# 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

Authors: Shihe Xu; Minhai Huang
Journal: Mathematical Problems in Engineering
 (2014)

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## 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].


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## Body

## 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 concentrationσ(r,t), which we present in dimensionless form:
(1)c∂σ∂t=1r2∂∂r(r2∂σ∂r)-F(σ)H(σ-σN),xxxxxxxxxxxxxxxx0<r<R(t),t>0,
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. σN is the minimal nutrient concentration needed for proliferation; that is, σN is a positive constant representing a threshold value for necrosis. In the region where σ>σN, nutrient is enough to sustain (at least a portion of) tumor cells alive, whereas in the region where σ≤σN, nutrient is insufficient to sustain any tumor cells alive and all cells there are dead cells.Letd=(1/c) and F(σ)=cf(σ); we have
(2)∂σ∂t=d1r2∂∂r(r2∂σ∂r)-f(σ)H(σ-σN),xxxxxxxxxxxxxxxx0<r<R(t),t>0.
The initial and boundary conditions for the necrotic phase of the tumor are
(3)∂σ∂r(0,t)=0,σ(R(t),t)=σ∞,t>0,∂σ∂r(ρ(t)-0,t)=∂σ∂r(ρ(t)+0,t),t>0,σ(ρ(t)-0,t)=σ(ρ(t)+0,t)=σN,t>0,σ(r,t)=ψ(r,t),0≤r≤R(t),-τ≤t≤0,ρ(t)=ϕ(t),-τ≤t≤0,R(t)=φ(t),-τ≤t≤0.
The second equation of the model
(4)dRdt=sR2(∫ρ(t-τ)R(t-τ)‍σ(r,t-τ)r2drnnnnmln-∫ρ(t)R(t)‍σ~r2dr-∫0ρ(t)‍λr2dr),nnnnnnnnnnnnnnnnnnnmnnnnnnt>0,
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):
(5)dRdt=sR2(∫ρ(t)R(t)‍S(σ(r,t))r2drnnnnlnn-∫ρ(t)R(t)‍σ~r2dr-∫0ρ(t)‍λr2dr),nnnnnnnnnnnnnnnnnnnnmnnnnnlt>0.
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(6)B¯1(0)={y∈R3,|y|≤1},Q¯T=B¯1(0)×[0,T],Wp2,1(Q¯T)={u∈Lp(Q¯T):∇u,∇2u,ut∈Lp(Q¯T)}.
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)], ψ0′(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)∈Wp2,1(QT)(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:
(7)z=rR(t),t=t,u(z,t)=σ(zR(t),t),R(t)=R(t).
Then the problem (2)–(4) is transformed into the following problem:
(8)∂u∂t=d1R2(t)1z2∂∂z(z2∂u∂z)+R′(t)R(t)zuz-f(u)H(u-σN),nnnnn0<z<1,t>0,(9)∂u∂z(0,t)=0,u(1,t)=σ∞,t>0,(10)dRdt=sR(t)(∫01‍Rτ3R3uτH(uτ-σN)z2nnnnnnnnn-σ~H(u-σN)z2nnnnnnnnnl∫01‍Rτ3R3uτH(uτ-σN)z2-λ[1-H(u-σN)]z2dz),nnnnnnnnnnnnnnnnnnnnnnnnmlnnnt>0,(11)u(z,t)=ψ(zR(t),t),0≤z≤1,-τ≤t≤0,(12)∂u∂z(zρ(t)-0,t)=∂u∂z(zρ(t)+0,t),n0≤z≤1,t>0,(13)u(zρ(t)-0,t)=u(zρ(t)+0,t)=σN,xxxxxxxxxxxxxxxxx0≤z≤1,t>0,(14)ρ(t)=ϕ(t),-τ≤t≤0,(15)R(t)=φ(t),-τ≤t≤0,
where Rτ=R(t-τ), uτ=u(z,t-τ). Let XT=C([0,1]×[0,T])×C([0,1]×[-τ,T]). Denote the function G as follows
(16)G(u,uτ,t)=s(∫01‍Rτ3R3uτH(uτ-σN)z2nnnnnnnnnnnnnn-σ~H(u-σN)z2nnnnnnnnnnnnnn-λ[1-H(u-σN)]z2dz∫01‍Rτ3R3)
for t>0. Using this notation, (10) can be simply written as follows:
(17)R′(t)=R(t)G(u,uτ,t),t>0.
