A new (and optimal) result for boundedness of solution of a quasilinear chemotaxis--haptotaxis model (with logistic source)
Jiashan Zheng
TL;DR
This paper establishes the first rigorous conditions relating parameters in a chemotaxis-haptotaxis model that guarantee the boundedness of solutions, extending and optimizing previous results in the field.
Contribution
It provides the first precise relationship between model parameters ensuring solution boundedness in a coupled chemotaxis-haptotaxis system.
Findings
Derived optimal parameter conditions for bounded solutions.
Extended previous results with rigorous proofs.
Established new relationships between parameters and solution behavior.
Abstract
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion under homogeneous Neumann boundary conditions in a smooth bounded domain , where and , and are given nonnegative parameters. As far as we know, this situation provides the first {\bf rigorous} result which (precisely) gives the relationship between and that yields to the boundedness of the solutions. Moreover, these results thereby significantly extending results of previous results of several authors (see Remarks 1.1 and 1.2)…
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Taxonomy
TopicsMathematical Biology Tumor Growth · Gene Regulatory Network Analysis · Cellular Mechanics and Interactions
A new (and optimal) result for boundedness of solution of a quasilinear chemotaxis–haptotaxis model (with logistic source)
Ling Liu 1, Jiashan Zhenga, Yu Lia, Weifang Yana
1 Department of Basic Science,
Jilin Jianzhu University, Changchun 130118, P.R.China
a School of Mathematics and Statistics Science,
Ludong University, Yantai 264025, P.R.China Corresponding author. E-mail address: [email protected] (J. Zheng)
Abstract
This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion
[TABLE]
under homogeneous Neumann boundary conditions in a smooth bounded domain , where and , and are given nonnegative parameters. The diffusivity is assumed to satisfy
[TABLE]
In the present work it is shown that if
[TABLE]
[TABLE]
or
[TABLE]
or
[TABLE]
then for all reasonably regular initial data, a corresponding initial-boundary value problem for possesses a unique global classical solution that is uniformly bounded in , where
[TABLE]
and
[TABLE]
Here and are the constants which are corresponding to the Gagliardo–Nirenberg inequality (see Lemma 2.1) and the maximal Sobolev regularity (see Lemma 2.2), respectively. Relying on a new -estimate techniques to raise the a priori estimate of a solution from , these results thereby significantly extending results of previous results of several authors (see Remarks 1.1 and 1.2) and some optimal results are obtained.
Key words: Boundedness; Chemotaxis–haptotaxis; Nonlinear diffusion; Global existence
2010 Mathematics Subject Classification: 92C17, 35K55, 35K59, 35K20
1 Introduction
In order to describe the cancer cell invasion into surrounding healthy tissue, in 2005, Chaplain and Lolas ([4]) proposed a pioneering mathematical model which is called chemotaxis–haptotaxis model
[TABLE]
where , and are the cancer cell random motility, the chemotactic coefficients, the haptotactic coefficients and the proliferation rate of the cells, respectively. Here is the physical domain which we assume to be bounded with smooth boundary, and the unknown quantities , and represent the density of cancer cells, the concentration of matrix degrading enzymes (MDE) and the density of extracellular matrix (ECM), respectively.
As a subsystem, (1.1) contains the celebrated Keller–Segel ([17]) chemotaxis system (with logistic source, )
[TABLE]
by setting . Over the last four decades, there is a wide variety of patterns associated Keller–Segel system (1.2) have been studied extensively, and the main interest lies in whether the solution is global or blow up (see e.g., Cieślak [6], Cieślak and Winkler [5], Ishida et al. [13], Painter and Hillen [27], Winkler [47, 51, 51, 49], Li and Xiang [19], Tello and Winkler [42], Wang et al. [44], Zheng et al. [61]). In fact, if the two behaviors (boundedness and blow-up) of solutions strongly depend on the space dimension and the total mass of cells ([2, 10, 11, 48]). When , Tello and Winkler ([42]) mainly proved that the global boundedness for model (1.1) exists under the condition , moreover, they gave the weak solutions for arbitrary small . Kang and Stevens [16] (see also [53, 12]) improve the results of Tello and Winkler ([42]) to the case . While if and (where is a positive constant), Zheng ([61]) proved that for any sufficiently smooth initial data, the corresponding initial-boundary value problem for (1.2) possesses a globally defined bounded solution, which give the lower bound estimation for the logistic source, so that, improves the result of [47]. Furthermore, some recent studies have shown that the blow-up of solutions can be inhibited by the nonlinear diffusion (see Ishida et al. [13], Winkler et al. [1, 36, 55, 54, 46, 52]) and nonlinear logistic term (see [54, 56]).
