1. Introduction
Chemotaxis is the ability of some living organisms to direct their movement in response to the presence of a chemical gradient. This response can be either positive (chemoattractant) or negative (chemorepellent). Mathematical models for chemotaxis have been studied since 1970 when Keller and Segel proposed a system of two parabolic equations involving nonlinear second order terms in the form
in the
u-equation. Since the publication of the model, an extensive mathematical literature has treated the topic, see also [
1]. To present an exhaustive literature review is not the aim of this article, therefore we refer to the reader to the survey works of Horstmann [
2,
3], and Bellomo et al. [
4] for more details (see also [
5]).
The mathematical model that we study in this article describes the behavior of a biological species “
u” in terms of a PDE of parabolic type. The problem is posed in a bounded domain
, with a regular boundary
as follows
The equation includes the linear diffusion of “u” which also moves following the direction of the chemical gradient of a non-diffusive substance “v”. The chemotactic coefficient is assumed to be constant and positive, i.e., the biological species moves to a higher concentration of “v”. The logistic term includes a carrying capacity that limits the growth of “u” and it presents a spatial and time dependence, . Then, the reaction part is given by the quadratic term where is a positive constant. The chemical substance “v” is considered non-diffusive, i.e., once it is secreted by the biological species “u”, it is maintained up to degradation. The evolution of “v” is given in terms of a general function “h” satisfying some technical assumptions presented in this section.
Function
f, in the reaction term, is a smooth bounded given function, fulfilling
with
being a time-periodic function independent of the space variable “
x”.
In Ref. [
6] the fully parabolic system is considered, i.e., the equation for the chemical includes a diffusive term and the equation for
v reads
The global existence of solutions for the fully parabolic system is achieved by employing an iterative method based on the Alikakos-Moser iteration. By using an energy method through a Lyapunov functional, the convergence of the solution to a homogeneous in space and periodic in time function
is given. The parabolic-elliptic case, i.e., for
v satisfying the equation
has been studied in Negreanu, Tello and Vargas [
7], where the global existence and similar asymptotic behavior are done. In that case, the proof follows a sub-super solutions method already featured in Pao [
8], Tello and Winkler [
9], Galakhov, Salieva and Tello [
10] and Negreanu and Tello [
11,
12] among others. In [
7], the problem is addressed for a non constant function
f and
satisfying the ODE
where
is a periodic in time function such that
In Issa and Shen [
13] the logistic term is
and the authors got the existence of periodic solutions when the coefficients
(for
) are periodic in time.
Parabolic-ODE systems with chemotactic terms have been considered from the last three decades, and after the pionering works of Levine and Sleeman [
14] and Anderson and Chaplain [
15] modeling tumor angiogenesis, a considerable number of authors have analyzed such models. In Othmer and Stevens [
16] and Stevens [
17], the authors address a Parabolic-ODE system of chemotaxis passing to the limit from a discrete to a continuous system of equations. Concerning angiogenesis, the model has been raised in Kubo and Suzuki [
18], Suzuki [
19] and Kubo, H. Hoshino and K. Kimura [
20]. Mathematical analysis of these models with two equations can be found in Fontelos, Friedman and Hu [
21], Friedman and Tello [
22] and Negreanu and Tello [
11] among others. Systems with three or more equations involving chemotaxis and diffusive or non-diffusive processes also appear in ecology and other biological applications (see [
12,
23]). In Ref. [
24] the authors study a similar Parabolic-Parabolic-ODE system
where the chemosensitivities
are non-constant. Global existence and convergence of the solution to a steady-state
satisfying
are presented under suitable assumptions on the coefficients and the spatial dimension of the domain. The results in [
24] have been improved in Mizukami and Yokota [
25] for a larger range of parameters.
