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Existence and uniqueness of global solutions for a mathematical model of antiangiogenesis in tumor growth
Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836 www.elsevier.com/locate/na
Existence and uniqueness of global solutions for a mathematical model of antiangiogenesis in tumor growth Xuemei Weia,∗ , Shangbin Cuib a Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou, Guangdong 510090, People’s Republic of China b Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, People’s Republic of China
Received 29 March 2007; accepted 25 May 2007
Abstract In this paper we study a mathematical model of antiangiogenesis in tumor growth. The model consists of three semi-linear parabolic PDEs. By applying the Banach fixed point method, the parabolic Schauder estimates, parabolic Lp estimates and the continuation method, we prove that this system has a unique global classical solution. 䉷 2007 Elsevier Ltd. All rights reserved. MSC: 35K35; 35Q80; 35R05 Keywords: Tumor growth; Global solution; Existence; Uniqueness
1. Introduction This paper is concerned with the problem of tumor angiogenesis. Angiogenesis is an essential stage in the process of a solid tumor growth. In the beginning stage of the tumor growth, the tumor does not have its own capillary network. It receives nutrition through simple diffusion of nutrient materials from the surrounding normal tissue of the host body, so that it cannot grow very large. A tumor in this stage is called an avascular tumor. A dormant avascular tumor generally has a size of only 1–3 mm in diameter. Such tumors are not very harmful to the body. Tumors which are really detrimental to health of the body are vascularized tumors. A dormant avascular tumor surmounts its avascular dormant state by initiating the formation of new blood vessels from nearby preexisting vasculature in the host tissue, and gradually form its own capillary network, becoming vascularized. The development of new blood vessels is known as angiogenesis. It is a significant stage for tumor growth. Hence, the study of the mechanism of angiogenesis in the process of tumor growth is very important. During the last two decades, a number of mathematical models describing the process of angiogenesis of in vivo and in vitro tumors have been presented in the literature [1–7,10–12]. In this paper we study a mathematical model of tumor angiogenesis proposed by Anderson et al. in Ref. [3]. This model describes the angiogenic response of endothelial cells to a secondary tumor. The model assumes that the endothelial cells respond chemotactically to two opposing ∗ Corresponding author.
E-mail addresses: [email protected] (X. Wei), [email protected] (S. Cui). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.05.013
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
chemical gradients: a gradient of tumor angiogenic factor (TAF); and a gradient of angiostatin. The model consists of three semi-linear parabolic partial differential equations complemented with boundary and initial conditions: jc j ja j j2 n jn = Dn 2 − (c)n − (a)n , 0 < x < L, t > 0, (1.1) jt jx jx jx jx jx jc j2 c = Dc 2 − 1 c − F (c, n, x), jt jx
0 < x < L, t > 0,
(1.2)
ja j2 a = Da 2 − 2 a − G(a, n, x), 0 < x < L, t > 0, jt jx jc ja n jn + (c) , x = 0, L, t > 0, (a) = Dn jx jx jx jc = 0, jx a = A,
x = 0; x = 0;
n = nI (x),
(1.3) (1.4)
c = c0 ,
x = L,
t > 0,
(1.5)
ja = 0, jx
x = L,
t > 0,
(1.6)
c = cI (x),
a = aI (x),
t = 0,
0 < x < L.
(1.7)
Here n(x, t), c(x, t) and a(x, t), respectively, represent the endothelial cell tip density, the TAF, and the angiostatin concentrations in the one-dimensional domain [0, L], (c) and (a) are, respectively, the TAF chemotactic and angiostatin chemotactic functions, Dn is the cell random motility coefficients, and Dc , Da , respectively, represent the TAF diffusion coefficient and the angiostatin diffusion coefficient. According to [3], (c) and (a), respectively, have the following expressions: (c) =
0 k1 , k1 + c
(a) = 0 a,
where 0 , k1 and 0 are positive constants. We now nondimensionalize (1.1)–(1.7) by using the following rescaling: x¯ =
x , L
=
L2 , Dc
t t¯ = ,
c¯ =
c , c0
a¯ =
c , a0
n¯ =
n , n0
where c0 is the TAF concentration at the tumor and n0 , a0 are appropriate reference variables. Setting f (c, ¯ n, ¯ x) ¯ = F (c, n, x),
g(a, ¯ n, ¯ x) ¯ = G(a, n, x)
and dropping all bars for clarity, we obtain the following nondimensional system: j2 n j jc j ja jn = D1 2 − n − an , 0 < x < 1, t > 0, jt jx 1 + kc jx jx jx jx jc j2 c = 2 − 1 c − f (c, n, x), jt jx
0 < x < 1, t > 0,
ja j2 a = D2 2 − 2 a − g(a, n, x), 0 < x < 1, t > 0, jt jx jc n ja jn = + a , x = 0, 1, t > 0, jx D1 1 + kc jx jx jc = 0, jx
x = 0;
c = 1,
x = 1,
t > 0,
(1.8) (1.9) (1.10) (1.11) (1.12)
X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
a = s,
x = 0;
n = n0 (x),
ja = 0, jx
c = c0 (x),
x = 1, a = a0 (x),
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t > 0,
(1.13)
t = 0, 0 < x < 1,
(1.14)
where D1 =
Dn , Dc
D2 =
Da , Dc
=
0 c0 , Dc
=
0 a02 , Dc
=
c0 , k1
1 = 1 ,
2 = 2 ,
s=
A . a0
We make the following assumptions: (A1) f (c, n, x) = cf 1 (c, n, x), g(a, n, x) = ag 1 (a, n, x), where f1 , g1 are defined and Lipschitz continuous in their arguments, f1 (c, n, x)0, g1 (a, n, x)0 for c 0, n 0, a 0, x 0, and f1 , g1 are bounded for n. (A2) n0 (x), c0 (x), a0 (x) ∈ C 2+ [0, 1], 0 c0 (x)s, 0 a0 (x) 1, c0 (0) = 0, a0 (0) = s, c0 (1) = 1, a0 (1) = 0. The main result of this paper is as follows: Theorem 1.1. The problem (1.8)–(1.14) has a unique classical solution which exists for all t 0. The structure of this paper is as follows. Section 2 is devoted to presenting some preliminary lemmas that will be used in later analysis. In Section 3 we prove local existence and uniqueness to (1.8)–(1.14). In the last section, we prove that the solution of the problem (1.8)–(1.14) is global, which proves Theorem 1.1. 2. Preliminary lemmas We denote: QT = (0, 1) × (0, T ),
¯ T = the closure of QT . Q
¯ T . Let Lemma 2.1. Let D be a constant, and a(x, t), b(x, t), f (x, t) be bounded continuous functions defined on Q Bu = (ju/jn) + u, where (1) = 0, = 1; (2) = 1, = 0, and c0 ∈ Dp (0, 1), for some 1 < p < ∞. Then the initial value problem jc j2 c jc = D 2 + a(x, t) + b(x, t)c + f (x, t), jt jx jx x = 0, 1: Bc = , c(x, 0) = c0 (x),
0 < x < 1, 0 < t < T ,
0 t T ,
(2.1) (2.2)
0 x 1,
(2.3)
has a unique solution c(x, t) ∈ Wp2,1 (QT ). Moreover, cW 2,1 (Q ) Cp (T )(c0 Dp (0,1) + W 1,p (0,T ) + f p ), p
T
(2.4)
Cp (T ) are constants depending only on p, T, D, a∞ , b∞ , and Cp (T ) are bounded for all T in a bounded set. Proof. See [9].
