇∶∇𝑢,∇²𝑢,𝑢_𝑡∈𝐿^𝑝𝑄_𝑇. (6) We will prove a global existence and uniqueness theorem for the problem (2)–(4) under the following assumptions:  (𝐴₁)  𝑓(𝜎)=𝜎𝑓₁(𝜎), where 𝑓₁ is defined and Lipschitz is continuous on [0,∞)×[0,∞), and 𝑓₁(𝜎)≥0, for 𝜎≥0;  (𝐴₂)  𝜑∈𝐶[−𝜏,0], 𝜑(0)>𝜌(0)>0, for −𝜏≤𝑡≤0,𝜓∈𝐶([0,∞]×[−𝜏,0]), and 𝜓(𝑟,𝑡)=𝜎_∞, for 𝑟≥𝑅(𝑡); 0≤𝜓(𝑟,𝑡)≤𝜎_∞, for 𝑟≤𝑅(𝑡);  (𝐴₃)  𝜓(𝑟,0)=𝜓₀(𝑟) is twice weakly differentiable on [0,𝑅(0)], 𝜓₀^′′∈𝐿^∞[0,𝑅(0)], 𝜓^′₀(0)=0, 𝜓₀(𝑅(0))=𝜎_∞.

Our main result is as follows.

Theorem 1. Assume that conditions (𝐴₁), (𝐴₂), and (𝐴₃) are satisfied. Then for any 𝑇>0, the problem (2)–(4) has a solution (𝜎(𝑟,𝑡),𝑅(𝑡)), for all 0≤𝑡≤𝑇, satisfying 𝜎(|𝑥|,𝑡)∈𝑊_𝑝^2,1(𝑄_𝑇) (5<𝑝<∞) and 0≤𝜎(|𝑥|,𝑡)≤𝜎_∞.

3. Transformation to a Fixed Domain

We introduce a transformation of (𝑟,𝑡,𝜎,𝑅)→(𝑧,𝑡,𝑢,𝑅) as follows: 𝑟𝑧=𝑅(𝑡)𝑢()((𝑡))(𝑡)(𝑡).,𝑡=𝑡,𝑧,𝑡=𝜎𝑧𝑅,𝑡,𝑅=𝑅 (7) Then the problem (2)–(4) is transformed into the following problem: 𝜕𝑢1𝜕𝑡=𝑑𝑅²(𝑡)1𝑧²𝜕𝑧𝜕𝑧²𝜕𝑢+𝑅𝜕𝑧^′(𝑡)𝑅(𝑡)𝑧𝑢_𝑧−𝑓(𝑢)𝐻𝑢−𝜎_𝑁,0<𝑧<1,𝑡>0,𝜕𝑢()()𝜕𝑧0,𝑡=0,𝑢1,𝑡=𝜎_∞,𝑡>0,𝑑𝑅(𝑡)𝑑𝑡=𝑠𝑅¹₀𝑅³_𝜏𝑅³𝑢_𝜏𝐻𝑢_𝜏−𝜎_𝑁𝑧²−𝜎𝐻𝑢−𝜎_𝑁𝑧²−𝜆1−𝐻𝑢−𝜎_𝑁𝑧²𝑑𝑧,𝑡>0,𝑢(𝑧,𝑡)=𝜓(𝑧𝑅(𝑡),𝑡),0≤𝑧≤1,−𝜏≤𝑡≤0,𝜕𝑢(𝑡)=𝜕𝑧𝑧𝜌−0,𝑡𝜕𝑢(𝑡),𝜕𝑧𝑧𝜌+0,𝑡0≤𝑧≤1,𝑡>0,𝑢𝑧𝜌(𝑡)−0,𝑡=𝑢𝑧𝜌(𝑡)+0,𝑡=𝜎_𝑁,0≤𝑧≤1,𝑡>0,𝜌(𝑡)=𝜙(𝑡),−𝜏≤𝑡≤0,𝑅(𝑡)=𝜑(𝑡),−𝜏≤𝑡≤0, (8)(9)(10)(11)(12)(13)(14)(15) where 𝑅_𝜏=𝑅(𝑡−𝜏), 𝑢_𝜏=𝑢(𝑧,𝑡−𝜏). Let 𝑋_𝑇=𝐶([0,1]×[0,𝑇])×𝐶([0,1]×[−𝜏,𝑇]). Denote the function 𝐺 as follows 𝐺𝑢,𝑢_𝜏,𝑡=𝑠¹₀𝑅³_𝜏𝑅³𝑢_𝜏𝐻𝑢_𝜏−𝜎_𝑁𝑧²−𝜎𝐻𝑢−𝜎_𝑁𝑧²−𝜆1−𝐻𝑢−𝜎_𝑁𝑧²𝑑𝑧 (16) for 𝑡>0. Using this notation, (10) can be simply written as follows: 𝑅^′(𝑡)=𝑅(𝑡)𝐺𝑢,𝑢_𝜏,𝑡,𝑡>0. (17) It is clear that 𝑅(𝑡) can be expressed by 𝑅(𝑡)=𝑅₀exp^𝑡₀𝐺𝑢,𝑢_𝜏,𝜉𝑑𝜉,𝑡>0. (18) Hence the problem (8)–(15) is reduced to the problem below: 𝜕𝑢𝜕𝑡=𝑑𝑅₀⁻²exp−2^𝑡₀𝐺𝑢,𝑢_𝜏×1,𝜉𝑑𝜉𝑧²𝜕𝑧𝜕𝑧²𝜕𝑢𝜕𝑧+𝐺𝑢,𝑢_𝜏,𝑡𝑧𝑢_𝑧−𝑓(𝑢)𝐻𝑢−𝜎_𝑁,0<𝑧<1,𝑡>0,𝜕𝑢()()𝜕𝑧0,𝑡=0,𝑢1,𝑡=𝜎_∞,𝑡>0,𝑢(𝑧,𝑡)=𝜓(𝑧𝑅(𝑡),𝑡),0≤𝑧≤1,−𝜏≤𝑡≤0,𝜕𝑢(𝑡)=𝜕𝑧𝑧𝜌−0,𝑡𝜕𝑢(𝑡)𝜕𝑧𝑧𝜌+0,𝑡,0≤𝑧≤1,𝑡>0,𝑢𝑧𝜌(𝑡)−0,𝑡=𝑢𝑧𝜌(𝑡)+0,𝑡=𝜎_𝑁,0≤𝑧≤1,𝑡>0,𝜌(𝑡)=𝜙(𝑡),−𝜏≤𝑡≤0,𝑅(𝑡)=𝜑(𝑡),−𝜏≤𝑡≤0. (19)(20)(21)(22)(23)(24)(25) Then we conclude the following.