Also in the context of cancer dynamics, chemotactic systems with non-diffusive equations have been recently used to model
cancer cell invasion in Stinner, Surulescu and Winkler [
26] with a model consisting of six equations where the cancer cells behavior is described by a parabolic equation with chemotactic terms. Denoting by “
u” the cancer cells density, by “
v” the fibers of the extracellular matrix (ECM) and by “
l”, “
” and “
” the concentration of chemoattractant, integrins bound to ECM fibers and integrins bound to proteolitic residuals then, their model is the following
The authors prove the existence of global weak solutions together with some boundedness properties. The proof is based on the properties of the functional
Notice that, in this case, three of the variables leading the movement satisfy ordinary differential equations (see also Stinner, Surulescu and Uatay [
27], Tao and Winkler [
28], Zhigun, Surulescu and Hunt [
29] and Zhigun, Surulescu and Uatay [
30] for similar models). Throughout the article we use the notation
, for
, we assume, without loss of generality, that
and we denote by
g the function
We work under the following hypotheses
The positive initial data
of (
1) satisfy,
and
for some
and
There exists a periodic function
verifying
and
Moreover, there exists a positive constant
c such that
with some
as in (
7).
For a given constant
(defined in Lemma 6) we have that
The functions given by and , with verify the hypotheses.
2. Main Results
Our particular analysis will address the initial-boundary value problem (
1) in a bounded open domain
, where the initial data are as in (
4). The issue of the global solvability is presented in the following theorem.
Theorem 1. Let Ω be a bounded open domain with regular boundary of and suppose that assumptions (4)–(11) hold. Then, there exists a unique pair of nonnegative functions u and v which forms a global solutionto the problem (1). In addition, is uniformly bounded in , that is, there exists a constant , such that Afterwards, we study the asymptotic properties of the solutions. We introduce the function
as set out by
for
defined by
and
as in (
6). Notice that
satisfies Equation (
2) and it is an homogeneous in space and periodic in time function. We denote by
, the solution of the ordinary differential equation
The following assertion is the main result on the asymptotic behavior of solutions of (
1).
Theorem 2. Assume (4)–(11) and let us denote by the corresponding solution of (1) from Theorem 1. Then, has the following asymptotic behaviorfor any , where and are given by (12) and (13), respectively. The paper is organized as follows: in
Section 3 we proof the existence of a unique pair of classical solutions. A first key step consists in findining a maximal weak solution following [
31], and we then get the boundedness of the solution. As a crucial ingredient in our derivation of a
bound for
u, we employ a Alikakos-Moser-type iterative procedure [
32]. By means of these and some further higher regularity properties will assert the statements on global existence and boundedness of
u and
v from Theorem 1. Our collection of estimates of
Section 3 will moreover turn out to be sufficient to derive the stabilization result from Theorem 2 in
Section 4 through an analysis into two steps. First, we prove that the solutions converge to their respective averages, i.e.,
using energy estimates to conclude that these averages converge to the functions
and
, respectively. Finally, in
Section 5 we perform a brief numerical study of the system under consideration. Some of the results presented in this paper were announced in [
33].
3. Global Existence of Solutions
The present section is devoted to the proof of Theorem 1. We study the local-in-time existence of classical solutions to (
1) and we prove some preliminary technical facts. In order to prove the global existence of the solutions, we first obtain the local existence using classical results on partial differential equations and then we conclude the proof by constructing uniform bounds.
Lemma 1. Let be an open bounded set with smooth boundary, assume that the initial data, , are as in (4) and hypotheses of Theorem 1 hold. Then, there exists a maximal existence time such that system (1) has an unique non-negative classical solutionas well as Proof. We consider the system (6.2) of [
31] where
and
We can rewrite then (
1) as follows
with the same initial data as (
1). We apply Theorem 6.4 in [
31] and consider maximal interval of existence. So, the local-in-time existence for (
1) is proved.
In order to see the non-negativity of
u we introduce the following change of variables:
Then we can rewrite the first equation in (
1) as
Now, deriving with respect to the spatial variable in the previous equation we get
and
Then, the first equation of (
1) becomes
we multiply by
to get
Notice that the equation for
v remains as an ordinary differential equation
So, the original system (
1) becomes (
16) and (
17) together with the initial data
and the Neumann boundary condition
Finally, the Maximum Principle for parabolic equations and the regularity of
h prove the non-negativity of
u, taking into account that
Hypotheses (
9) and (
10) on
h and the Maximum Principle applied to (
17) prove
This completes the proof. □
Let us now collect some basic properties thereof which in our subsequent analysis will play important roles not only by providing some useful fundamental regularity features, but also by establishing the first quantitative information (
18) on large time behavior. We remember that
for the next matches, and also,
.