¯ T ), and c0 (x) ∈ C 2+ [0, 1]. Then the initial value problem Lemma 2.2. Let a(x, t), b(x, t), f (x, t) ∈ C ,/2 (Q jc j2 c jc = D 2 + a(x, t) + b(z, t)c + f (x, t), jt jx jx x = 0, 1 : Bc = , c(x, 0) = c0 (x),
0 < x < 1, 0 < t < T ,
0 t T , 0 x 1,
(2.5) (2.6) (2.7)
has a unique solution, and when (1) = 0, = 1, we have cC 2+,(1+)/2 (Q¯ T ) c0 C 2+ [0,1] + C (T )( C (1+)/2 [0,T ] + f C ,/2 (Q¯ T ) );
(2.8)
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when (2) = 1, = 0, we have cC 2+,(1+)/2 (Q¯ T ) c0 C 2+ [0,1] + C (T )( C (1+)/2 [0,T ] + f C ,/2 (Q¯ T ) ).
(2.9)
Here C (T ) are constants depending only on D, T, a(x, t)C ,/2 (Q¯ T ) , b(x, t)C ,/2 (Q¯ T ) .
Proof. See (3.10) in [8].
Lemma 2.3. Let u(x, t) ∈ C 2+,(1+)/2 (QT ). Then u(x, t) − u(x, 0)C 1+,(1+)/2 (QT ) C (T )uC 2+,(1+)/2 (QT ) .
(2.10)
where (T ) = max{T /2 , T (1−)/2 }. Proof. By the define of Hölder norm, we have [u(x, t) − u(x, 0)]C 1+,0 (QT ) CDx2 u(x, t) − Dx2 u(x, 0)L∞ (QT )
C[Dx2 u]C 0,/2 (QT ) · |T |/2 C[u]C 2,/2 (QT ) · |T |/2 ,
(2.11)
and |u(x, t) − u(x, 0)| |Dt u| · |T |(1−)/2 . |t|(1+)/2 Combining (2.11) and (2.12) we obtain (2.10).
(2.12)
3. Local existence and uniqueness For a given T > 0 and a positive number M to be specified later, we introduce a metric space (XM , d) as follows: The set XM consists of vector functions (n(x, t), c(x, t), a(x, t)), satisfying the following conditions: n, c, a ∈ C 1+,(1+)/2 (QT ), n, c, a satisfy conditions (1.12).(1.14), and nC 1+,(1+)/2 M,
cC 1+,(1+)/2 M,
aC 1+,(1+)/2 M.
(3.1)
For convenience we set u = (n, c, a),
u˜ = (n, ˜ c, ˜ a). ˜
The metric d is defined by d(u1 , u2 ) = u1 − u2 C 1+,(1+)/2 . It is obvious that (XM , d) is a complete metric space. ˜ c, ˜ a), ˜ where u˜ satisfies the following problem: Given any (n, c, a) ∈ XM , we define a mapping F : (n, c, a) → (n, jc˜ j2 c˜ = 2 − 1 c˜ − cf ˜ 1 (c, n, x), jt jx x=0:
jc˜ = 0; jx
t = 0 : c˜ = c0 (x),
0 < x < 1, t > 0,
x = 1 : c˜ = 1, 0 < x < 1,
ja˜ j2 a˜ = D2 2 − 2 a˜ − ag ˜ 1 (a, n, x), jt jx x = 0 : a˜ = s;
t > 0,
x=1:
ja˜ = 0, jx
(3.2) (3.3) (3.4)
0 < x < 1, t > 0, t > 0,
(3.5) (3.6)
X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
t = 0 : a˜ = a0 (x),
0 < x < 1.
(3.7)
jc˜ j ja˜ n˜ − a˜ n˜ , 1 + k c˜ jx jx jx jn˜ jc˜ n ja˜ x = 0, 1 : = + a , t > 0, jx D1 1 + kc jx jx j j2 n˜ jn˜ = D1 2 − jt jx jx
t = 0 : n˜ = n0 (x),
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0 < x < 1, t > 0,
0 < x < 1.
(3.8) (3.9) (3.10)
¯ T ) satisfying (i) By Lemma 2.2, (3.2)–(3.4) has a solution c˜ ∈ C 2+,(1+)/2 (Q c ˜ C 2+,(1+)/2 (Q¯ T ) c0 C 2+ [0,1] + C(T ) ≡ C1 (T ). Hence, by Lemma 2.3 we have c ˜ C 1+,(1+)/2 (Q¯ T ) c0 C (1+)/2 [0,1] + c˜ − c0 C 1+,(1+)/2 (Q¯ T ) c0 C (1+)/2 [0,1] + C (T )c ˜ C 2+,(1+)/2 (Q¯ T ) c0 C (1+)/2 [0,1] + C (T )C1 (T ). Since limT →0 (T )=0, from the above result we see easily that if M > c0 C (1+)/2 [0,1] and if we take T > 0 sufficiently small then c ˜ C 1+,(1+)/2 (Q¯ T ) M.
(3.11)
Moreover, applying the maximum principle to (3.2)–(3.4) we get 0 c˜ 1.
(3.12)
¯ T ) satisfying (ii) Similarly, by Lemma 2.2, (3.5)–(3.7) has a solution a˜ ∈ C 2+,(1+)/2 (Q a ˜ C 2+,(1+)/2 (Q¯ T ) a0 (x)C 2+ [0,1] + C(T ) ≡ C2 (T ). In the same way, by Lemma 2.3 we have a ˜ C 1+,(1+)/2 (Q¯ T ) a0 C (1+)/2 [0,1] + a˜ − a0 C 1+,(1+)/2 (Q¯ T ) a0 C (1+)/2 [0,1] + C (T )a ˜ C 2+,(1+)/2 (Q¯ T ) a0 C (1+)/2 [0,1] + C (T )C2 (T ). Since limT →0 (T )=0, from the above result we see easily that if M > a0 C (1+)/2 [0,1] and if we take T > 0 sufficiently small then a ˜ C 1+,(1+)/2 (Q¯ T ) M.
(3.13)
Also applying the maximum principle to (3.5)–(3.7) we get 0 a˜ s.