There have been large literature on the global existence and the large time behavior of solutions to the system (1.1). We refer to [3, 22, 33, 37, 38, 41, 60] and the references therein. In fact, when , MDEs diffuses much faster than cells (see [15, 38]), Tao and Wang [33] proved that model (1.1) possesses a unique global bounded classical solution for any in two space dimensions, and for large in three space dimensions. In [38],Tao and Winkler improved the condition on (), so that it coincides with the best one known for the parabolic-elliptic Keller-Segel system (1.2) (see Tello and Winkler[42]), moreover, in additional explicit smallness on , they gave the exponential decay of in the large time limit. However, this problem is left open for the critical case . While, if , refined approaches involving a more subtle analysis of (1.1), Tao ([32]) and Cao ([3]) obtained the boundedness of global solution for the 2D and 3D space respectively, especially, for the 3D space, similar to the chemotaxis-only system ([61, 47]), the global solution is obtained only for large , and it remains open for small .
The diffusion of cancer cell may depend nonlinearly on their densities ([9, 30, 35]), and so we are led to consider the cell motility as a nonlinear function of the cancer cell density, . Introducing this into the model (1.1) leads to the following chemotaxis-haptotaxis system with nonlinear diffusion
[TABLE]
where and are denoted as before, , the diffusion function fulfills
[TABLE]
and there exist constants and such that
[TABLE]
This parabolic-parabolic-ODE system ( in (1.3)) and its parabolic-elliptic-ODE simplifications ( in (1.3)) have been objects of extensive studies in recent decades. In fact, in , Zheng et al. ([62]) studied the global boundedness for model (1.3) with satisfies (1.4)–(1.5) and , moreover, in additional explicit smallness on , they gave the exponential decay of in the large time limit. Moreover, if satisfies (1.4)–(1.5) with and
[TABLE]
Tao and Winkler ([35]) proved that model (1.3) possesses at least one nonnegative global classical solution, however, their boundedness is left as an open problem. Using the boundedness of , Wang ([45]) and Li, Lankeit ([20]) proved that the global solvability and boundedness of classical solution (or weak solution) for any satisfies (1.4)–(1.5) and . Recently, Zheng ([58]) and Jin ([14]) extended these results to the case and (as well as large ), respectively. But the cases remain unknown. Other variants of the model that are commonly treated including the (nonlinear) logistic types and the re-establishment of ECM components, please refer to [29, 26, 39, 57], etc, and references therein. Thus it is meaningful to analyze the following question:
: Which size of , and are sufficient to ensure boundedness of solutions to (1.3)?
It is our goal in this work to give answers to . To the best of our knowledge, this is the first result which gives a explicit condition between and that yields to the boundedness of the solution.
Motivated by the above works, the aim of the present paper is to study the quasilinear parabolic–elliptic–ODE ( in (1.3)) and parabolic–parabolic–ODE ( in (1.3)) chemotaxis–haptotaxis model (1.3) under the conditions (1.4)–(1.5). Our main result is the following:
Theorem 1.1**.**
Let be a bounded domain with smooth boundary . Assume that satisfy (1.4)–(1.5) and the initial data fulfills
[TABLE]
with some
If one of the following cases holds:
(i) with ;
(ii) with ;
(iii) with ;
(iv) and ;
then there exists a triple which solves (1.3) in the classical sense. Here is a positive constant which is corresponding to the Gagliardo–Nirenberg inequality (see Lemma 2.1). Moreover, both , and are bounded in .