Lemma 2. Under the variable transformation (7), the free boundary problem (2)–(4) is equivalent to the initial-boundary value problem (19)–(25).

4. The Proof of Main Results

First we consider the following approximation problem: 𝜕𝑢𝜕𝑡=𝑑𝑅₀⁻²exp−2^𝑡₀𝐺_𝜀𝑢,𝑢_𝜏×1,𝜉𝑑𝜉𝑧²𝜕𝑧𝜕𝑧²𝜕𝑢𝜕𝑧+𝐺_𝜀𝑢,𝑢_𝜏,𝑡𝑧𝑢_𝑧−𝑓(𝑢)𝐻𝑢−𝜎_𝑁,0<𝑧<1,𝑡>0, (26) with initial and boundary condition as (20)–(25), where 𝜀 is an arbitrary positive number, 𝐺_𝜀𝑢,𝑢_𝜏,𝑡=𝑠¹₀𝑅³_𝜏𝑅³𝑢_𝜏𝐻_𝜀𝑢_𝜏−𝜎_𝑁𝑧²−𝜎𝐻_𝜀𝑢−𝜎_𝑁𝑧²−𝜆1−𝐻_𝜀𝑢−𝜎_𝑁𝑧²,𝐻𝑑𝑧_𝜀(𝑥)=𝜀1,𝑥≥𝜀,𝑥,0<𝑥<𝜀,0,𝑥≤0. (27) It is clear that 𝐻_𝜀 is a Lipschitz continuous function.

Recall that 𝑢(𝑧,𝑡)=𝜓(𝑧𝑅(𝑡),𝑡), 0≤𝑧≤1, −𝜏≤𝑡≤0; by assumption (𝐴₃) we know that 𝑢(𝑟,0)=𝜓₀(𝑟)=𝜓(𝑟𝑅(0),0) is twice weakly differentiable on [0,1], 𝜓₀^′′∈𝐿^∞[0,1],  𝜓^′₀(0)=0, and 𝜓₀(1)=𝜎_∞.

Lemma 3. If the assumptions (𝐴₁)--(𝐴₃) hold, then for any 𝑇>0 the problem (26), (20)–(25) has a unique solution 𝑢_𝜀 on [0,1]×[−𝜏,𝑇] satisfying the following assertions:(i)𝑢_𝜀(𝑦,𝑡)∈𝑊_𝑝^2,1(𝑄_𝑇), 𝑝∈(1,∞) and for any 𝜀>0 there holds 𝑢_𝜀𝑦,𝑡_{𝑊𝑝2,1(𝑄𝑇)}≤𝐶𝑇,𝑝𝑢₀𝑦,0_{𝑊2,∞(0,1)}+𝜎_∞, (28) where 𝐶(𝑇,𝑝) is a constant independent of 𝜀,𝑢₀(𝑦,0)=𝜓(𝑧𝑅(0),0);(ii)For any 𝜀>0, there holds 0≤|𝑢_𝜀(|𝑦|,𝑡)|≤𝜎_∞, 0≤𝑦≤1, −𝜏≤𝑡≤𝑇.