(3.14)
(iii) We deduce that the coefficients of Eq. (3.8) are in C ,/2 from (i) and (ii). Hence, by Lemma 2.2, we see that ¯ T ) satisfying (3.8)–(3.10) has a solution n˜ ∈ C 2+,(1+)/2 (Q jn˜ n ˜ C 2+,(1+)/2 (Q¯ T ) n0 C 2+ [0,1] + C(T ) jx x=0,1 (1+)/2 C [0,T ] ˜ jc˜ 2 ja n0 C 2+ [0,1] + C(T ) M + M jx x=0 C (1+)/2 [0,T ] jx x=1 C (1+)/2 [0,T ] ≡ C3 (T , M).
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
Hence, by Lemma 2.3 we have n ˜ C 1+,(1+)/2 (Q¯ T ) n0 C (1+)/2 [0,1] + n˜ − n0 C 1+,(1+)/2 (Q¯ T ) ˜ C 2+,(1+)/2 (Q¯ T ) n0 C (1+)/2 [0,1] + C (T )n n0 C (1+)/2 [0,1] + C (T )C3 (T , M). Since limT →0 (T )=0, from the above result we see easily that if M > n0 C (1+)/2 [0,1] and if we take T > 0 sufficiently small then n ˜ C 1+,(1+)/2 (Q¯ T ) M.
(3.15)
Combining (3.11), (3.13) with (3.15), we conclude that if M > max{c0 C (1+)/2 [0,1] , a0 C (1+)/2 [0,1] , n0 C (1+)/2 [0,1] } and if we take T > 0 sufficiently small, then u˜ ∈ XM , i.e. F maps XM to itself. In the following we prove that F is a contraction. Take u1 , u2 ∈ XM , and let u˜ 1 = F u1 , u˜ 2 = F u2 , u˜ ∗ = u˜ 1 − u˜ 2 . Setting = u1 − u2 C 1+,(1+)/2 (Q¯ T ) . Denoting c˜∗ = c˜1 − c˜2 , we have jc˜∗ j2 c˜∗ − (1 + f1 (c1 , n1 , x))c˜∗ − h1 , = jt jx 2 x=0:
jc˜∗ = 0; jx
t = 0 : c˜∗ = 0,
0 < x < 1, t > 0,
x = 1 : c˜∗ = 0, t > 0, 0 < x < 1,
where h1 = c˜2 (f1 (c1 , n1 , x) − f1 (c2 , n2 , x)). From Lemma 2.2, we deduce that c˜∗ C 2+,(1+)/2 (Q¯ T ) C(T )h1 C ,/2 (Q¯ T ) ,
(3.16)
and by the assumption (A1), we have h1 C ,/2 (Q¯ T ) c˜2 C ,/2 (Q¯ T ) c1 − c2 C ,/2 (Q¯ T ) + n1 − n2 C ,/2 (Q¯ T ) .
(3.17)
Substituting (3.11), (3.17) into (3.16), we get c˜∗ C 2+,(1+)/2 (Q¯ T ) C(T )M.
(3.18)
Next, denoting a˜ ∗ = a˜ 1 − a˜ 2 , we have ja˜ ∗ j2 a˜ ∗ = D2 2 − (2 + g1 (c1 , n1 , x))a˜ ∗ − h2 , jt jx ja˜ ∗ = 0, jx
x = 0 : a˜ ∗ = 0;
x=1:
t = 0 : a˜ ∗ = 0,
0 < x < 1,
t > 0,
where h2 = a˜ 2 (g1 (c1 , n1 , x) − g1 (c2 , n2 , x)).
0 < x < 1, t > 0,
X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
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From Lemma 2.2, we deduce that a˜ ∗ C 2+,(1+)/2 (Q¯ T ) C(T )h2 C ,/2 (Q¯ T ) ,
(3.19)
and by the assumption (A1), we have
h2 C ,/2 (Q¯ T ) a˜ 2 C ,/2 (Q¯ T ) c1 − c2 C ,/2 (Q¯ T ) + n1 − n2 C ,/2 (Q¯ T ) .
(3.20)
Substituting (3.13), (3.20) into (3.19), we get a˜ ∗ C 2+,(1+)/2 (Q¯ T ) C(T )M.
(3.21)
Finally, denoting n˜ ∗ = n˜ 1 − n˜ 2 , we have jn˜ ∗ j2 n˜ ∗ jc˜∗ ja˜ ∗ j j n˜ 1 = D1 2 − − a˜ 1 n˜ 1 − h3 , jt jx 1 + k c˜1 jx jx jx jx jn˜ ∗ ja˜ 1 ja˜ 2 x=0: n1 a1 = − n 2 a2 , t > 0, jx D1 jx jx n2 jc˜2 n1 jc˜1 jn˜ ∗ = − , t > 0, x=1: jx D1 1 + kc1 jx 1 + kc2 jx t = 0 : n˜ ∗ = 0, where j h3 = jx
0 < x < 1, t > 0,
0 < x < 1,
n˜ 2 n˜ 1 − 1 + k c˜1 1 + k c˜2
jc˜2 jx
j + jx
ja˜ 2 (a˜ 1 n˜ 1 − a˜ 2 n˜ 2 ) jx
.
By Lemma 2.2, we deduce that
1 x=0 C(1+)/2[0,T ] j c ˜ j c ˜ n n 1 1 2 2 − + + h3 C ,/2 (Q¯ T ) . D 1 + kc2 jx x=1 C (1+)/2 [0,T ] 1 1 + kc1 jx
ja˜ 1 ja˜ 2 n1 a1 − n 2 a2 n˜ C 2+,(1+)/2 (Q¯ T ) C(T ) D jx jx ∗
Obviously, the first terms of the right-hand side of (3.22) is n1 a1 ja˜ 1 − n2 a2 ja˜ 2 C(T )M. jx jx x=0 C (1+)/2 [0,T ] Similarly, the second terms of the right-hand side of (3.22) is n1 jc˜1 n2 jc˜2 C(T )M. 1 + kc jx − 1 + kc jx (1+)/2 1 2 x=1 C [0,T ] In the same way, the third terms of the right-hand side of (3.22) is
jc˜2 h3 C ,/2 (Q¯ T ) C(T ) · (n˜ 1 − n˜ 2 C 1+,(1+)/2 (Q¯ T ) + c˜1 − c˜2 C 1+,(1+)/2 (Q¯ T ) ) jx C 1+,(1+)/2 (Q¯ T ) ja˜ 2 + · (a˜ 1 − a˜ 2 C 1+,(1+)/2 (Q¯ T ) + n˜ 1 − n˜ 2 C 1+,(1+)/2 (Q¯ T ) . jx C 1+,(1+)/2 (Q¯ T )
(3.22)
(3.23)
(3.24)
(3.25)
Substituting (3.23)–(3.25) into (3.22) and applying (3.11), (3.13), we get n˜ ∗ C 2+,(1+)/2 (Q¯ T ) C(T )M.
(3.26)
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
Combining (3.16), (3.21) and (3.26), we deduce that u˜ ∗ C 2+,(1+)/2 (Q¯ T ) C(T ) .
(3.27)
By u˜ ∗ (z, 0) = 0 and Lemma 2.3, we conclude that u˜ ∗ C 1+,(1+)/2 (Q¯ T ) (T )u˜ ∗ C 2+,(1+)/2 (QT ) .