Before we prove theorem 1.1, there exist a few remarks in order.
Remark 1.1**.**
(i) Theorem 1.1 extends the results of Theorem 1.1 of Tao and Winkler ([38]) for the critical case and
(ii) If in comparison to the result for the corresponding haptotaxis-free system ([52], ), it is easy to see that the restriction on here is optimal.
(iii) Observing that if and , then , therefore, Theorem 1.1 also extends the results of Theorem 1.1 of Tello and Winkler ([42]).
(iv) Obviously, if , then , so that, Theorem 1.1 extends the results of Theorem 1.1 of Tao and Winkler ([37]).
(v) Obviously, if and then , so that, Theorem 1.1 also partly extends the results of Theorem 1.1 of Wang et al. ([44]).
(vi) Obviously, if , and , then , so that, Theorem 1.1 is consistent with the results of Theorem 3 of [16].
Theorem 1.2**.**
Let be a bounded domain with smooth boundary . Assume that satisfy (1.4)–(1.5) and the initial data fulfills
[TABLE]
with some If one of the following cases holds:
(i) with ;
(ii) with ;
(iii) with ;
(iv) and ;
then there exists a pair which solves (1.3) in the classical sense, where and are the constants which are corresponding to the Gagliardo–Nirenberg inequality (see Lemma 2.1) and the maximal Sobolev regularity (see Lemma 2.2), respectively. Moreover, both and are bounded in .
Remark 1.2**.**
(i) Obviously, if , hence, Theorem 1.2 extends the results of Ke and Zheng ([60]) and partly extends the result of Liu et al ([21]).
(ii) Obviously, if , , hence Theorem 1.2 extends the results of Wang ([45]) and Li and Lankeit ([20]).
(iii) Theorem 1.2 extends the results of Zheng et al. ([59]) for the critical case as well as and
(iv) If and , then (1.3) possess some solutions which blow up in finite time provided that satisfy (1.4)–(1.5) with (see e.g. [46, 6]). Therefore, In comparison to the result for the corresponding haptotaxis-free system ( in (1.3)), it is easy to see that the restriction on here is optimal.
(v) If , then
[TABLE]
therefore, Theorem 1.1 extends the results of Wang et al. ([62]), who proved the possibility of global and bounded, in the cases, satisfies (1.4)–(1.5) with .
The main novelty and difficulty of the paper is how to control the chemotaxis term , haptotaxis term and strong degeneracies caused by system (1.3). To overcome this difficulty, the purpose of the present paper is to demonstrate how far an adequate combination of maximal Sobolev regularity theory and develop new -estimate techniques (see Lemmas 3.4–3.11) can be used to obtain the global existence and boundedness of solutions to (1.3).
The rest of the paper is organized in the following way. Section 2 will be concerned with preliminaries, including some basic facts and a local existence result. In section 3, by careful analysis, this paper develops some -estimate techniques to raise the a priori estimate of a solution from , where
[TABLE]
and
[TABLE]
To this end, by using the maximal Sobolev regularity and the standard estimate for the solution, we may derive entropy-like inequalities (see (3.26) and (3.23)). Then in order to estimate the right term and on the rightmost of (3.26) and (3.23), we need to deal with for two steps from (see the proof of Lemmas 3.4 and 3.5), which plays a key rule in obtaining the main results. Then employing a bootstrap argument (see (3.48) and (3.49)), one could derive the boundedness of (see Lemma 3.6). Relying on this, we develop new -estimate techniques to raise the a priori estimate of solutions from (see Lemmas 3.7–3.11). Finally, applying the standard Alikakos–Moser iteration, we prove the main results of this paper in the last part.
2 Preliminaries
In this section, we will recall some lemmas and elementary inequalities which will be used frequently later. To begin with, let us collect some basic solution properties which essentially have already been used in [18].