Remark 4. From (i) and assumption (𝐴₂) it is easy to get 𝑢_𝜀𝑦,𝑡−𝜏_{𝑊𝑝2,1(𝑄𝑇)}≤𝐶𝑇,𝑝𝑢₀𝑦,0_{𝑊2,∞(0,1)}+𝜎_∞. (29)

Proof. Endow 𝑋_𝑇 with the following norm: 𝑢,𝑢_𝜏=max_{[0,1]×[−𝜏,𝑇]}|𝑢|. (30) Then 𝑋_𝑇 is clearly a Banach space. Define a mapping 𝐹∶𝑋_𝑇→𝑋_𝑇 as follows: for any (𝑢,𝑢_𝜏)∈𝑋_𝑇, define 𝐹(𝑢,𝑢_𝜏)=(𝑢,𝑢_𝜏), where 𝑢(𝑧,𝑡)=𝑢(𝑧𝑦^𝑜,𝑡), 𝑦^𝑜∈𝑅³, |𝑦^𝑜|=1, 𝑢_𝜏=𝑢(𝑧,𝑡−𝜏), and 𝑢 is the solution of the following problem: 𝜕𝑢𝜕𝑡=𝑑𝑅₀⁻²exp−2^𝑡₀𝐺_𝜀𝑢,𝑢_𝜏,𝜉𝑑𝜉Δ𝑢+𝐺_𝜀𝑢,𝑢_𝜏𝑢−,𝑡𝑧⋅∇𝑢𝑓₁(𝑢)𝐻𝑢−𝜎_𝑁,𝑢𝑦,𝑡=𝜎_∞,𝑦𝜕𝑢=1,𝑡>0,(𝑡)=𝜕𝑢𝜕𝑧𝑧𝜌−0,𝑡(𝑡),𝑢(𝑡)=𝑢(𝑡)𝜕𝑧𝑧𝜌+0,𝑡0≤𝑧≤1,𝑡>0,𝑧𝜌−0,𝑡𝑧𝜌+0,𝑡=𝜎_𝑁,𝑢0≤𝑧≤1,𝑡>0,𝑦,𝑡=𝑢₀𝑦,𝑡,𝑦≤1,−𝜏≤𝑡≤0, (31) where 𝑢₀(𝑦,𝑡)=𝜓(𝑧𝑅(𝑡),𝑡). First we prove that 𝐹 is well defined. Since all coefficients in the first equation of (31) are bounded continuous functions and 𝑅₀⁻²exp−2^𝑡₀𝐺_𝜀𝑢,𝑢_𝜏,𝜉𝑑𝜉≥𝜆_𝜀𝑇,𝑢,𝑢_𝜏,𝜏>0,0≤𝑡≤𝑇. (32) Using the 𝐿^𝑝 theory linear parabolic equations [29] we infer that the problem (31) has a unique solution 𝑢. Since the initial and boundary values are symmetric in space variable 𝑦, it is easy to see that, for any orthogonal transformation 𝑂∶𝑅³→𝑅³, 𝑢₁(𝑦,𝑡)=𝑢(𝑂𝑦,𝑡) are also solutions of this problem which implies, by uniqueness of the solution, that there exists functions 𝑢∈𝐶([0,1]×[−𝜏,𝑇]) such that 𝑢𝑦,𝑡=𝑢𝑦,𝑡,𝑦≤1,−𝜏≤𝑡≤𝑇. (33) By 𝐿^𝑝 estimates we have, for 𝑝∈(1,∞), 𝑢𝑦,𝑡≤𝐶𝑇,𝑝,‖𝑢‖𝑢₀𝑦,0_{𝑊2,∞(𝐵1(0))}+𝜎_∞+𝑢𝑦,𝑡_{𝐿∞(𝑄𝑇)}. (34) By the maximum principle one can get 0≤𝑢𝑦,𝑡≤𝜎_∞,𝑦≤1,0≤𝑡≤𝑇. (35) Hence 𝑢𝑦,𝑡_{𝑊𝑝2,1(𝑄𝑇)}≤𝐶𝑇,𝑝,𝑢,𝑢_𝜏×𝑢₀𝑦,0_{𝑊2,∞(0,1)}+𝜎_∞,0≤𝑢𝑦,𝑡≤𝜎_∞,𝑦≤1,−𝜏≤𝑡≤𝑇. (36)(37) Taking particularly 𝑝>(5/2), by the embedding 𝑊_𝑝^2,1(𝑄_𝑇)↪↪𝐶(𝑄_𝑇), (𝑝>(5/2)), we infer that (𝑢,𝑢_𝜏)∈𝑋_𝑇. Hence the mapping 𝐹 is well defined.