(3.28)
Substituting (3.27) into (3.28), we get u˜ ∗ C 1+,(1+)/2 (Q¯ T ) C(T ) (T ) .
(3.29)
Taking T such that C(T ) (T ) < 1, we conclude that F is a contraction. By the contraction mapping theorem F has a unique fixed point u = (n, c, a), which is the unique solution. T depends only on an upper bound on the size of the C 2+ norm of u(x, 0). We summarize the above result in the following theorem: Theorem 3.1. There exists T > 0, T depending on an upper bound on the size of the C 2+ norm of u(x, 0), such that the problem (3.2)–(3.9) has a unique solution for 0 t T . 4. Global existence and uniqueness Lemma 4.1. The following holds: 0 c 1,
0 a s,
n 0,
cW 2,1 (Q ) Cq (T ),
aW 2,1 (Q ) Cq (T ).
T
q
(4.1) (4.2)
T
q
Proof. The inequalities 0 c 1, 0 a s follow from the maximum principle. Next, we consider Eq. (1.8). We can easily prove that n = 0 is the lower solution. Hence n 0. Finally, by the assumption (A1) and Lemma 2.1, we conclude that (4.2) holds. Lemma 4.2. For any 1 < q < ∞, there exists a constant Cq (T ), depending on T, such that nLq (QT ) Cq (T ).
(4.3)
Proof. Multiplying (1.8) by nk and integrating over Qt , we get t 0
1 0
jn k n dx dt= jt
t 0
1 0
j2 n D1 2 nk dx dt− jx
1
t 0
0
j jx
jc n 1+kc jx
j + jx
ja an jx
nk dx dt.
(4.4)
The left-hand side of (4.4) is equivalent to t 0
1 0
jn k 1 n dx dt = jt k+1
1 0
d dt
t
nk+1 dx dt.
(4.5)
0
By parts, the first terms of the right-hand side of (4.4) are transformed into t
1
D1 0
0
j2 n k n dx dt = jx 2
t
D1 0
jn k 1 n | dt − D1 jx 0
1 jn 2
t 0
0
jx
nk−1 dx dt,
(4.6)
X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
1835
similarly, the second terms of the right-hand side of (4.4) are transformed into t 1 j jc j ja n + an nk dx dt jx 1 + kc jx jx jx 0 0 t ja jc = + an nk |10 dt n jx 1 + kc jx 0 t 1 jc ja jn k −k + a n dx dt. 1 + kc jx jx jx 0 0
(4.7)
Substituting (4.5)–(4.7) into (4.4), we get 1 k+1
1 0
d dt
t
n
k+1
0
t
dx dt − D1 0 t +k 0
1 jn 2
0 1 0
nk−1 dx dt jx jc ja jn k + a n dx dt. 1 + kc jx jx jx
(4.8)
By (4.2) and Young inequality, we deduce that t 1 t 1 jc ja jn k jn k + a n dx dt kC q n dx dt k 1 + kc jx jx jx 0 0 0 0 jx t 1 2 t 1 jn 1 k−1 kC q n dx dt + kC q nk+1 dx dt. 2 jx 2 0 0 0 0
(4.9)
Substituting (4.9) into (4.8) and taking sufficiently small, such that ( /2)kC q D1 , we conclude that 1 k+1
1 0
d dt
t
nk+1 dx dt
0
1 kC q 2
t 0
1
nk+1 dx dt.
(4.10)
0
By Gronwall’s Lemma, we get nLk+1 Cq (T ). Therefore, Lemma 4.2 holds.
(4.11)
Lemma 4.3. For any 1 < q < ∞, there exists a constant C˜ q , depending only on T, such that nW 2,1 (Q ) C˜ q (T ). q
T
Proof. We consider the equation of n jn j2 n j jc j ja = D1 2 − n − an . jt jx 1 + kc jx jx jx jx
(4.12)
(4.13)
By a direction calculation, we see that j jc ja jc ja jn − n + an = − n + a jx 1 + kc jx jx 1 + kc jx jx jx j jc ja jn − + a n ≡ (x, t) + (x, t)n. jx 1 + kc jx jx jx Obviously, (x, t), (x, t) are bounded continuous functions defined on QT . Eq. (4.13) can be transformed into jn j2 n jn = D1 2 + (x, t) + (x, t)n. jt jx jx
(4.14)
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
By Lemma 4.2 and parabolic Lq estimates, we conclude that nW 2,1 (Q ) C˜ q (T ). q
(4.15)
T
Hence Lemma 4.3 holds.
Lemma 4.4. There exists a constant C(T ), depending on T, such that uC 2+,(1+)/2 (Q¯ T ) C(T ). Proof. From Lemma 4.3 and Soblev embedding, it follows that uC ,/2 (Q¯ T ) C(T ),
0 < < 1.
Hence the coefficients of (1.8)–(1.10) satisfy the condition of Lemma 2.2. Applying Lemma 2.2, we get uC 2+,(1+)/2 (Q¯ T ) C(T ). Lemma 4.4 holds.
(4.16)
By Theorem 3.1 and Lemma 4.4, we see that Theorem 1.1 is proved. Acknowledgments This work is financially supported by the China National Natural Science Foundation under the Grant no. 10471157. It is also supported in part by the Ph.D. Foundation of Guangdong University of Technology under the Grant no. 063043. References [1] A. Anderson, M. Chaplain, Continuous and discrete mathematical modela of tumour-induced antiangiogenesis, Bull. Math. Biol. 60 (1998) 857–900. [2] A. Anderson, M. Chaplain, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett. 11 (1998) 109–114. [3] A. Anderson, M. Chaplain, C. García-Reimbert, C. Vargas, A gradient-driven mathematical model of antiangiogenesis, Math. Comput. Model. 32 (2000) 1141–1152. [4] D. Balding, D.L.A. McElwain, A mathematical model of tumour-induced capillary growth, J. Theor. Biol. 114 (1985) 53–73. [5] M. Chaplain, Avascular growth angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development, Math. Comput. Modelling 23 (1996) 47–88. [6] M. Chaplain, M. Stuart, A model mechanism for the chemotactic response of tumour cells to tumour angiogenesis factor, IMA. J. Math. Appl. Med. Biol. 10 (1993) 149–168. [7] J. Folkman, M. Klagsburn, Angiogenic factors, Science 235 (1987) 442–447. [8] A. Friedman, G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis, Math. Models Methods Appl. Sci. 15 (2005) 95–107. [9] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and quasilinear partial differential equations of parabolic type. Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1968. [10] M.E. Orme, M. Chaplain, Two-dimensional models of tumour angiogenesis, IMA J. Math. Appl. Med. Biol. 14 (1987) 189–205. [11] M.E. Orme, M. Chaplain, A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching, IMA J. Math. Appl. Med. Biol. 13 (1989) 73–98. [12] C.L. Stokes, D.A. Lauffenburger, Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis, J. Theor. Biol. 152 (1991) 377–403.