Lemma 2.1**.**
([8]) Let . There exists a positive constant such that for all ,
[TABLE]
*is valid with . *
Lemma 2.2**.**
*([18]) Suppose that and . Consider the following evolution equation *
[TABLE]
For each such that and any , there exists a unique solution In addition, if , with then there exists a positive constant such that
[TABLE]
The local-in-time existence of classical solutions to the chemotaxis–haptotaxis model (1.3) is quite standard; see similar discussions in [35, 21]. Therefore we omit it.
Lemma 2.3**.**
Assume that the nonnegative functions and satisfies (1.8) (or (1.7), if ) for some satisfies (1.4) and (1.5). Then there exists a maximal existence time and a triple of nonnegative functions
[TABLE]
which solves (1.3) classically and satisfies in . Moreover, if , then
[TABLE]
According to the above existence theory, for any , . Without loss of generality, we can assume that there exists a positive constant such that
[TABLE]
Employing the same arguments as in the proof of Lemma 2.3 in [41] (see also [32]), we derive the following Lemma:
Lemma 2.4**.**
*Let solve (1.3) in . Then *
[TABLE]
where
[TABLE]
3 A priori estimates
In this section, we are going to establish an iteration step to develop the main ingredient of our result. The iteration depends on a series of a priori estimates. Firstly, the following two lemmas provide some elementary material that will be essential to our bootstrap procedure.
Lemma 3.1**.**
Let then the solution of (1.3) satisfies
[TABLE]
In contrast to the situation without source terms ( in (1.3)), we cannot hope for mass conservation in the first component. Nevertheless, the following inequality still holds:
Lemma 3.2**.**
(see e.g. [60]) Assume that There exists a positive constant such that the solution of (1.3) satisfies
[TABLE]
*and *
[TABLE]
*where *
[TABLE]
Now, we now proceed to derive a uniform upper bound for , which turns out to be the key to obtain all the higher order estimates and thus to extend the classical solution globally. To do this, employing the maximal Sobolev regularity, in light of Lemma 3.1, as a first conclusion towards global existence of the classical solutions is the following a priori estimate which asserts that, in sharp contrast to the case (see also [47]) is a priori uniformly bounded in for some larger than one. In order to deal with the critical case (), the novelty of paper, we first obtain the bounded of where . And then by some careful analysis, one can finally derive the bounded of the critical case, which are the following Lemmas:
Lemma 3.3**.**
Let be a solution to (1.3) on . Then for any one can find positive constants and such that
[TABLE]
*where is the same as (1.9). *
Proof.
Multiplying the first equation in (1.3) by , and integrating in space and using , we get
[TABLE]
for all .
Case Integrating by parts to the first term on the right hand side of (3.6) and from we obtain
[TABLE]
where we have used the fact that Summing up (2.3) and (2.4) yields to
[TABLE]
where is give by (2.4) and we have used the fact that in (2.3).
Here, by some basic calculation, we deduce that
[TABLE]
Therefore, combined with (3.8), (3.9), and (3.6) and (1.5), we have
[TABLE]
where
[TABLE]
Here we have used the fact that (by ). For any , applying the Gronwall Lemma to the above inequality, we have
[TABLE]
with
[TABLE]
Case Integrating by parts to the first term on the right hand side of (3.6) and from we obtain for any
[TABLE]
where
[TABLE]
Due to (2.3) and (2.4), it follows that for any
[TABLE]
where
[TABLE]
Here is give by (2.4) and we have used the fact that in (2.3).
On the other hand, in view of the Young inequality, we also derive that
[TABLE]
by using Therefore, combined with (3.14), (3.16), (3.6) as well as (3.18) and (1.5), we have
[TABLE]
where is the same as (3.11). For any , applying the Gronwall Lemma to the above inequality shows that
[TABLE]
where is given by (3.13). Next, a use of Lemma 2.2 leads to
[TABLE]
and
[TABLE]
for all . On the other hand, choosing and , with the help of (3.15) and (3.17), a simple calculation shows that
[TABLE]
and
[TABLE]
so that, substituting (3.21)–(3.22) into (3.20) implies that
[TABLE]
with
[TABLE]
Finally, choosing and , using (3.12) and (3.23), applying (1.9), we derive that (3.5) holds. ∎
Lemma 3.4**.**
Let be a solution to (1.3) on . If and then for any
[TABLE]
one can find a positive constant such that
[TABLE]
*holds. *
Proof.
