1. Introduction
As a physiological process, angiogenesis involves the formation of a new capillary network sprouting from a pre-existing vascular network. It has been recognized that the capillary growth through angiogenesis leads to vascularization of tumor, providing it with its own blood supply. During this process, the endothelial cells are induced by the fibronectin, which are bounded in Extracellular Matrix (ECM) and gathered under the effect of fibronection to form new vessels. In [
1], Anderson et al. proposed a reaction–diffusion model to describe those procedures. In [
2], Stevens and Othmer developed the so-called reinforced random walk to gain the understanding of the mechanism that causes the aggregation of myxobacteria. By the methodology established in [
2], Levine et al. [
3,
4] derived models of the angiogenesis based on analysis of the relevant biochemical processes. We refer to [
5,
6] for the related research.
In this paper, we are concerned with the initial-boundary problem
.
Here is a bounded domain with smooth boundary and is unit outward normal vector. p denotes the density of endothelial cells and w represents the density of concentration of chemical substance such as fibronectin; and are positive constant; . denotes proliferation of endothelial cells; and is called chemosensitivity. In addition to random motion, endothelial cells have haptotaxis migration, which responds to the gradient of attractant such as the fibronectin that is non-diffusible in the extracellular matrix.
Mathematical models with haptotaxis have attracted lots of attention [
7,
8,
9,
10,
11,
12]. For example, Corrias et al. [
7] considered the system in Equation (
1) with
and
in the bounded domain and whole space, respectively. In fact, they proved that there exists a global-in-time
-bounded weak solution when the domain is bounded and the sensitivity function
,
. Moreover, the solution
p converges to
in
weakly and
c decays to zero in
(
) in strong topology. For the unbounded domain and the initial data satisfying
, they obtained the existence of weak solution and the self-similar solutions thereof. Furthermore, they extended their previous result to the case of
in [
8]. Guarguaglini et al. [
9] considered the variant of Equation (
1) with
,
, and
w-equation
in whole line. Under some suitable assumption, it is shown that the corresponding system admits a global weak solution and local classic solution for sufficiently regular initial data. On the other hand, Rascle [
12] showed the local existence and uniqueness of classic solutions of the system in Equation (
1) with boundary condition
instead of
where
is a positive constant and particularly
with some
for all
,
. Liţcanu and Morales-Rodrigo [
10] studied the system with
, and standard logistic growing source. Indeed, they proved the system in Equation (
1) admits a globally
-bounded classic solution which converges to a constant with polynomial rate when
and exponential rate if
in
-norm, respectively, when the initial data have a positive lower bound.
It is noted that modeling approaches indicate that, in situations of significantly heterogeneous environment, adequate macroscopic limits of random walk rather lead to certain non-Fickian diffusion operator. We refer to [
13,
14] for the fractional diffusion and refer to [
15,
16] for the nonlinear diffusion. It is should be mentioned that Winkler [
17] considered the related haptotaxis system of Equation (
1), which describes the glioma spread in heterogeneous tissue, and proved the system has global weak solution with few initial data and the solution component
p stabilizes towards a state involving infinite densities and other component
w tends to zero. Finally, we would like mention some papers [
18,
19] where
w satisfies
and assumptions on
g are much more restricted.
Our aim is to consider the system in Equation (
1) with
and a standard logistic growing source, namely
We assume the initial data satisfy
The main result can be stated as follows:
Theorem 1. Let γλ, ρ be positive parameter and . Then the problem in Equations (2) and (3) admits a globally -bounded positive classic solution which satisfies Moreover,
(1) If , then for any , there exists such that:where is the first nonzero eigenvalue of in Ω with the homogeneous Neumann boundary condition. (2) If , then for any , there exist a constant and such thatfor all . Remark 1. The system under consideration is very similar to the problem in [10]. However, we remove the condition that the initial data must have a positive lower bound to reach the -convergence of solution. The crucial idea towards to the proof of Theorem 1 in our approach is to derive a bound for
from the bound of
(
) with some constant
and the inequality
with
for some
. Furthermore, with the help of estimate of
and Gagliardo–Nirenberg interpolation theorem, we can show the component
converges to 1 in
as
with exponential rate if
and polynomial rate in
-norm if
, respectively. This paper is organized as follows. In
Section 2, we prove the system exists a globally
-bounded classic solution for any
by the iterated method. In
Section 3, we show the solution converge to stationary solution in
-norm when
and establish the explicit decay rate of solution in case of
and
, respectively. In
Section 4, we prove the same result in one-dimensional setting.