Existence and uniqueness of global solutions for a mathematical model of antiangiogenesis in tumor growth Xuemei Weia,∗ , Shangbin Cuib a Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou, Guangdong 510090, People’s Republic of China b Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, People’s Republic of China
Received 29 March 2007; accepted 25 May 2007
Abstract In this paper we study a mathematical model of antiangiogenesis in tumor growth. The model consists of three semi-linear parabolic PDEs. By applying the Banach fixed point method, the parabolic Schauder estimates, parabolic Lp estimates and the continuation method, we prove that this system has a unique global classical solution. 䉷 2007 Elsevier Ltd. All rights reserved. MSC: 35K35; 35Q80; 35R05 Keywords: Tumor growth; Global solution; Existence; Uniqueness
1. Introduction This paper is concerned with the problem of tumor angiogenesis. Angiogenesis is an essential stage in the process of a solid tumor growth. In the beginning stage of the tumor growth, the tumor does not have its own capillary network. It receives nutrition through simple diffusion of nutrient materials from the surrounding normal tissue of the host body, so that it cannot grow very large. A tumor in this stage is called an avascular tumor. A dormant avascular tumor generally has a size of only 1–3 mm in diameter. Such tumors are not very harmful to the body. Tumors which are really detrimental to health of the body are vascularized tumors. A dormant avascular tumor surmounts its avascular dormant state by initiating the formation of new blood vessels from nearby preexisting vasculature in the host tissue, and gradually form its own capillary network, becoming vascularized. The development of new blood vessels is known as angiogenesis. It is a significant stage for tumor growth. Hence, the study of the mechanism of angiogenesis in the process of tumor growth is very important. During the last two decades, a number of mathematical models describing the process of angiogenesis of in vivo and in vitro tumors have been presented in the literature [1–7,10–12]. In this paper we study a mathematical model of tumor angiogenesis proposed by Anderson et al. in Ref. [3]. This model describes the angiogenic response of endothelial cells to a secondary tumor. The model assumes that the endothelial cells respond chemotactically to two opposing ∗ Corresponding author.
E-mail addresses: [email protected] (X. Wei), [email protected] (S. Cui). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.05.013
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
chemical gradients: a gradient of tumor angiogenic factor (TAF); and a gradient of angiostatin. The model consists of three semi-linear parabolic partial differential equations complemented with boundary and initial conditions: jc j ja j j2 n jn = Dn 2 − (c)n − (a)n , 0 < x < L, t > 0, (1.1) jt jx jx jx jx jx jc j2 c = Dc 2 − 1 c − F (c, n, x), jt jx
0 < x < L, t > 0,
(1.2)
ja j2 a = Da 2 − 2 a − G(a, n, x), 0 < x < L, t > 0, jt jx jc ja n jn + (c) , x = 0, L, t > 0, (a) = Dn jx jx jx jc = 0, jx a = A,
x = 0; x = 0;
n = nI (x),
(1.3) (1.4)
c = c0 ,
x = L,
t > 0,
(1.5)
ja = 0, jx
x = L,
t > 0,
(1.6)
c = cI (x),
a = aI (x),
t = 0,
0 < x < L.
(1.7)
Here n(x, t), c(x, t) and a(x, t), respectively, represent the endothelial cell tip density, the TAF, and the angiostatin concentrations in the one-dimensional domain [0, L], (c) and (a) are, respectively, the TAF chemotactic and angiostatin chemotactic functions, Dn is the cell random motility coefficients, and Dc , Da , respectively, represent the TAF diffusion coefficient and the angiostatin diffusion coefficient. According to [3], (c) and (a), respectively, have the following expressions: (c) =
0 k1 , k1 + c
(a) = 0 a,
where 0 , k1 and 0 are positive constants. We now nondimensionalize (1.1)–(1.7) by using the following rescaling: x¯ =
x , L
=
L2 , Dc
t t¯ = ,
c¯ =
c , c0
a¯ =
c , a0
n¯ =
n , n0
where c0 is the TAF concentration at the tumor and n0 , a0 are appropriate reference variables. Setting f (c, ¯ n, ¯ x) ¯ = F (c, n, x),
g(a, ¯ n, ¯ x) ¯ = G(a, n, x)
and dropping all bars for clarity, we obtain the following nondimensional system: j2 n j jc j ja jn = D1 2 − n − an , 0 < x < 1, t > 0, jt jx 1 + kc jx jx jx jx jc j2 c = 2 − 1 c − f (c, n, x), jt jx
0 < x < 1, t > 0,
ja j2 a = D2 2 − 2 a − g(a, n, x), 0 < x < 1, t > 0, jt jx jc n ja jn = + a , x = 0, 1, t > 0, jx D1 1 + kc jx jx jc = 0, jx
x = 0;
c = 1,
x = 1,
t > 0,
(1.8) (1.9) (1.10) (1.11) (1.12)
X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
a = s,
x = 0;
n = n0 (x),
ja = 0, jx
c = c0 (x),
x = 1, a = a0 (x),
1829
t > 0,
(1.13)
t = 0, 0 < x < 1,
(1.14)
where D1 =
Dn , Dc
D2 =
Da , Dc
=
0 c0 , Dc
=
0 a02 , Dc
=
c0 , k1
1 = 1 ,
2 = 2 ,
s=
A . a0
We make the following assumptions: (A1) f (c, n, x) = cf 1 (c, n, x), g(a, n, x) = ag 1 (a, n, x), where f1 , g1 are defined and Lipschitz continuous in their arguments, f1 (c, n, x)0, g1 (a, n, x)0 for c 0, n 0, a 0, x 0, and f1 , g1 are bounded for n. (A2) n0 (x), c0 (x), a0 (x) ∈ C 2+ [0, 1], 0 c0 (x)s, 0 a0 (x) 1, c0 (0) = 0, a0 (0) = s, c0 (1) = 1, a0 (1) = 0. The main result of this paper is as follows: Theorem 1.1. The problem (1.8)–(1.14) has a unique classical solution which exists for all t 0. The structure of this paper is as follows. Section 2 is devoted to presenting some preliminary lemmas that will be used in later analysis. In Section 3 we prove local existence and uniqueness to (1.8)–(1.14). In the last section, we prove that the solution of the problem (1.8)–(1.14) is global, which proves Theorem 1.1. 2. Preliminary lemmas We denote: QT = (0, 1) × (0, T ),
¯ T = the closure of QT . Q
¯ T . Let Lemma 2.1. Let D be a constant, and a(x, t), b(x, t), f (x, t) be bounded continuous functions defined on Q Bu = (ju/jn) + u, where (1) = 0, = 1; (2) = 1, = 0, and c0 ∈ Dp (0, 1), for some 1 < p < ∞. Then the initial value problem jc j2 c jc = D 2 + a(x, t) + b(x, t)c + f (x, t), jt jx jx x = 0, 1: Bc = , c(x, 0) = c0 (x),
0 < x < 1, 0 < t < T ,
0 t T ,
(2.1) (2.2)
0 x 1,
(2.3)
has a unique solution c(x, t) ∈ Wp2,1 (QT ). Moreover, cW 2,1 (Q ) Cp (T )(c0 Dp (0,1) + W 1,p (0,T ) + f p ), p
T
(2.4)
Cp (T ) are constants depending only on p, T, D, a∞ , b∞ , and Cp (T ) are bounded for all T in a bounded set. Proof. See [9].