Firstly, applying Lemma 3.3, using and (1.9), we conclude that
[TABLE]
where and are the same as Lemma 3.3.
Case :
**Step 1. The boundedness of for all and . **
To this end, for any , pick in (3.26), then, (by ), so that, (3.26) implies that there exists a positive constant such that
[TABLE]
by using the Young inequality. Applying the Gronwall lemma to (3.27), we derive
[TABLE]
which combined with the arbitrariness of and the Hölder inequality yields to for any ,
[TABLE]
Step 2. The boundedness of for all and .
To achieve this, by step 1, we may pick such that
[TABLE]
Now, set in (3.26), then, (by ), so that, (3.26) implies that
[TABLE]
Now, observe that and implies that
[TABLE]
therefore, in view of (3.30), a use of the Gagliardo-Nirenberg inequality to (3.31) implies that there exist positive constants and such that
[TABLE]
where combined with (by ) implies that
[TABLE]
for some positive constant . Substituting the above inequality into (3.31), one can easily deduce that there exists a positive constant such that
[TABLE]
Case In view of
[TABLE]
so that, (3.26) and the Young inequality yields to
[TABLE]
This completes the proof. ∎
Lemma 3.5**.**
Let be a solution to (1.3) on and . If and then for any
[TABLE]
there exists a positive constant which depends on such that
[TABLE]
*holds. *
Proof.
Firstly, due to Lemma 3.3 as well as and (1.9), we have
[TABLE]
where and are the same as Lemma 3.3. In the sequel, we wish to bound the terms on the right-hand side of (3.36) in terms of the dissipation term on its left-hand side.
Case :
Step 1. The boundedness of for all and with
To this end, for any , pick in (3.36), then,
[TABLE]
so that, (3.36) implies that for some positive constant such that
[TABLE]
by using the Young inequality. This combined with the arbitrariness of and the Hölder inequality yields to for any ,
[TABLE]
and some positive constant .
Step 2. The boundedness of for all and , where .
To achieve this, we pick in (3.36), then, , so that, by (3.36), we have
[TABLE]
In the following, we shall apply the Gagliardo-Nirenberg interpolation inequality to control the second integral on the right-hand side of (3.39). To this end, in view of and implies that
[TABLE]
therefore, in view of (3.38), we deduce from the Gagliardo–Nirenberg inequality that there exist positive constants and such that
[TABLE]
which, together with the fact
[TABLE]
immediately gives that
[TABLE]
and some positive constant . Substituting the above inequality into (3.39), we can get (3.35).
Case : For any we choose
[TABLE]
Then
[TABLE]
so that, (3.36) yields to for some positive constant such that
[TABLE]
by using Young inequality. The proof Lemma 3.5 is completed. ∎
.
Lemma 3.6**.**
Let be a solution to (1.3) on . If and , then there exist positive constants and which satisfy
[TABLE]
where
[TABLE]
*and is the same as (1.9). *
Proof.
Firstly, by (3.42) and (1.9), in view of Lemma 3.4 and Lemma 3.5 , we deduce that
[TABLE]
and some positive constant where is given by (3.42). On the other hand, by Lemma 3.3, we obtain that
[TABLE]
where and are the same as Lemma 3.3. This combined with the Young inequality implies that for any
[TABLE]
and some positive constant . Next, in view of (3.43), we conclude from the Gagliardo–Nirenberg inequality that there exist positive constants and such that
[TABLE]
where is the same as (3.42). Since , one can easily see that
[TABLE]
so that, (3.45) yields to
[TABLE]
for some positive constant Substituting (3.46) into (3.44), we obatin that
[TABLE]
and some positive constant . Let . Then by some basic calculation, we derive that
[TABLE]
and
[TABLE]
where we have used the fact that . Collecting (3.46)–(3.49), we may choose which is close to such that
[TABLE]
Next, substitute (3.50) into (3.47) and choose suitablely small (e.g. ), then we have
[TABLE]
and the proof of Lemma 3.6 is thus completed. ∎
Along with the basic estimate from Lemmas 3.4–3.6, this immediately implies the following Lemma:
Lemma 3.7**.**
*Assume that . If *
[TABLE]
then for , there exists a positive constant such that the solution of (1.3) from Lemma 2.3 satisfies
[TABLE]
where and are given by (3.42) and (1.9), respectively.