It is clear that R(t) can be expressed by
(18)R(t)=R0exp⁡{∫0t‍G(u,uτ,ξ)dξ},t>0.
Hence the problem (8)–(15) is reduced to the problem below:
(19)∂u∂t=dR0-2exp⁡{-2∫0t‍G(u,uτ,ξ)dξ}×1z2∂∂z(z2∂u∂z)+G(u,uτ,t)zuz-f(u)H(u-σN),nnnn0<z<1,t>0,(20)∂u∂z(0,t)=0,u(1,t)=σ∞,t>0,(21)u(z,t)=ψ(zR(t),t),0≤z≤1,-τ≤t≤0,(22)∂u∂z(zρ(t)-0,t)=∂u∂z(zρ(t)+0,t),0≤z≤1,t>0,(23)u(zρ(t)-0,t)=u(zρ(t)+0,t)=σN,0≤z≤1,t>0,(24)ρ(t)=ϕ(t),-τ≤t≤0,(25)R(t)=φ(t),-τ≤t≤0.
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:(26)∂u∂t=dR0-2exp⁡{-2∫0t‍Gɛ(u,uτ,ξ)dξ}×1z2∂∂z(z2∂u∂z)+Gɛ(u,uτ,t)zuz-f(u)H(u-σN),nnnnn0<z<1,t>0,
with initial and boundary condition as (20)–(25), where ɛ is an arbitrary positive number,
(27)Gɛ(u,uτ,t)=s(∫01‍Rτ3R3uτHɛ(uτ-σN)z2nnnnn-σ~Hɛ(u-σN)z2nnnnn-λ[1-Hɛ(u-σN)]z2dz∫01‍Rτ3R3),Hɛ(x)={1,x≥ɛ,ɛx,0<x<ɛ,0,x≤0.
It is clear that Hɛ is a Lipschitz continuous function.Recall thatu(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],  ψ0′(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 uɛ on [0,1]×[-τ,T] satisfying the following assertions:(i)
uɛ(y,t)∈Wp2,1(Q¯T), p∈(1,∞) and for any ɛ>0 there holds
(28)∥uɛ(|y|,t)∥Wp2,1(Q¯T)≤C(T,p)(∥u0(|y|,0)∥W2,∞(0,1)+σ∞),whereC(T,p) is a constant independent of ɛ,u0(y,0)=ψ(zR(0),0);(ii)
For anyɛ>0, there holds 0≤|uɛ(|y|,t)|≤σ∞, 0≤y≤1, -τ≤t≤T.Remark 4.
From (i) and assumption(A2) it is easy to get
(29)∥uɛ(|y|,t-τ)∥Wp2,1(Q¯T)≤C(T,p)(∥u0(|y|,0)∥W2,∞(0,1)+σ∞).Proof.
EndowXT with the following norm:
(30)∥(u,uτ)∥=max⁡[0,1]×[-τ,T]|u|.
Then XT is clearly a Banach space. Define a mapping F:XT→XT as follows: for any (u,uτ)∈XT, define F(u,uτ)=(u~,u~τ), where u~(z,t)=u~~(zyo,t), yo∈R3, |yo|=1, uτ=u~(z,t-τ), and u~~ is the solution of the following problem:
(31)∂u~~∂t=dR0-2exp⁡{-2∫0t‍Gɛ(u,uτ,ξ)dξ}Δu~~+Gɛ(u,uτ,t)z·∇u~~-u~~f1(u)H(u-σN),u~~(y,t)=σ∞,|y|=1,t>0,∂u~~∂z(zρ(t)-0,t)=∂u~~∂z(zρ(t)+0,t),cccccccccccccccccc0≤z≤1,t>0,u~~(zρ(t)-0,t)=u~~(zρ(t)+0,t)=σN,cccccccccccccccccccc0≤z≤1,t>0,u~~(y,t)=u0(y,t),|y|≤1,-τ≤t≤0,
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
(32)R0-2exp⁡{-2∫0t‍Gɛ(u,uτ,ξ)dξ}≥λɛ(T,∥(u,uτ)∥,τ)>0,cccccccccccccccccccccccccccccccccccccccccccccc0≤t≤T.