¯ T ), and c0 (x) ∈ C 2+ [0, 1]. Then the initial value problem Lemma 2.2. Let a(x, t), b(x, t), f (x, t) ∈ C ,/2 (Q jc j2 c jc = D 2 + a(x, t) + b(z, t)c + f (x, t), jt jx jx x = 0, 1 : Bc = , c(x, 0) = c0 (x),
0 < x < 1, 0 < t < T ,
0 t T , 0 x 1,
(2.5) (2.6) (2.7)
has a unique solution, and when (1) = 0, = 1, we have cC 2+,(1+)/2 (Q¯ T ) c0 C 2+ [0,1] + C (T )( C (1+)/2 [0,T ] + f C ,/2 (Q¯ T ) );
(2.8)
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
when (2) = 1, = 0, we have cC 2+,(1+)/2 (Q¯ T ) c0 C 2+ [0,1] + C (T )( C (1+)/2 [0,T ] + f C ,/2 (Q¯ T ) ).
(2.9)
Here C (T ) are constants depending only on D, T, a(x, t)C ,/2 (Q¯ T ) , b(x, t)C ,/2 (Q¯ T ) .
Proof. See (3.10) in [8].
Lemma 2.3. Let u(x, t) ∈ C 2+,(1+)/2 (QT ). Then u(x, t) − u(x, 0)C 1+,(1+)/2 (QT ) C (T )uC 2+,(1+)/2 (QT ) .
(2.10)
where (T ) = max{T /2 , T (1−)/2 }. Proof. By the define of Hölder norm, we have [u(x, t) − u(x, 0)]C 1+,0 (QT ) CDx2 u(x, t) − Dx2 u(x, 0)L∞ (QT )
C[Dx2 u]C 0,/2 (QT ) · |T |/2 C[u]C 2,/2 (QT ) · |T |/2 ,
(2.11)
and |u(x, t) − u(x, 0)| |Dt u| · |T |(1−)/2 . |t|(1+)/2 Combining (2.11) and (2.12) we obtain (2.10).
(2.12)
3. Local existence and uniqueness For a given T > 0 and a positive number M to be specified later, we introduce a metric space (XM , d) as follows: The set XM consists of vector functions (n(x, t), c(x, t), a(x, t)), satisfying the following conditions: n, c, a ∈ C 1+,(1+)/2 (QT ), n, c, a satisfy conditions (1.12).(1.14), and nC 1+,(1+)/2 M,
cC 1+,(1+)/2 M,
aC 1+,(1+)/2 M.
(3.1)
For convenience we set u = (n, c, a),
u˜ = (n, ˜ c, ˜ a). ˜
The metric d is defined by d(u1 , u2 ) = u1 − u2 C 1+,(1+)/2 . It is obvious that (XM , d) is a complete metric space. ˜ c, ˜ a), ˜ where u˜ satisfies the following problem: Given any (n, c, a) ∈ XM , we define a mapping F : (n, c, a) → (n, jc˜ j2 c˜ = 2 − 1 c˜ − cf ˜ 1 (c, n, x), jt jx x=0:
jc˜ = 0; jx
t = 0 : c˜ = c0 (x),
0 < x < 1, t > 0,
x = 1 : c˜ = 1, 0 < x < 1,
ja˜ j2 a˜ = D2 2 − 2 a˜ − ag ˜ 1 (a, n, x), jt jx x = 0 : a˜ = s;
t > 0,
x=1:
ja˜ = 0, jx
(3.2) (3.3) (3.4)
0 < x < 1, t > 0, t > 0,
(3.5) (3.6)
X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
t = 0 : a˜ = a0 (x),
0 < x < 1.
(3.7)
jc˜ j ja˜ n˜ − a˜ n˜ , 1 + k c˜ jx jx jx jn˜ jc˜ n ja˜ x = 0, 1 : = + a , t > 0, jx D1 1 + kc jx jx j j2 n˜ jn˜ = D1 2 − jt jx jx
t = 0 : n˜ = n0 (x),
1831
0 < x < 1, t > 0,
0 < x < 1.
(3.8) (3.9) (3.10)
¯ T ) satisfying (i) By Lemma 2.2, (3.2)–(3.4) has a solution c˜ ∈ C 2+,(1+)/2 (Q c ˜ C 2+,(1+)/2 (Q¯ T ) c0 C 2+ [0,1] + C(T ) ≡ C1 (T ). Hence, by Lemma 2.3 we have c ˜ C 1+,(1+)/2 (Q¯ T ) c0 C (1+)/2 [0,1] + c˜ − c0 C 1+,(1+)/2 (Q¯ T ) c0 C (1+)/2 [0,1] + C (T )c ˜ C 2+,(1+)/2 (Q¯ T ) c0 C (1+)/2 [0,1] + C (T )C1 (T ). Since limT →0 (T )=0, from the above result we see easily that if M > c0 C (1+)/2 [0,1] and if we take T > 0 sufficiently small then c ˜ C 1+,(1+)/2 (Q¯ T ) M.
(3.11)
Moreover, applying the maximum principle to (3.2)–(3.4) we get 0 c˜ 1.
(3.12)
¯ T ) satisfying (ii) Similarly, by Lemma 2.2, (3.5)–(3.7) has a solution a˜ ∈ C 2+,(1+)/2 (Q a ˜ C 2+,(1+)/2 (Q¯ T ) a0 (x)C 2+ [0,1] + C(T ) ≡ C2 (T ). In the same way, by Lemma 2.3 we have a ˜ C 1+,(1+)/2 (Q¯ T ) a0 C (1+)/2 [0,1] + a˜ − a0 C 1+,(1+)/2 (Q¯ T ) a0 C (1+)/2 [0,1] + C (T )a ˜ C 2+,(1+)/2 (Q¯ T ) a0 C (1+)/2 [0,1] + C (T )C2 (T ). Since limT →0 (T )=0, from the above result we see easily that if M > a0 C (1+)/2 [0,1] and if we take T > 0 sufficiently small then a ˜ C 1+,(1+)/2 (Q¯ T ) M.
(3.13)
Also applying the maximum principle to (3.5)–(3.7) we get 0 a˜ s.
(3.14)
(iii) We deduce that the coefficients of Eq. (3.8) are in C ,/2 from (i) and (ii). Hence, by Lemma 2.2, we see that ¯ T ) satisfying (3.8)–(3.10) has a solution n˜ ∈ C 2+,(1+)/2 (Q jn˜ n ˜ C 2+,(1+)/2 (Q¯ T ) n0 C 2+ [0,1] + C(T ) jx x=0,1 (1+)/2 C [0,T ] ˜ jc˜ 2 ja n0 C 2+ [0,1] + C(T ) M + M jx x=0 C (1+)/2 [0,T ] jx x=1 C (1+)/2 [0,T ] ≡ C3 (T , M).