Proof.
Firstly, due to Lemma 3.6, we derive that there exists a positive constant such that
[TABLE]
where is the same as Lemma 3.6. Next, Lemma 3.3 implies that
[TABLE]
where and are the same as Lemma 3.3. Therefore, we derive from the Young inequality that there exists a positive constant such that
[TABLE]
For any , together with yields to
[TABLE]
so that, in particular, according to by the Gagliardo–Nirenberg inequality and (3.54), one can get there exist positive constants and such that
[TABLE]
In view of and , by some basic calculation, we derive that
[TABLE]
so that, which returns, using again the Young inequality, for any ,
[TABLE]
Combining the above three estimates and choosing appropriately small, we arrive at for some positive constant such that
[TABLE]
from which we readily infer (3.53). The proof of Lemma 3.7 is completed. ∎
Lemma 3.8**.**
*Assume that . If *
[TABLE]
then for all , there exists a positive constant such that the solution of (1.3) from Lemma 2.3 satisfies
[TABLE]
*where and are given by (3.42) and (1.9), respectively. *
Proof.
Firstly, in view of Lemma 3.5, there exists a positive constant such that
[TABLE]
where with . Next, by (3.55), we also have
[TABLE]
where is the same as (3.55). On the other hand, since yields to so that, in particular, according to by the Gagliardo–Nirenberg inequality and (3.61), one can get there exist positive constants and such that
[TABLE]
This together with (by ) and the Young ineuqality implies that for any ,
[TABLE]
which combined with (3.62) implies that
[TABLE]
by picking appropriately small in (3.64). Finally, using the Hölder inequality, we can get (3.60). The proof of Lemma 3.8 is completed. ∎
Lemma 3.9**.**
Let
[TABLE]
and
[TABLE]
where is the same as (1.9), and are positive constants. Then there exists a positive constant such that
[TABLE]
Proof.
The idea comes from [59]. Indeed, due to (3.66), it is not difficult to verify that . Next, by basic calculation, we derive that for any Therefore, from the monotonicity of , there exists a positive constant such that (3.67) holds. ∎
Lemma 3.10**.**
*Assume that . If *
[TABLE]
or
[TABLE]
then there exists a positive constant such that the solution of (1.3) from Lemma 2.3 satisfies
[TABLE]
*where is the same as (1.9). *
Proof.
Without loss of generality, we may assume that
[TABLE]
since, can be proved similarly and easily. Assume that is the same as lemma 3.9 and let . Then due to (3.5) and , we also derive that for the above
[TABLE]
where and are the same as Lemma 3.3. Here, in order to estimate the rightmost term appropriately, we employ the Gagliardo-Nirenberg inequality to obtain such that
[TABLE]
by using (3.1), where in the last inequality we have used and is the same as Lemma 2.1. In combination with (3.71) and (3.72), this shows that
[TABLE]
Therefore, by (3.69) and ,
[TABLE]
Hence, using Lemma 3.9, we derive from the Young inequality and (3.73) that there exist positive constants and such that
[TABLE]
The proof of Lemma 3.10 is completed. ∎
Our next goal is to make sure that Lemma 3.10 is sufficient to enforce boundedness of for all and , which plays a key step in the derivation of our main results.
Lemma 3.11**.**
Suppose that the conditions of Lemma 3.10 hold. Then for any there exists a positive constant such that
[TABLE]
Proof.