Let 𝐸={(𝑢,𝑢_𝜏)∶(𝑢,𝑢_𝜏)∈𝐶([0,1]×[0,𝑇])×𝐶([0,1]×[−𝜏,𝑇]),0≤𝑢,𝑢_𝜏≤𝜎_∞}. 𝐸 is clearly a bounded closed convex subset of 𝑋_𝑇. We first prove that 𝐹(𝐸) is precontract in 𝑋_𝑇. Actually, since, for any (𝑢,𝑢_𝜏)∈𝐸, ‖(𝑢,𝑢_𝜏)‖≤𝜎_∞ by (36) we infer that for any 1<𝑝<∞ there hold 𝑢𝑦,𝑡≤𝐶𝑇,𝑝. (38) Taking 𝑝>(5/2) and using the compact embedding 𝑊_𝑝^2,1(𝑄_𝑇)↪↪𝐶(𝑄_𝑇), (𝑝>(5/2)), we conclude that 𝐹(𝐸) is precontract in 𝑋_𝑇.
Next we prove that 𝐹 is continuous. Let (𝑢₁,𝑢_1𝜏), (𝑢₂,𝑢_2𝜏)∈𝑋_𝑇 and denote (𝑢₁,𝑢_1𝜏)=𝐹(𝑢₁,𝑢_1𝜏), (𝑢₂,𝑢_2𝜏)=𝐹(𝑢₂,𝑢_2𝜏), 𝑢^∗=𝑢₁−𝑢₂, 𝑢^∗_1𝜏=𝑢_1𝜏−𝑢_2𝜏. It is obvious that 𝑢^∗ is a solution of the following problem: 𝜕𝑢^∗𝜕𝑡=𝑑𝑅₀⁻²exp−2^𝑡₀𝐺_𝜀𝑢₁,𝑢_1𝜏,𝜉𝑑𝜉Δ𝑢^∗+𝐺_𝜀𝑢₁,𝑢_1𝜏,𝑡𝑦⋅∇𝑢^∗−𝑢^∗𝑓𝑢₁𝐻_𝜀𝑢₁−𝜎_𝑁,𝑢𝑦𝜕𝑢+ℎ𝑦,𝑡𝑦,𝑡=0,=1,𝑡>0,(𝑡)=𝜕𝑢𝜕𝑧𝑧𝜌−0,𝑡(𝑡),𝑢(𝑡)=𝑢(𝑡)𝑢𝑦𝜕𝑧𝑧𝜌+0,𝑡0≤𝑧≤1,𝑡>0,𝑧𝜌−0,𝑡𝑧𝜌+0,𝑡=0,0≤𝑧≤1,𝑡>0,𝑦,𝑡=0,≤1,−𝜏≤𝑡≤0, (39) where ℎ(𝑦,𝑡)=𝑑𝑅₀⁻²[exp{−2∫^𝑡₀𝐺_𝜀(𝑢₁,𝑢_1𝜏,𝜉)𝑑𝜉}  −  exp{−2∫^𝑡₀𝐺_𝜀(𝑢₂,𝑢_2𝜏,𝜉)𝑑𝜉}]𝑢₂  +  [𝐺_𝜀(𝑢₁,𝑢_1𝜏,𝑡)  −  𝐺_𝜀(𝑢₂,𝑢_2𝜏,𝑡)](𝑦⋅∇𝑢₂)  −  𝑢₂[𝑓₁(𝑢₁)𝐻_𝜀(𝑢₁−𝜎_𝑁)−𝑓₁(𝑢₂)𝐻_𝜀(𝑢₂−𝜎_𝑁)].
Clearly, 𝐺_𝜀𝑢,𝑢_𝜏,𝑡≤𝐶𝑢,𝑢_𝜏,𝑇 (40) for (𝑢,𝑢_𝜏)∈𝑋_𝑇, 𝑡∈[0,𝑇], where 𝐶(‖(𝑢,𝑢_𝜏)‖) is independent of 𝜀. Then one can get 𝑒^{−2𝑇𝐶(‖(𝑢,𝑢𝜏)‖,𝑇)}≤exp−2^𝑡₀𝐺_𝜀𝑢,𝑢_𝜏,𝜉𝑑𝜉≤𝑒^{2𝑇𝐶(‖(𝑢,𝑢𝜏)‖,𝑇)},exp−2^𝑡₀𝐺_𝜀𝑢,𝑢_𝜏,𝜉𝑑𝜉−exp−2^𝑡′₀𝐺_𝜀𝑢,𝑢_𝜏,𝜉𝑑𝜉≤𝐶𝑢,𝑢_𝜏𝑡−𝑡^′ (41) for 𝑢,𝑣∈𝑋_𝑇, 𝑡∈[0,𝑇]. Besides, it is also clear that 0≤𝐻_𝜀𝑢−𝜎_𝑁≤1,𝑦≤1,0≤𝑡≤𝑇. (42) Hence, by the 𝐿^𝑝 estimate one can get 𝑢^∗_{𝑊𝑝2,1(𝑄𝑇)}≤𝐶𝑇,𝑝,𝑢₁,𝑢_1𝜏‖ℎ‖_𝐿^𝑝_(𝑄𝑇). (43) By the fact that 𝐻_𝜀 is Lipschitz continuous and (36), one can verify ‖ℎ‖∣_{𝐿𝑝(𝑄𝑇)}≤𝐶𝑇,𝑝,𝑢₁,𝑢_1𝜏,𝑢₂,𝑢_2𝜏×𝑢,𝜀₁−𝑢₂,𝑢_1𝜏−𝑢_2𝜏. (44) Substituting these estimates into (43) we have 𝑢^∗_{𝑊𝑝2,1(𝑄𝑇)}≤𝐶𝑇,𝑝,𝑢₁,𝑢_1𝜏,𝑢₂,𝑢_2𝜏×𝑢,𝜀₁−𝑢₂,𝑢_1𝜏−𝑢_2𝜏. (45) Taking 𝑝>(5/2) and using the embedding 𝑊_𝑝^2,1(𝑄_𝑇)↪↪𝐶(𝑄_𝑇) we get 𝑢₁−𝑢₂,𝑢_1𝜏−𝑢_2𝜏≤𝑢^∗_{𝑊𝑝2,1(𝑄𝑇)}≤𝐶𝑇,𝑝,𝑢,𝑢_𝜏,𝜀𝑢₁−𝑢₂,𝑢_1𝜏−𝑢_2𝜏. (46) Hence, 𝐹 is continuous.