Lemma 2. Under Hypotheses (4)–(11), the total mass of the component of the solution to (1) is bounded: Proof. We integrate the first equation of (
1) directly over
to get
Applying the Cauchy-Schwarz inequality, since
,
we directly obtain
Finally, (
18) is a consequence of the Maximum Principle applied to (
20), i.e.,
□
Lemma 3. Under the same assumptions of Lemma 2, the solution to (1) satisfies for all , where and Proof. We integrate (
19) over the interval
for
to obtain
, or, equivalently,
By the previous lemma it follows
Finally, since
we have
thereby completes the proof. □
Lemma 4. Under hypotheses of Theorem 1, the following assertion is verified: there exists a positive constant defined by for such that Proof. Notice that
for some
, such that
. Then, we get
□
We are now prepared to perform an iterative argument of Alikakos-Moser type in order to derive bounds for u and v.
The proof starts with the following lemma.
Lemma 5. Let be defined by (15), then, for the following estimate holdswhere c is the constant given in assumption (10). Proof. We proceed by induction in
p, then, for
we have
For the first integral in (
23) we infer that
From the expression of the above identity, we deduce
We look now at the last term of (
23). By the Mean Value Theorem and assumption (
10) we have
then,
Moreover, for the restant term of (
23), we have
We now replace (
24)–(
26) in (
23) to get
which yields (
22) and the proof ends. □
Lemma 6. Let us consider and as in (15). Let ϵ be a positive constant defined by then, there exist and given by for as in Lemma 4, such that Proof. For
the result is a consequence of Lemmas 2 and 3. For
we proceed by induction and assume the result for
, i.e.,
Taking
, thanks to (
22), we have
We first recall the Young’s inequality:
multiplying it by
we get
which is equivalent to
we integrate in space over
, and in view of
, we get
Thanks to the definition (
27) of
, we have
We replace (
32) into (
31) to get
By solving the differential Equation (
33) after integration in time, we obtain
Dropping the nonpositive term and making use of a favorable cancellation, it yields
and
By definition of
it follows
and due to (
35), the following inequality holds
□
Lemma 7. Under the assumptions of Theorem 1, we have where has been defined in Lemma 6.
Proof. According to Lemma 6 we have that
and therefore
Since
we take limits when
, to obtain (
36). □
Lemma 8. Suppose that (9)–(11) hold. Then there exists a positive constant such that the solution v of (1) satisfies Proof. By contradiction, we assume that for any
there exists
such that
which is the first
fulfilling this condition. Since by assumption (
4)
v must be an increasing function in a neighborhood of
. Then, by applying (
11), we obtain
then, since
h is increasing in the first variable, we have
Thanks to assumption (
11) we have that for
large enough
which is a contradiction and the proof ends. □
The above results entail the claimed qualitative properties of u:
Lemma 9. Under assumptions of Theorem 1, the solution u is uniformly bounded by Proof. The result is a consequence of Lemmas 7 and 8. □
Proof of Theorem 1. The global existence of
over
is a direct consequence of the local existence (Lemma 3.1, Theorem 6.4 in [
31]) and the uniform boundedness of
in
established in the previous Lemmas.□
4. Asymptotic Behavior
The main propose of this section is to demonstrate Theorem 2, i.e., to obtain the convergence of the solution
to
. We proceed in two steps: first of all we get the convergence of the solution
u to its average
, to get later the convergence of the average to the periodic function
given by (
12). For it, we need to prove the boundedness of
in
. The result is enclosed in the following lemma.
Lemma 10. Suppose that the assumptions of Theorem 2 hold. Then, there exists , independent of t, such thatwhere v is the solution of (1). Proof. We consider Equation (
22), for
, and integrate over
to obtain, after routinary computations and thanks to Lemmas 4, 6 and 7
for any
. Recalling that
v satisfies
then taking gradients we get
Now, we multiply (
38) by
and integrate over
to obtain, in view of assumptions (
9)
and therefore, by the Young’s inequality
After integration in time we get
and due to (
37), we conclude the lemma. □
Lemma 11. Under the assumptions of Theorem 2, there exists a positive constant such that Proof. We proceed as in Mizukami-Yokota [
25] (Lemma 4.2.) and multiply the equation of
u by
for some
, after integration by parts we obtain
Since
, we have that
Notice that, thanks to the Mean Value Theorem it yields
for some
. Assumptions (
8)–(
10) imply
with
. Therefore we have that
and then
In view of assumption over
h, for
close enough to 2, we get, by the Maximum Principle that
and the non-negativity of
v implies
with positive constants
c and
. Moreover, the Hölder inequality implies
After some computations, the proof ends. □
Similar results can be found in Tao and Winkler [
28] (Theorem 1.1) and [
34] for parabolic-elliptic and fully parabolic systems.