Firstly, let , where is the same as Lemma 3.10. In view of (3.71), we have
[TABLE]
where and are the same as Lemma 3.3. Next, observe that and yields to so that, in view of the Gagliardo–Nirenberg inequality, (3.70) and using the Young inequality, one can get there exist positive constants and such that for any
[TABLE]
where we have used together with and . Inserting (3.77) into (3.76), choosing appropriately small and using the Hölder inequality, we can get (3.75). ∎
4 The proof of main results
In this section, we are going to prove our main result. To this end, we will proceed in two steps. Firstly, applying the standard regularity theory of partial differential equation, we turn the bounds from Lemma 3.11 into a higher order bound for .
Lemma 4.1**.**
Suppose that the conditions of Theorem 1.1 (or Theorem 1.2) hold. Let and be the solution of (1.3). Then there exists a constant independent of such that the component of satisfies
[TABLE]
Proof.
Due to is bounded for any large (see Lemma 3.11), we infer from the standard regularity theory of parabolic equation (or elliptic equation, ) (see e.g. [7]) that (4.1) holds. ∎
The previous lemmas at hand, we can now pass to the proof of our main result. Its proof is based on a Moser-type iteration (see e.g. [36] and [18]).
Lemma 4.2**.**
Under the assumptions of Theorem 1.1 (or Theorem 1.2), one can find a positive constant such that for every
[TABLE]
Proof.
Throughout the proof of Lemma 4.2, we use to denote the different positive constants independent of and (
Case For any multiplying both sides of the first equation in (1.3) by , integrating over , integrating by parts and using the Young inequality and (4.1) and (2.3), we derive that
[TABLE]
with , where in the last inequality we have used the fact that and . Due to (4.3), we deduce that
[TABLE]
Now, we let and
[TABLE]
We now invoke the Gagliardo–Nirenberg inequality ensures that
[TABLE]
where
[TABLE]
and
[TABLE]
Therefore, an application of the Young inequality yields
[TABLE]
Here we have used the fact that and (by ). Thus, in light of by means of (4.5)–(4.7),
[TABLE]
with some Here we have used the fact that
[TABLE]
and
[TABLE]
Integrating (4.8) over with , we derive
[TABLE]
If for any large then we obtain (4.2) directly. Otherwise, by a straightforward induction, we have
[TABLE]
Combined with the boundedness of and in light of for all , so that, taking -th roots on both sides of (4.10), we can easily get (4.2).
Case : Due to Lemmas 3.7, (3.8) and 3.11, we may choose
[TABLE]
such that
[TABLE]
Next, testing the first equation in (1.3) by , integrating over , integrating by parts and applying the Young inequality and (4.1), we derive that
[TABLE]
with . Here we have used the fact that and . Therefore, (4.13) yields to
[TABLE]
for all Let and
[TABLE]
where is given by (4.11). As moreover by the Gagliardo CNirenberg inequality, we have
[TABLE]
where
[TABLE]
and
[TABLE]
Therefore, in light of the Young inequality, we conclude that
[TABLE]
where we have utilized the following facts
[TABLE]
The fact then ensures
[TABLE]
so that, in light of (4.11), (4.15)–(4.17),
[TABLE]
with some where
[TABLE]
Here we note that for , where satisfies for all with some . Next, we integrate (4.18) over with , then yields to
[TABLE]
If for any large then we derive (4.2) holds. Otherwise, by a straightforward induction, we have
[TABLE]
On the other hand, due to the fact that (for all ), a simple computation yields
[TABLE]
This together with (4.20) entails that
[TABLE]
which after taking readily implies that (4.2) is valid. ∎
The previous lemmas at hand, we can conclude main results in a straightforward manner. The proof of main results Theorem 1.2 (and Theorem 1.1) will be proved if we can show . Suppose on contrary that . In view of (4.2), we apply Lemma 2.3 to reach a contradiction. Hence the classical solution of (1.3) is global in time and bounded. ∎
Acknowledgement: Acknowledgement: This work is partially supported by Shandong Provincial Science Foundation for Outstanding Youth (No. ZR2018JL005).
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