Since we have proven that 𝐹 maps 𝐸 into itself, using the Schauder fixed point theorem we conclude that 𝐹 has a fixed point in 𝐸. Therefore, the problem (19)–(25) has a solution 𝑢=𝑢_𝜀.
Next, we prove uniqueness. Let 𝑢^∗=𝑢₁−𝑢₂. Then 𝑢^∗ satisfy 𝜕𝑢^∗𝜕𝑡=𝑑𝑅₀⁻²exp−2^𝑡₀𝐺_𝜀𝑢₁,𝑢_1𝜏,𝜉𝑑𝜉Δ𝑢^∗+𝐺_𝜀𝑢₁,𝑢_1𝜏,𝑡𝑦⋅𝑢^∗+𝑝𝑦,𝑡 (47) for |𝑦|<1,𝑡>0, where 𝑝(𝑦,𝑡)=𝑑𝑅₀⁻²[exp{−2∫^𝑡₀𝐺_𝜀(𝑢₁,𝑢_1𝜏,𝜉)𝑑𝜉}  −  exp{−2∫^𝑡₀𝐺_𝜀(𝑢₂,𝑢_2𝜏,𝜉)𝑑𝜉}]Δ𝑢₂  +  [𝐺_𝜀(𝑢₁,𝑢_1𝜏,𝑡)−𝐺_𝜀(𝑢₂,𝑢_2𝜏,𝑡)](𝑦⋅∇𝑢₂)  −  [𝑓(𝑢₁)−𝑓(𝑢₂)]𝐻_𝜀(𝑢₁(|𝑦|,𝑡)  −  𝜎_𝑁)−𝑓(𝑢₂)[𝐻_𝜀(𝑢₁(|𝑦|,𝑡)−𝜎_𝑁)  −  𝐻_𝜀(𝑢₂(|𝑦|,𝑡)−𝜎_𝑁)].
Multiplying (47) with 𝑢^∗ and integrating in 𝑦, one can get 12_{‖𝑦‖≤1}𝑢^∗𝑦,𝑡²𝑑𝑦=−𝑑𝑅₀⁻²exp−2^𝑡₀𝐺_𝜀𝑢₁,𝑢_1𝜏×,𝜉𝑑𝜉_{‖𝑦‖≤1}∇𝑢^∗𝑦,𝑡²−3𝑑𝑦2𝐺_𝜀𝑢₁,𝑢_1𝜏,𝑡_{‖𝑦‖≤1}𝑢^∗𝑦,𝑡²+𝑑𝑦_{‖𝑦‖≤1}𝑝𝑦,𝑡𝑢^∗𝑦,𝑡𝑑𝑦≤−𝑑𝑅₀⁻²exp−2𝑇𝐶𝑢₁,𝑢_1𝜏∇𝑢^∗(⋅,𝑡)²_{𝐿2(𝐵1(0))}+32𝐶𝑢₁,𝑢_1𝜏𝑢^∗(⋅,𝑡)²_{𝐿2(𝐵1(0))}+𝐼(𝑡), (48) where 𝐼(𝑡)=𝐼₁+𝐼₂+𝐼₃+𝐼₄ and 𝐼₁=𝑑𝑅₀⁻²exp−2^𝑡₀𝐺_𝜀𝑢₁,𝑢_1𝜏,𝜉𝑑𝜉−exp−2^𝑡₀𝐺_𝜀𝑢₂,𝑢_2𝜏×,𝜉𝑑𝜉_{‖𝑦‖≤1}Δ𝑢₂⋅∇𝑢^∗𝐼𝑦,𝑡𝑑𝑦,₂=𝐺_𝜀𝑢₁,𝑢_1𝜏,𝑡−𝐺_𝜀𝑢₂,𝑢_2𝜏×,𝑡_{‖𝑦‖≤1}𝑦⋅∇𝑢₂∇𝑢^∗𝐼𝑦,𝑡𝑑𝑦,₃=−_{‖𝑦‖≤1}𝑓𝑢₁−𝑓𝑢₂𝐻_𝜀𝑢₁𝑦,𝑡−𝜎_𝑁×𝑢^∗𝐼𝑦,𝑡𝑑𝑦,₄=−_{‖𝑦‖≤1}𝑓𝑢₂𝐻_𝜀𝑢₁𝑦,𝑡−𝜎_𝑁−𝐻_𝜀𝑢₂𝑦,𝑡−𝜎_𝑁𝑢^∗𝑦,𝑡𝑑𝑦. (49) By the mean value theorem and the fact that 𝑓(𝑢)≥0, we have 𝐼₄=−_{‖𝑦‖≤1}𝑓𝑢₂¹₀𝐻^′_𝜀𝑑𝜃𝑢^∗2𝑑𝑦≤0. (50) Using (28) and embedding 𝑊_𝑝^2,1(𝑄_𝑇)↪↪𝐶(𝑄_𝑇), (𝑝>(5/2)), we get max_𝑄𝑇∇𝑢₂𝑦,𝑡≤𝐶(𝑇),max_𝑄𝑇∇𝑢_2𝜏𝑦,𝑡≤𝐶(𝑇). (51) From definition of 𝐺_𝜀 and Lipschitz continuous of 𝐻_𝜀, we have 𝐺_𝜀𝑢₁,𝑢_1𝜏,𝑡−𝐺_𝜀𝑢₂,𝑢_2𝜏,𝑡≤𝐶(𝜀)𝑢^∗₂+𝑢^∗_𝜏2. (52) Then by using (51) one can get 