Lemma 12. Under assumption (6), the solution to (2) defined in (12) admits a lower boundfor some . Proof. We divide by
in (
2) and integrate over
to obtain the result. □
We now define the positive function
thus we achieve the following.
Lemma 13. Under the assumptions of Theorem 2, there exists a positive constant independent of t such that the following estimate holds Proof. We integrate the first equation of (
1) over
and in view of
we obtain
Since
f and
are uniformly bounded, we have
and
for any
. We take
and then
We divide by
to get
Since
satisfies
we have
Now, we consider the following functions
Functionals of quite a similar form have previously been used in several works on related chemotaxis problems, e.g., in [
35]. Notice that
and
. Let
be defined in (
18), then
and also
We take gradients in the equation of
v, multiply the obtained equation by
and integrate over
to get
Now we add both expressions to obtain
We apply the Cauchy-Swartz inequality to the term
then, operating we achieve
which is reduced to
Due to the discriminant of the polynomial
is given by
, which is positive, we have two different roots
and
that are both positive. Since
we have that
Then, we take
to obtain
Through the inequality (
43) it results
After integration over and taking limits when we conclude the lemma. □
Lemma 14. Under the assumptions of Theorem 2 the following estimate holdswith a positive constant. Proof. We first notice that
defined in (
45) achieves its minimum at
and
where
is a compact set of
. Due to (
44) we get
We now proceed as in Lemma 13 and we obtain
After integration over
, in view of
we end the proof. □
We have the following boundedness
Lemma 15. Under the assumptions of Theorem 2, there exists a positive constant such that the following inequality holds Proof. After integration in the time variable the expression
by Lemma 14, we obtain
with
. In view of the boundedness of
u we have
and the proof ends. In Negreanu, Tello and Vargas [
6], a similar problem is studied for the fully parabolic system. □
Lemma 16. Under assumptions (2)–(9), there exists a positive constant independent of t such thatwhere is defined in (39). Proof. The following relations hold:
and
By applying the Young’s inequality we have
The boundedness of u and Lemma 14 imply the result. □
The following lemma is used to prove the behavior of the solution. The proof follows Lemma 5.1 in Friedman-Tello [
22], where
is uniformly bounded, i.e.,
. Here, the boundedness of
is replaced by a weaker assumption given in
.
Lemma 17. Let a function satisfying
- (i)
for any
- (ii)
- (iii)
for any ,
then, as .
Proof. By contradiction, we assume that there exists a sequence
such that
and
. Then, there exist a subsequence
such that
and
Then,
in the interval
for
. So
and taking limits when
we reach the contradiction. □
Lemma 18. Under assumptions of Theorem 2 we have Proof. We consider
defined in (
39), then, thanks to Lemmas 13 and 16 we have
Now, we define
as follows
By recalling the definition of
as in (
42)
due to (
39), we get
We multiply by
and due to the Mean Value Theorem we claim
for some
if
or
otherwise. After integration it results
Notice that Lemma 2 implies
for some positive constant
. Therefore, there exists
such that
In view of Lemma 2, assumption (
6) and Lemma 9 it is easy to see that
Now, by Lemma 17, (
48) and (
49) we obtain
Since
by taking into account (
47) and (
50), we get
and the proof ends. □
In order to obtain
we proceed as before in the following lemma.
Lemma 19. Under assumptions (4)–(8), the solution v fulfills Proof. By the Mean Value Theorem, it follows
We call
, by multiplying by
z the above equation and after integrating over
, it yields
where we have applied the Hölder inequality to the last term. Now, by assumption (
9) it results
where
is uniformly bounded. We obtain the result by solving the differential inequality. □
Proof. (
End of the proof of Theorem 2.) The asymptotic behavior (
14) of
is a direct consequence of Lemmas 18 and 19 and the uniform bounds of
u and
v established therein. □