3. -Convergence of Solution in Two Dimensions
In this section, we discuss the asymptotic behavior of solution in the spatially two-dimension of problem in Equations (
2) and (
3). First, we define the Lyapunov functional
Lemma 7. The equalityis valid for all . Proof. The simple computation shows that
and
Combining above two equalities, we complete the proof. ☐
3.1. Asymptotic Behavior When
Lemma 8. There exists constant such that Proof. By solving the
w-equation, we obtain
and thus
by the positivity of
p. Now, combining the above inequalities with the fact that
for all
, we arrive at Equation (
34). ☐
Lemma 9. There exists constant such that Proof. Then, integrating over time variable, we have
Since
, we conclude that
which implies
☐
Lemma 10. If the initial data satisfy Equation (3), then, for any , Proof. We define
and
By the proof in [
20], it is easy to check that
, which implies
. The Gagliardo–Nirenberg inequality indicates that
We can get Equation (
41) immediately with Lemmas 4 and 9. ☐
Lemma 11. If the initial data satisfy Equation (3), then, for any , we haveand Proof. The proof of Equation (
44) is the same as the one in [
20] for Lemmas 5.10 and 5.11. We only need to prove Equation (
43). With the Poincaré–Wirtinger inequality, we can choose
such that
Then, Lemma 10 implies that
as
. Now, we define
. Then, Equation (
45) indicates that
in
as
. Now, we choose subsequence (
) with
such that
for every
with
,
as
and a constant
such that for all
where
. Because of Equation (
44), we can pick some
such that for all
Then, for any
where
. Now, we can conclude that
as
. On the other hand,
implies that
by the dominated convergence theorem. ☐
Then, we use the well-known - estimates of heat semigroup and the Gagliardo–Nirenberg inequality to show the -convergence of solution.
Lemma 12. There exists a constant C such that Proof. We notice that
satisfies
Now, we define
. Applying variation-of-constant formula to
q-equation yields
Then, the well-known
-
estimate of heat semigroup shows that
Then, the Hölder inequality with the boundedness of
w and
p show that
where
and
are independent of
t. Then, Gronwall inequality yields
where
represents the gamma function. ☐
Lemma 13. If the initial data satisfy Equation (3), then we have Proof. The
-convergence of
p can be proved by the argument in Lemmas 5.9, 5.10, and 5.11 in [
20]. Furthermore, the Gagliardo–Nirenberg inequality implies that
Then, Equation (
53) follows by the boundedness of
p and Lemma 12. ☐
Now, we establish the explicit decay rate of and with respect to -norm.
Lemma 14. For any , there exist and constant such that for all Proof. The explicit expression of
shows that
Then, we can choose positive
and
such that
for all
whenever
. Now, we multiply the term
on both side and integrate over the interval
to obtain
which leads to Equation (
54). ☐
Lemma 15. For any , there exist constant and such thatfor all . Proof. We choose
and
such that
for all
. By multiplying the
p-equation with
and integrating over
, we have
Now, we choose
. Then, for any
This implies that
which leads to Equation (
56). ☐
Lemma 16. For any , there exist constant and such that for all for all . Proof. By the Gagliardo–Nirenberg inequality in the two-dimensional setting, we can obtain
The Gagliardo–Nirenberg inequality still has a similar form in the one-dimensional setting. Then, Lemmas 5, 12, and 15 imply Equation (
57). ☐
Lemma 17. For any , there exist constant and such thatfor all . Proof. Now, we choose the same
in Lemma 14. Then, the explicit expression of
w implies that
Then, we obtain Equation (
58). ☐
Collecting all the lemmas above, we infer that
Theorem 3. If the initial data satisfy Equation (3), then the solution satisfies Moreover, for some , there exists such that for any and 3.2. Asymptotic Behavior When
By checking the proof of Lemmas 10 and 11, a similar result is also valid for which can be stated as follows
At this position, we focus on the explicit decay rate of solutions.