𝐼₂≤𝐶(𝜀)𝑢^∗₂+𝑢^∗_{𝜏2‖𝑦‖≤1}𝑢^∗𝑦,𝑡𝑑𝑦. (53) By a direct computation and (52) one can get 𝐼₁≤𝐶(𝜀,𝑇)^𝑡₀𝑢^∗₂+𝑢^∗_𝜏2𝑑𝜉_{‖𝑦‖≤1}∇𝑢^∗𝑦,𝑡𝑑𝑦. (54) Lipschitz continuity of 𝑓(𝑢) combined with (51) yields 𝐼₃≤𝐶(𝜀,𝑇)_{‖𝑦‖≤1}𝑢^∗𝑦,𝑡²𝑑𝑦. (55) Summing up (51), (53)–(55) into 𝐼(𝑡), one can get 𝐼(𝑡)≤𝐶(𝜀)𝑢^∗₂+𝑢^∗_{𝜏2‖𝑦‖≤1}𝑢^∗()𝑦,𝑡𝑑𝑦+𝐶𝜀,𝑇^𝑡₀𝑢^∗₂+𝑢^∗_𝜏2𝑑𝜉_{‖𝑦‖≤1}∇𝑢^∗()𝑦,𝑡𝑑𝑦+𝐶𝜀,𝑇_{‖𝑦‖≤1}𝑢^∗𝑦,𝑡²𝑑𝑦. (56) Using Cauchy inequality and summing up (51), (53)–(55) into 𝐼(𝑡), one can get 12𝑑𝑢𝑑𝑡^∗(⋅,𝑡)²₂≤−𝐶₁(𝑇)∇𝑢^∗(⋅,𝑡)²₂+𝐶₂𝑢^∗(⋅,𝑡)²₂+𝐶(𝜀)𝑢^∗₂+𝑢^∗_𝜏2×_{‖𝑦‖≤1}𝑢^∗()𝑦,𝑡𝑑𝑦+𝐶𝜀,𝑇^𝑡₀𝑢^∗₂+𝑢^∗_𝜏2𝑑𝜉_{‖𝑦‖≤1}∇𝑢^∗()𝑦,𝑡𝑑𝑦+𝐶𝜀,𝑇_{‖𝑦‖≤1}𝑢^∗𝑦,𝑡²𝑑𝑦. (57) Denoting 𝑤(𝑡)=‖𝑢^∗(⋅,𝑡)‖²₂ and 𝑤₁(𝑡)=‖∇𝑢^∗(⋅,𝑡)‖²₂, one can get 12𝑑𝑤𝑑𝑡≤−𝐶₁(𝑇)𝑤₁(𝑡)+𝐶₂(𝑇)𝑤(𝑡)+𝐽(𝑡)+𝐻(𝑡)+𝐶(𝜀,𝑇)𝑤(𝑡), (58) where 𝐽(𝑡)=𝐶(𝜀)(‖𝑢^∗‖₂+‖𝑢^∗_𝜏‖₂)∫_{‖𝑦‖≤1}|𝑢^∗(𝑦,𝑡)|𝑑𝑦  =  𝐶(𝜀)[𝑤(𝑡)+𝑤(𝑡−𝜏)]𝑤(𝑡), 𝐻(𝑡)=𝐶(𝜀,𝑇)∫^𝑡₀(‖𝑢^∗‖₂  +  ‖𝑢^∗_𝜏‖₂)𝑑𝜉∫_{‖𝑦‖≤1}|∇𝑢^∗(𝑦,𝑡)|𝑑𝑦  =  𝐶(𝜀,𝑇)∫^𝑡₀𝑤(𝑡)  +  𝑤(𝑡−𝜏)𝑑𝜉𝑤₁(𝑡). By the fact that 𝑢^∗=0, for −𝜏≤𝑡≤0, then 𝑤(𝑡−𝜏)=𝑢^∗(⋅,𝑡−𝜏)²₂=_{‖𝑦‖≤1,𝑡>0}𝑢^∗(⋅,𝑡−𝜏)²𝑑𝑦=_{‖𝑦‖≤1,𝑡>0}𝑢^∗(⋅,𝑡)²𝑑𝑦+_{‖𝑦‖≤1,−𝜏≤𝑡≤0}𝑢^∗(⋅,𝑡)²=𝑑𝑦_{‖𝑦‖≤1,𝑡>0}𝑢^∗(⋅,𝑡)²𝑑𝑦=𝑤(𝑡)=𝑢^∗(⋅,𝑡)²₂. (59) It follows that 𝐽(𝑡)(𝜀)𝑤(𝑡),𝐻(𝑡)()𝐶(𝛿)≤2𝐶≤𝐶𝜀,𝑇^𝑡₀𝑤(𝜉)𝑑𝜉+𝐶(𝜀,𝑇)𝛿𝑤₁(𝑡), (60) where we have used the 𝛿-Cauchy inequality, 𝛿 is an arbitrary positive number, and 𝐶(𝛿) is a positive constant depending on 𝛿. Then we have 12𝑑𝑤𝑑𝑡≤−𝐶₁(𝑇)𝑤₁(𝑡)+𝐶₃(𝑇)𝑤(𝑡)+2𝐶(𝜀)𝐶(𝛿)^𝑡₀𝑤(𝜉)𝑑𝜉+𝐶(𝜀,𝑇)𝛿𝑤₁(𝑡), (61) where 𝐶₃(𝑇)=𝐶₂(𝑇)+2𝐶(𝜀)+𝐶(𝜀,𝑇). Therefore, by choosing 𝛿 sufficiently small such that 𝐶(𝜀,𝑇)𝛿<𝐶₁(𝑇) one can get 12𝑑𝑤𝑑𝑡≤𝐶₃(𝑇)𝑤(𝑡)+2𝐶(𝜀)𝐶(𝛿)^𝑡₀𝑤(𝜉)𝑑𝜉. (62) Since 𝑤(0)=0, by Gronwall lemma we conclude that 𝑤(𝑡)≡0, for 0≤𝑡≤𝑇. Hence 𝑢₁=𝑢₂.