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
Hence, by Lemma 2.3 we have n ˜ C 1+,(1+)/2 (Q¯ T ) n0 C (1+)/2 [0,1] + n˜ − n0 C 1+,(1+)/2 (Q¯ T ) ˜ C 2+,(1+)/2 (Q¯ T ) n0 C (1+)/2 [0,1] + C (T )n n0 C (1+)/2 [0,1] + C (T )C3 (T , M). Since limT →0 (T )=0, from the above result we see easily that if M > n0 C (1+)/2 [0,1] and if we take T > 0 sufficiently small then n ˜ C 1+,(1+)/2 (Q¯ T ) M.
(3.15)
Combining (3.11), (3.13) with (3.15), we conclude that if M > max{c0 C (1+)/2 [0,1] , a0 C (1+)/2 [0,1] , n0 C (1+)/2 [0,1] } and if we take T > 0 sufficiently small, then u˜ ∈ XM , i.e. F maps XM to itself. In the following we prove that F is a contraction. Take u1 , u2 ∈ XM , and let u˜ 1 = F u1 , u˜ 2 = F u2 , u˜ ∗ = u˜ 1 − u˜ 2 . Setting = u1 − u2 C 1+,(1+)/2 (Q¯ T ) . Denoting c˜∗ = c˜1 − c˜2 , we have jc˜∗ j2 c˜∗ − (1 + f1 (c1 , n1 , x))c˜∗ − h1 , = jt jx 2 x=0:
jc˜∗ = 0; jx
t = 0 : c˜∗ = 0,
0 < x < 1, t > 0,
x = 1 : c˜∗ = 0, t > 0, 0 < x < 1,
where h1 = c˜2 (f1 (c1 , n1 , x) − f1 (c2 , n2 , x)). From Lemma 2.2, we deduce that c˜∗ C 2+,(1+)/2 (Q¯ T ) C(T )h1 C ,/2 (Q¯ T ) ,
(3.16)
and by the assumption (A1), we have h1 C ,/2 (Q¯ T ) c˜2 C ,/2 (Q¯ T ) c1 − c2 C ,/2 (Q¯ T ) + n1 − n2 C ,/2 (Q¯ T ) .
(3.17)
Substituting (3.11), (3.17) into (3.16), we get c˜∗ C 2+,(1+)/2 (Q¯ T ) C(T )M.
(3.18)
Next, denoting a˜ ∗ = a˜ 1 − a˜ 2 , we have ja˜ ∗ j2 a˜ ∗ = D2 2 − (2 + g1 (c1 , n1 , x))a˜ ∗ − h2 , jt jx ja˜ ∗ = 0, jx
x = 0 : a˜ ∗ = 0;
x=1:
t = 0 : a˜ ∗ = 0,
0 < x < 1,
t > 0,
where h2 = a˜ 2 (g1 (c1 , n1 , x) − g1 (c2 , n2 , x)).
0 < x < 1, t > 0,
X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
1833
From Lemma 2.2, we deduce that a˜ ∗ C 2+,(1+)/2 (Q¯ T ) C(T )h2 C ,/2 (Q¯ T ) ,
(3.19)
and by the assumption (A1), we have
h2 C ,/2 (Q¯ T ) a˜ 2 C ,/2 (Q¯ T ) c1 − c2 C ,/2 (Q¯ T ) + n1 − n2 C ,/2 (Q¯ T ) .
(3.20)
Substituting (3.13), (3.20) into (3.19), we get a˜ ∗ C 2+,(1+)/2 (Q¯ T ) C(T )M.
(3.21)
Finally, denoting n˜ ∗ = n˜ 1 − n˜ 2 , we have jn˜ ∗ j2 n˜ ∗ jc˜∗ ja˜ ∗ j j n˜ 1 = D1 2 − − a˜ 1 n˜ 1 − h3 , jt jx 1 + k c˜1 jx jx jx jx jn˜ ∗ ja˜ 1 ja˜ 2 x=0: n1 a1 = − n 2 a2 , t > 0, jx D1 jx jx n2 jc˜2 n1 jc˜1 jn˜ ∗ = − , t > 0, x=1: jx D1 1 + kc1 jx 1 + kc2 jx t = 0 : n˜ ∗ = 0, where j h3 = jx
0 < x < 1, t > 0,
0 < x < 1,
n˜ 2 n˜ 1 − 1 + k c˜1 1 + k c˜2
jc˜2 jx
j + jx
ja˜ 2 (a˜ 1 n˜ 1 − a˜ 2 n˜ 2 ) jx
.
By Lemma 2.2, we deduce that
1 x=0 C(1+)/2[0,T ] j c ˜ j c ˜ n n 1 1 2 2 − + + h3 C ,/2 (Q¯ T ) . D 1 + kc2 jx x=1 C (1+)/2 [0,T ] 1 1 + kc1 jx
ja˜ 1 ja˜ 2 n1 a1 − n 2 a2 n˜ C 2+,(1+)/2 (Q¯ T ) C(T ) D jx jx ∗
Obviously, the first terms of the right-hand side of (3.22) is n1 a1 ja˜ 1 − n2 a2 ja˜ 2 C(T )M. jx jx x=0 C (1+)/2 [0,T ] Similarly, the second terms of the right-hand side of (3.22) is n1 jc˜1 n2 jc˜2 C(T )M. 1 + kc jx − 1 + kc jx (1+)/2 1 2 x=1 C [0,T ] In the same way, the third terms of the right-hand side of (3.22) is
jc˜2 h3 C ,/2 (Q¯ T ) C(T ) · (n˜ 1 − n˜ 2 C 1+,(1+)/2 (Q¯ T ) + c˜1 − c˜2 C 1+,(1+)/2 (Q¯ T ) ) jx C 1+,(1+)/2 (Q¯ T ) ja˜ 2 + · (a˜ 1 − a˜ 2 C 1+,(1+)/2 (Q¯ T ) + n˜ 1 − n˜ 2 C 1+,(1+)/2 (Q¯ T ) . jx C 1+,(1+)/2 (Q¯ T )
(3.22)
(3.23)
(3.24)
(3.25)
Substituting (3.23)–(3.25) into (3.22) and applying (3.11), (3.13), we get n˜ ∗ C 2+,(1+)/2 (Q¯ T ) C(T )M.
(3.26)
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
Combining (3.16), (3.21) and (3.26), we deduce that u˜ ∗ C 2+,(1+)/2 (Q¯ T ) C(T ) .
(3.27)
By u˜ ∗ (z, 0) = 0 and Lemma 2.3, we conclude that u˜ ∗ C 1+,(1+)/2 (Q¯ T ) (T )u˜ ∗ C 2+,(1+)/2 (QT ) .
(3.28)
Substituting (3.27) into (3.28), we get u˜ ∗ C 1+,(1+)/2 (Q¯ T ) C(T ) (T ) .