Using the Lp 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, u~~1(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
(33)u~~(y,t)=u~(|y|,t),|y|≤1,-τ≤t≤T.
By Lp estimates we have, for p∈(1,∞),
(34)∥u~~(y,t)∥≤C(T,p,∥u∥)∥u0(|y|,0)∥W2,∞(B1(0))+σ∞+∥u~~(|y|,t)∥L∞(Q¯T).
By the maximum principle one can get
(35)0≤u~~(y,t)≤σ∞,|y|≤1,0≤t≤T.
Hence
(36)∥u~(|y|,t)∥Wp2,1(Q¯T)≤C(T,p,∥(u,uτ)∥)×(∥u0(|y|,0)∥W2,∞(0,1)+σ∞),(37)0≤u~(y,t)≤σ∞,|y|≤1,-τ≤t≤T.
Taking particularly p>(5/2), by the embedding Wp2,1(Q¯T)↪↪C(Q¯T), (p>(5/2)), we infer that (u~,u~τ)∈XT. Hence the mapping F is well defined.
LetE={(u,uτ):(u,uτ)∈C([0,1]×[0,T])×C([0,1]×[-τ,T]),0≤u,uτ≤σ∞}. E is clearly a bounded closed convex subset of XT. We first prove that F(E) is precontract in XT. Actually, since, for any (u,uτ)∈E, ∥(u,uτ)∥≤σ∞ by (36) we infer that for any 1<p<∞ there hold
(38)∥u~(|y|,t)∥≤C(T,p).
Taking p>(5/2) and using the compact embedding Wp2,1(Q¯T)↪↪C(Q¯T), (p>(5/2)), we conclude that F(E) is precontract in XT.
Next we prove thatF is continuous. Let (u1,u1τ), (u2,u2τ)∈XT and denote (u~1,u~1τ)=F(u1,u1τ), (u~2,u~2τ)=F(u2,u2τ), u~~*=u~1-u~2, u~~1τ*=u~1τ-u~2τ. It is obvious that u~~* is a solution of the following problem:
(39)∂u~~*∂t=dR0-2exp⁡{-2∫0t‍Gɛ(u1,u1τ,ξ)dξ}Δu~~*+Gɛ(u1,u1τ,t)(y·∇u~~*)-u~~*f(u1)Hɛ(u1-σN)+h(y,t),u~~(y,t)=0,|y|=1,t>0,∂u~~∂z(zρ(t)-0,t)=∂u~~∂z(zρ(t)+0,t),cccccccccccccccccc0≤z≤1,t>0,u~~(zρ(t)-0,t)=u~~(zρ(t)+0,t)=0,ccccccccccccccccccc0≤z≤1,t>0,u~~(y,t)=0,|y|≤1,-τ≤t≤0,
where h(y,t)=dR0-2[exp⁡⁡{-2∫0t‍Gɛ(u1,u1τ,ξ)dξ}  −  exp⁡⁡{-2∫0t‍Gɛ(u2,u2τ,ξ)dξ}]u~~2  +  [Gɛ(u1,u1τ,t)  −  Gɛ(u2,u2τ,t)](y·∇u~~2)  −  u~~2[f1(u1)Hɛ(u1-σN)-f1(u2)Hɛ(u2-σN)].