Lemma 5. If conditions (𝐴₁)--(𝐴₃) are satisfied, then for any 𝑇>0 the problem (19)–(25) has a unique solution (𝑢,𝑢_𝜏) on 𝑋_𝑇 satisfying 𝑢𝑦,𝑡,𝑢𝑦,𝑡−𝜏∈𝑊_𝑝^2,1𝑄_𝑇(),(),𝑝∈1,∞0≤𝑢𝑧,𝑡≤𝜎_∞,0≤𝑧≤1,−𝜏≤𝑡≤𝑇. (63)

Proof. By Lemma 3 for any 𝜀>0 the problem (26), (20)–(25) has a unique solution 𝑢_𝜀 on [0,1]×[−𝜏,𝑇]. Take a number 𝑝>5, by compact embedding 𝑊_𝑝^2,1(𝑄_𝑇)↪↪𝐶^{𝛼,(𝛼/2)}(𝑄_𝑇)(0<𝛼<2−(5/𝑝)). It follows that one can find a sequence of positive numbers 𝜀_𝑘→0,𝑘→∞, and function 𝑢∈𝑊_𝑝^2,1(𝑄_𝑇) such that if we denote 𝑢_𝑘=𝑢_𝜀𝑘, then 𝑢_𝑘(|𝑦|,𝑡)→𝑢(|𝑦|,𝑡), ∇𝑢_𝑘(|𝑦|,𝑡)→∇𝑢(|𝑦|,𝑡) uniformly for (𝑦,𝑡)∈𝑄_𝑇, 𝑢_𝑘(|𝑦|,𝑡−𝜏)→𝑢(|𝑦|,𝑡−𝜏), ∇𝑢_𝑘(|𝑦|,𝑡−𝜏)→∇𝑢(|𝑦|,𝑡−𝜏) uniformly for (𝑦,𝑡)∈𝑄_𝑇, 𝑢_𝑘𝑡(|𝑦,𝑡|)𝑤^→𝑢_𝑡(|𝑦|,𝑡)(𝐿^𝑝(𝑄_𝑇)), (𝜕²𝑢_𝑘/𝜕𝑦_𝑖𝜕𝑦_𝑗)𝑤^→(𝜕²𝑢/𝜕𝑦_𝑖𝜕𝑦_𝑗)(𝐿^𝑝(𝑄_𝑇)).
Clearly {𝐻_𝜀𝑘(𝑢−𝜎_𝑁)} is a bounded sequence in 𝐿^∞(𝑄_𝑇), so we can find a subsequence of {𝜀_𝑘} which is still denoted as {𝜀_𝑘} and a function ℎ∈𝐿^∞(𝑄_𝑇), such that 𝐻_𝜀𝑘𝑢−𝜎𝑤_𝑁^→ℎ𝐿^∞𝑄_𝑇,𝐺_𝜀𝑘𝑢_𝜀𝑘,𝑢_𝜏𝜀𝑘,𝑡∗^→𝑤𝑚(𝑡)𝐿^∞𝑄_𝑇, (64) where 𝑚(𝑡)=𝑠¹₀𝑅³_𝜏𝑅³𝑢_𝜏ℎ(𝑧,𝑡−𝜏)𝑧²−𝜎ℎ(𝑧,𝑡)𝑧²−𝜆[1−ℎ(𝑧,𝑡)]𝑧²𝑑𝑧. (65) Taking 𝜀=𝜀_𝑘 in (26), replacing 𝑢,𝑢_𝜏 with 𝑢_𝑘,𝑢_𝜏𝑘, respectively, and letting 𝑘→∞, one can get 𝜕𝑢𝜕𝑡=𝑑𝑅₀⁻²exp−2^𝑡₀𝑚(𝜉)1𝑑𝜉𝑧²𝜕𝑧𝜕𝑧²𝜕𝑢(𝑡)𝜕𝑧+𝑚𝑧𝑢_𝑧−𝑓(𝑢)ℎ,0<𝑧<1,𝑡>0,𝜕𝑢()()𝜕𝑧0,𝑡=0,𝑢1,𝑡=𝜎_∞,𝑡>0,𝑢(𝑧,𝑡)=𝜓(𝑧𝑅(𝑡),𝑡),0≤𝑧≤1,−𝜏≤𝑡≤0𝜕𝑢(𝑡)=𝜕𝑧𝑧𝜌−0,𝑡𝜕𝑢(𝑡)𝜕𝑧𝑧𝜌+0,𝑡,0≤𝑧≤1,𝑡>0,𝑢𝑧𝜌(𝑡)−0,𝑡=𝑢𝑧𝜌(𝑡)+0,𝑡=𝜎_𝑁,0≤𝑧≤1,𝑡>0,𝜌(𝑡)=𝜙(𝑡),−𝜏≤𝑡≤0,𝑅(𝑡)=𝜑(𝑡),−𝜏≤𝑡≤0. (66)(67)(68)(69)(70)(71)(72) We assert that ℎ𝑦=𝑦,𝑡1,𝑢,𝑡>𝜎_𝑁,0,𝑢𝑦,𝑡<𝜎_𝑁,a.e. (73) Actually, it is easy to verify that, for any 𝛿>0,ℎ(|𝑦|,𝑡)=1 for a.e. (𝑦,𝑡)∈{(𝑦,𝑡)∈𝑄_𝑇)∶𝑢(|𝑦|,𝑡)≥𝜎_𝑁+𝛿}. By the arbitrariness of 𝛿, we infer that ℎ(|𝑦|,𝑡)=1 for a.e. (𝑦,𝑡)∈{(𝑦,𝑡)∈𝑄_𝑇)∶𝑢(|𝑦|,𝑡)>𝜎_𝑁}. Similarly we can prove that ℎ(|𝑦|,𝑡)=0 for a.e. (𝑦,𝑡)∈{(𝑦,𝑡)∈𝑄_𝑇∶𝑢(|𝑦|,𝑡)<𝜎_𝑁}. This proves the assertion