Lemma 18. For any , there exists such that Proof. Now, Equation (
63) implies that there exist
and
such that
whenever
for all
. By integrating Equation (
66) over
, we get
which implies Equation (
65). ☐
Lemma 19. For any , there exists such that Proof. We pick
such that
for all
with
. Hence, we multiply the
p-equation by
and integrate the result over
to obtain
for all
. The Gronwall inequality shows that
Then, we have
which completes the proof. ☐
By checking the proof of Lemmas 13, 15 and 16, we can establish the following explicit decay rate.
Theorem 5. If the initial data satisfy Equation (3), for any , we haveandfor any . 4. -Convergence in One Dimensions
In this section, we establish the explicit decay rate of and for and in one-dimensional setting respectively. In should be mentioned here that the results in previous subsection are still valid in the one-dimensional case.
The following lemma plays a crucial role in establishing the uniformly convergence of when .
Lemma 20. There exists a constant such that:where . Proof. By using the Cauchy inequality, we have
Then, we integrate it over
to obtain
which implies Equation (
71). ☐
Lemma 21. There exists a constant C such thatfor all . Proof. By differentiating
q-equation with respect to time variable, we can easily obtain
Thus, the inequality in Equation (
72) can be deduced by Equations (
73) and (
74) and Lemma 5 by choosing
. ☐
Lemma 22. If the initial data satisfy Equation (3), then we have Proof. By integrating the
w-equation in time variable, we have
for arbitrary
, which leads to Equation (
75). ☐
4.1. Asymptotic Behavior When
Now, we focus our attention on the decay property of solutions for .
Lemma 23. If the initial data satisfy Equation (3), then we have: Proof. By the Poincaré–Wirtinger inequality, we have
This indicates that the term
is bounded. Now, we choose
such that
for all
We notice
Then, for any
, we pick
to get
Then, the assertion now follows from Equation (
78). ☐
Lemma 24. If the initial data satisfy Equation (3), then we have Proof. We define
and
. Then, Lemma 21 implies
as
.
Now, the Poincaré–Wirtinger inequality and the Sobolev embedding theorem show that
Then, Lemma 20 implies that
and Lemmas 10 and 11 show
for sequence
with
. Meanwhile, the
-boundedness of
p implies that
Then, the dominated convergence theorem and inequality below
yield
Thus, Lemma 24 and Equations (
82) and (
85) yield Equation (
79). ☐
The explicit polynomial decay rate can be established by the argument in two-dimensional setting which can be stated as
Lemma 25. If the initial data satisfy Equation (3), then, for any , there exist such that for all and Now, from all the above lemmas, we get
Theorem 6. If the initial data satisfy Equation (3) and , then the solution satisfies Moreover, for any , there exist such that for 4.2. Asymptotical Behavior When
Lemma 26. If and the initial data satisfy Equation (3), then we haveand Proof. Equation (
91) can be checked by repeating the proof in Lemma 24. Now, we focus on the proof of Equation (
92).
By solving the
w-equation, we obtain
Then, the Poincaré-iWirtinger inequality and the Sobolev embedding theorem imply that
By applying the above inequality, we have
Equation (
48) implies that we can choose
such that for
Thus, Equation (
93) indicates that
Then, we pick
to obtain Equation (
92). ☐
Lemma 27. For any , there exists such thatwhere is the first nonzero eigenvalue of with homogeneous Neumann boundary condition. Proof. Notice that and estimate are still valid in the one-dimensional setting.
By integrating the
p-equation over
, we deduce
By applying the variation-of-constant formula to
, we get
With the Sobolev inequality, we can obtain
Then, the well-known
-
estimates of heat semigroup(e.g., [
22]) imply that
Hence, the result of Lemmas 16 and 17 show that
where
. ☐
Lemma 28. For any , there exists constant such that Proof. By integrating the
p-equation over spatial variable, we have
Now, we choose some
and
such that
whenever
. By applying the Gronwall inequality, we have
For the last term on the right side of all the above inequalities, it holds that
Along all inequalities above, we get
where
. ☐
Lemma 29. For any , there exists such that Proof. Then, Lemmas 27 and 28 imply Equation (
102). ☐
Lemma 30. For any , there exists such that Proof. It results from Equation (
92) and estimate
. ☐
Now, we conclude all the above lemmas as the following result
Theorem 7. If the initial data satisfy Equation (3), then the solution Moreover, for any , there exists such that andwhere is the first nonzero eigenvalue of in Ω with the homogeneous Neumann boundary condition.