Clearly,(40)Gɛ(u,uτ,t)≤C(∥(u,uτ)∥,T)
for (u,uτ)∈XT, t∈[0,T], where C(∥(u,uτ)∥) is independent of ɛ. Then one can get
(41)e-2TC(∥(u,uτ)∥,T)≤exp⁡{-2∫0t‍Gɛ(u,uτ,ξ)}dξ≤e2TC(∥(u,uτ)∥,T),|exp⁡{-2∫0t‍Gɛ(u,uτ,ξ)}dξ-exp⁡{-2∫0t′‍Gɛ(u,uτ,ξ)}dξ|≤C(∥(u,uτ)∥)|t-t′|
for u,v∈XT, t∈[0,T]. Besides, it is also clear that
(42)0≤Hɛ(u-σN)≤1,|y|≤1,0≤t≤T.
Hence, by the Lp estimate one can get
(43)∥u~~*∥Wp2,1(Q¯T)≤C(T,p,∥(u1,u1τ)∥)∥h∥Lp(Q¯T).
By the fact that Hɛ is Lipschitz continuous and (36), one can verify
(44)∥h∥∣Lp(Q¯T)≤C(T,p,∥(u1,u1τ)∥,∥(u2,u2τ)∥,ɛ)×∥(u1-u2,u1τ-u2τ)∥.
Substituting these estimates into (43) we have
(45)∥u~~*∥Wp2,1(Q¯T)≤C(T,p,∥(u1,u1τ)∥,∥(u2,u2τ)∥,ɛ)×∥(u1-u2,u1τ-u2τ)∥.
Taking p>(5/2) and using the embedding Wp2,1(Q¯T)↪↪C(Q¯T) we get
(46)∥(u~1-u~2,u~1τ-u~2τ)∥≤∥u~~*∥Wp2,1(Q¯T)≤C(T,p,∥(u,uτ)∥,ɛ)∥(u1-u2,u1τ-u2τ)∥.
Hence, F is continuous.
Since we have proven thatF 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=uɛ.
Next, we prove uniqueness. Letu*=u1-u2. Then u* satisfy
(47)∂u*∂t=dR0-2exp⁡{-2∫0t‍Gɛ(u1,u1τ,ξ)dξ}Δu*+Gɛ(u1,u1τ,t)(y·u*)+p(y,t)
for |y|<1,t>0, where p(y,t)=dR0-2[exp⁡⁡{-2∫0t‍Gɛ(u1,u1τ,ξ)dξ}  −  exp⁡⁡{-2∫0t‍Gɛ(u2,u2τ,ξ)dξ}]Δu2  +  [Gɛ(u1,u1τ,t)-Gɛ(u2,u2τ,t)](y·∇u2)  −  [f(u1)-f(u2)]Hɛ(u1(|y|,t)  −  σN)-f(u2)[Hɛ(u1(|y|,t)-σN)  −  Hɛ(u2(|y|,t)-σN)].
Multiplying (47) with u* and integrating in y, one can get
(48)12∫∥y∥≤1‍(u*(y,t))2dy=-dR0-2exp⁡{-2∫0t‍Gɛ(u1,u1τ,ξ)dξ}×∫∥y∥≤1‍|∇u*(y,t)|2dy-32Gɛ(u1,u1τ,t)∫∥y∥≤1‍(u*(y,t))2dy+∫∥y∥≤1‍p(y,t)u*(y,t)dy≤-dR0-2exp⁡(-2TC(∥(u1,u1τ)∥))∥∇u*(·,t)∥L2(B1(0))2+32C(∥(u1,u1τ)∥)∥u*(·,t)∥L2(B1(0))2+I(t),
where I(t)=I1+I2+I3+I4 and
(49)I1=dR0-2[exp⁡{-2∫0t‍Gɛ(u1,u1τ,ξ)dξ}nnnnnnnnn-exp⁡{-2∫0t‍Gɛ(u2,u2τ,ξ)dξ}]×∫∥y∥≤1‍Δu2·∇u*(y,t)dy,I2=[Gɛ(u1,u1τ,t)-Gɛ(u2,u2τ,t)]×∫∥y∥≤1‍(y·∇u2)∇u*(y,t)dy,I3=-∫∥y∥≤1‍[f(u1)-f(u2)]Hɛ(u1(|y|,t)-σN)xxxxxxxx×u*(y,t)dy,I4=-∫∥y∥≤1‍f(u2)[Hɛ(u1(|y|,t)-σN)nnnnnnnnnmmn-Hɛ(u2(|y|,t)-σN)]u*(y,t)dy.