Since 𝑢_𝑡=0, ∇𝑢=0, and Δ𝑢=0 a.e. on the set {(𝑦,𝑡)∈𝑄_𝑇∶𝑢(|𝑦|,𝑡)=𝜎_𝑁}, by (66) it follows that ℎ=0 a.e. on the set {(𝑦,𝑡)∈𝑄_𝑇∶𝑢(|𝑦|,𝑡)=𝜎_𝑁}. Hence ℎ𝑦,𝑡=𝐻𝑢𝑦,𝑡−𝜎_𝑁,𝑦,𝑡∈𝑄_𝑇. (74)

Substituting (74) into (65), one can get 𝑚(𝑡)=𝐺𝑢,𝑢_𝜏,𝑡. (75) By (66), (74), and (75) we conclude that 𝑢=𝑢(𝑧,𝑡) is a solution of (19)–(25). This proves the existence of a solution. The rest of the assertions of Lemma 5 follow immediately from taking the weak limit in (26).

By Lemmas 2 and 5, we see that Theorem 1 immediately follows.

5. Conclusions

In this paper a mathematical model for the growth of necrotic tumors with time delays in proliferation is studied. By transforming this problem into an initial-boundary value problem in fixed domain of a coupled system of a parabolic equation and one integrodifferential equation with time delays, in which all equations involve discontinuous terms, and using the approximation method combined with Schauder fixed point theorem, we prove that this problem has a unique global solution in any time interval [0,𝑇].

The model we study in this paper is established by modifying the model studied in [15] by introducing a time delay in proliferation as that in Byrne [1]. It should be pointed out that, in [15], the authors only studied the existence and uniqueness of stationary solutions to the model (2)–(4), but in our study we mainly discuss the existence and uniqueness of the solution to the model (2)–(4) under some assumptions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their thanks to anonymous referees for their careful comments and valuable suggestions on modification of the original paper. This work is supported by the NSF of China (11226182, 11301474, and 11171295) and the Foundation for Distinguished Young Teacher in Higher Education of Guangdong, China (Yq2013163).