(3.29)
Taking T such that C(T ) (T ) < 1, we conclude that F is a contraction. By the contraction mapping theorem F has a unique fixed point u = (n, c, a), which is the unique solution. T depends only on an upper bound on the size of the C 2+ norm of u(x, 0). We summarize the above result in the following theorem: Theorem 3.1. There exists T > 0, T depending on an upper bound on the size of the C 2+ norm of u(x, 0), such that the problem (3.2)–(3.9) has a unique solution for 0 t T . 4. Global existence and uniqueness Lemma 4.1. The following holds: 0 c 1,
0 a s,
n 0,
cW 2,1 (Q ) Cq (T ),
aW 2,1 (Q ) Cq (T ).
T
q
(4.1) (4.2)
T
q
Proof. The inequalities 0 c 1, 0 a s follow from the maximum principle. Next, we consider Eq. (1.8). We can easily prove that n = 0 is the lower solution. Hence n 0. Finally, by the assumption (A1) and Lemma 2.1, we conclude that (4.2) holds. Lemma 4.2. For any 1 < q < ∞, there exists a constant Cq (T ), depending on T, such that nLq (QT ) Cq (T ).
(4.3)
Proof. Multiplying (1.8) by nk and integrating over Qt , we get t 0
1 0
jn k n dx dt= jt
t 0
1 0
j2 n D1 2 nk dx dt− jx
1
t 0
0
j jx
jc n 1+kc jx
j + jx
ja an jx
nk dx dt.
(4.4)
The left-hand side of (4.4) is equivalent to t 0
1 0
jn k 1 n dx dt = jt k+1
1 0
d dt
t
nk+1 dx dt.
(4.5)
0
By parts, the first terms of the right-hand side of (4.4) are transformed into t
1
D1 0
0
j2 n k n dx dt = jx 2
t
D1 0
jn k 1 n | dt − D1 jx 0
1 jn 2
t 0
0
jx
nk−1 dx dt,
(4.6)
X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
1835
similarly, the second terms of the right-hand side of (4.4) are transformed into t 1 j jc j ja n + an nk dx dt jx 1 + kc jx jx jx 0 0 t ja jc = + an nk |10 dt n jx 1 + kc jx 0 t 1 jc ja jn k −k + a n dx dt. 1 + kc jx jx jx 0 0
(4.7)
Substituting (4.5)–(4.7) into (4.4), we get 1 k+1
1 0
d dt
t
n
k+1
0
t
dx dt − D1 0 t +k 0
1 jn 2
0 1 0
nk−1 dx dt jx jc ja jn k + a n dx dt. 1 + kc jx jx jx
(4.8)
By (4.2) and Young inequality, we deduce that t 1 t 1 jc ja jn k jn k + a n dx dt kC q n dx dt k 1 + kc jx jx jx 0 0 0 0 jx t 1 2 t 1 jn 1 k−1 kC q n dx dt + kC q nk+1 dx dt. 2 jx 2 0 0 0 0
(4.9)
Substituting (4.9) into (4.8) and taking sufficiently small, such that ( /2)kC q D1 , we conclude that 1 k+1
1 0
d dt
t
nk+1 dx dt
0
1 kC q 2
t 0
1
nk+1 dx dt.
(4.10)
0
By Gronwall’s Lemma, we get nLk+1 Cq (T ). Therefore, Lemma 4.2 holds.
(4.11)
Lemma 4.3. For any 1 < q < ∞, there exists a constant C˜ q , depending only on T, such that nW 2,1 (Q ) C˜ q (T ). q
T
Proof. We consider the equation of n jn j2 n j jc j ja = D1 2 − n − an . jt jx 1 + kc jx jx jx jx
(4.12)
(4.13)
By a direction calculation, we see that j jc ja jc ja jn − n + an = − n + a jx 1 + kc jx jx 1 + kc jx jx jx j jc ja jn − + a n ≡ (x, t) + (x, t)n. jx 1 + kc jx jx jx Obviously, (x, t), (x, t) are bounded continuous functions defined on QT . Eq. (4.13) can be transformed into jn j2 n jn = D1 2 + (x, t) + (x, t)n. jt jx jx
(4.14)
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X. Wei, S. Cui / Nonlinear Analysis: Real World Applications 9 (2008) 1827 – 1836
By Lemma 4.2 and parabolic Lq estimates, we conclude that nW 2,1 (Q ) C˜ q (T ). q
(4.15)
T
Hence Lemma 4.3 holds.
Lemma 4.4. There exists a constant C(T ), depending on T, such that uC 2+,(1+)/2 (Q¯ T ) C(T ). Proof. From Lemma 4.3 and Soblev embedding, it follows that uC ,/2 (Q¯ T ) C(T ),
0 < < 1.
Hence the coefficients of (1.8)–(1.10) satisfy the condition of Lemma 2.2. Applying Lemma 2.2, we get uC 2+,(1+)/2 (Q¯ T ) C(T ). Lemma 4.4 holds.
(4.16)
By Theorem 3.1 and Lemma 4.4, we see that Theorem 1.1 is proved. Acknowledgments This work is financially supported by the China National Natural Science Foundation under the Grant no. 10471157. It is also supported in part by the Ph.D. Foundation of Guangdong University of Technology under the Grant no. 063043. References [1] A. Anderson, M. Chaplain, Continuous and discrete mathematical modela of tumour-induced antiangiogenesis, Bull. Math. Biol. 60 (1998) 857–900. [2] A. Anderson, M. Chaplain, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett. 11 (1998) 109–114. [3] A. Anderson, M. Chaplain, C. García-Reimbert, C. Vargas, A gradient-driven mathematical model of antiangiogenesis, Math. Comput. Model. 32 (2000) 1141–1152. [4] D. Balding, D.L.A. McElwain, A mathematical model of tumour-induced capillary growth, J. Theor. Biol. 114 (1985) 53–73. [5] M. Chaplain, Avascular growth angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development, Math. Comput. Modelling 23 (1996) 47–88. [6] M. Chaplain, M. Stuart, A model mechanism for the chemotactic response of tumour cells to tumour angiogenesis factor, IMA. J. Math. Appl. Med. Biol. 10 (1993) 149–168. [7] J. Folkman, M. Klagsburn, Angiogenic factors, Science 235 (1987) 442–447. [8] A. Friedman, G. Lolas, Analysis of a mathematical model of tumor lymphangiogenesis, Math. Models Methods Appl. Sci. 15 (2005) 95–107. [9] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and quasilinear partial differential equations of parabolic type. Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1968. [10] M.E. Orme, M. Chaplain, Two-dimensional models of tumour angiogenesis, IMA J. Math. Appl. Med. Biol. 14 (1987) 189–205. [11] M.E. Orme, M. Chaplain, A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching, IMA J. Math. Appl. Med. Biol. 13 (1989) 73–98. [12] C.L. Stokes, D.A. Lauffenburger, Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis, J. Theor. Biol. 152 (1991) 377–403.