By the mean value theorem and the fact that f(u)≥0, we have
(50)I4=-∫∥y∥≤1‍f(u2)∫01‍Hɛ′dθ(u*)2dy≤0.
Using (28) and embedding Wp2,1(Q¯T)↪↪C(Q¯T), (p>(5/2)), we get
(51)max⁡Q¯T|∇u2(|y|,t)|≤C(T),max⁡Q¯T|∇u2τ(|y|,t)|≤C(T).
From definition of Gɛ and Lipschitz continuous of Hɛ, we have
(52)|Gɛ(u1,u1τ,t)-Gɛ(u2,u2τ,t)|≤C(ɛ)(∥u*∥2+∥uτ*∥2).
Then by using (51) one can get
(53)I2≤C(ɛ)(∥u*∥2+∥uτ*∥2)∫∥y∥≤1‍|u*(y,t)|dy.
By a direct computation and (52) one can get
(54)I1≤C(ɛ,T)∫0t‍(∥u*∥2+∥uτ*∥2)dξ∫∥y∥≤1‍|∇u*(y,t)|dy.
Lipschitz continuity of f(u) combined with (51) yields
(55)I3≤C(ɛ,T)∫∥y∥≤1‍|u*(y,t)|2dy.
Summing up (51), (53)–(55) into I(t), one can get
(56)I(t)≤C(ɛ)(∥u*∥2+∥uτ*∥2)∫∥y∥≤1‍|u*(y,t)|dy+C(ɛ,T)∫0t‍(∥u*∥2+∥uτ*∥2)dξ∫∥y∥≤1‍|∇u*(y,t)|dy+C(ɛ,T)∫∥y∥≤1‍|u*(y,t)|2dy.
Using Cauchy inequality and summing up (51), (53)–(55) into I(t), one can get
(57)12ddt∥u*(·,t)∥22≤-C1(T)∥∇u*(·,t)∥22+C2∥u*(·,t)∥22+C(ɛ)(∥u*∥2+∥uτ*∥2)×∫∥y∥≤1‍|u*(y,t)|dy+C(ɛ,T)∫0t‍(∥u*∥2+∥uτ*∥2)dξ∫∥y∥≤1‍|∇u*(y,t)|dy+C(ɛ,T)∫∥y∥≤1‍|u*(y,t)|2dy.
Denoting w(t)=∥u*(·,t)∥22 and w1(t)=∥∇u*(·,t)∥22, one can get
(58)12dwdt≤-C1(T)w1(t)+C2(T)w(t)+J(t)+H(t)+C(ɛ,T)w(t),
where J(t)=C(ɛ)(∥u*∥2+∥uτ*∥2)∫∥y∥≤1‍|u*(y,t)|dy  =  C(ɛ)[w(t)+w(t-τ)]w(t), H(t)=C(ɛ,T)∫0t‍(∥u*∥2  +  ∥uτ*∥2)dξ∫∥y∥≤1‍|∇u*(y,t)|dy  =  C(ɛ,T)∫0t‍w(t)  +  w(t-τ)dξw1(t). By the fact that u*=0, for -τ≤t≤0, then
(59)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)|2dy=∫∥y∥≤1,t>0‍|u*(·,t)|2dy=w(t)=∥u*(·,t)∥22.
It follows that
(60)J(t)≤2C(ɛ)w(t),H(t)≤C(ɛ,T)C(δ)∫0t‍w(ξ)dξ+C(ɛ,T)δw1(t),
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
(61)12dwdt≤-C1(T)w1(t)+C3(T)w(t)+2C(ɛ)C(δ)∫0t‍w(ξ)dξ+C(ɛ,T)δw1(t),
where C3(T)=C2(T)+2C(ɛ)+C(ɛ,T). Therefore, by choosing δ sufficiently small such that C(ɛ,T)δ<C1(T) one can get
(62)12dwdt≤C3(T)w(t)+2C(ɛ)C(δ)∫0t‍w(ξ)dξ.
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,uτ) on XT satisfying
(63)u(|y|,t),u(|y|,t-τ)∈Wp2,1(Q¯T),p∈(1,∞),0≤u(z,t)≤σ∞,0≤z≤1,-τ≤t≤T.Proof.
By Lemma3 for any ɛ>0 the problem (26), (20)–(25) has a unique solution uɛ on [0,1]×[-τ,T]. Take a number p>5, by compact embedding Wp2,1(Q¯T)↪↪Cα,(α/2)(Q¯T)(0<α<2-(5/p)). It follows that one can find a sequence of positive numbers εk→0,k→∞, and function u∈Wp2,1(Q¯T) such that if we denote uk=uεk, thenuk(|y|,t)→u(|y|,t), ∇uk(|y|,t)→∇u(|y|,t) uniformly for (y,t)∈Q¯T,uk(|y|,t-τ)→u(|y|,t-τ), ∇uk(|y|,t-τ)→∇u(|y|,t-τ) uniformly for (y,t)∈Q¯T,ukt(|y,t|)w→ut(|y|,t)(Lp(Q¯T)), (∂2uk/∂yi∂yj)w→(∂2u/∂yi∂yj)(Lp(Q¯T)).
Clearly{Hɛk(u-σN)} is a bounded sequence in L∞(Q¯T), so we can find a subsequence of {εk} which is still denoted as {εk} and a function h∈L∞(Q¯T), such that
(64)Hɛk(u-σN)w→h(L∞(Q¯T)),Gɛk(uεk,uτεk,t)*w→m(t)(L∞(Q¯T)),
where
(65)m(t)=s(∫01‍Rτ3R3uτh(z,t-τ)z2-σ~h(z,t)z2nnnnnn-λ[1-h(z,t)]z2dz∫01‍Rτ3R3).
Taking ɛ=εk in (26), replacing u,uτ with uk,uτk, respectively, and letting k→∞, one can get
(66)∂u∂t=dR0-2exp⁡{-2∫0t‍m(ξ)dξ}1z2∂∂z(z2∂u∂z)+m(t)zuz-f(u)h,nnnnn0<z<1,t>0,(67)∂u∂z(0,t)=0,u(1,t)=σ∞,t>0,(68)u(z,t)=ψ(zR(t),t),0≤z≤1,-τ≤t≤0(69)∂u∂z(zρ(t)-0,t)=∂u∂z(zρ(t)+0,t),0≤z≤1,t>0,(70)u(zρ(t)-0,t)=u(zρ(t)+0,t)=σN,0≤z≤1,t>0,(71)ρ(t)=ϕ(t),-τ≤t≤0,(72)R(t)=φ(t),-τ≤t≤0.
We assert that
(73)h(|y|,t)={1,u(|y|,t)>σN,0,u(|y|,t)<σN,a.e.
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)≥σN+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)>σN}. Similarly we can prove that h(|y|,t)=0 for a.e. (y,t)∈{(y,t)∈Q¯T:u(|y|,t)<σN}. This proves the assertionSinceut=0, ∇u=0, and Δu=0 a.e. on the set {(y,t)∈Q¯T:u(|y|,t)=σN}, by (66) it follows that h=0 a.e. on the set {(y,t)∈Q¯T:u(|y|,t)=σN}. Hence
(74)h(|y|,t)=H(u(|y|,t)-σN),(y,t)∈Q¯T.Substituting (74) into (65), one can get
(75)m(t)=G(u,uτ,t).
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 Lemmas2 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.



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