In this paper, we consider the following competition-diffusion-advection system
where and denote the densities of two competing species at position and time . represents a local population growth rate that depends on location. In some sense, can reflect the quality and quantity of resources available at the location x, where the favorable region acts as a source and the unfavorable part is a sink region . and account for random diffusion, and and represent movement upward along the environmental gradient. The two non-negative constants and measure the tendency of the two species to move up along the gradient of , and and represent the random diffusion rates of two species, respectively. The positive constants b and c are the intraspecific and k the interspecific competition rates. is a bounded domain in with smooth boundary . The Zero-flux boundary condition in (1) means that no individuals cross the boundary of the habitat.
From the mathematical viewpoint, qualitative properties of non-negative solutions of system (1) have been extensively studied. We will briefly review some of them, for a more complete and detailed discussion, see . For the case when , , Cantrell et al.   showed that if is convex and , then for positive small the semi-trivial steady state of (1) is globally asymptotically stable. In contrast, Cantrell et al.  and Chen et al.  proved that for large values system (1) can have a stable positive steady state and two competing species coexist for large . For the case when , , Chen et al.  showed that if the ratio is suitable related, then the two species coexist for sufficiently large .
For the case when k is sufficiently large, we proved in  that system is expected to approach a limiting configuration where all the populations survive but have disjoint habitats. Precisely, we proved that k-dependent solutions of (1) are uniformly bounded in Hölder spaces and they converge to the positive and negative parts of a solution of a scalar limit problem. The objective of this paper is to improve the result in , proving the uniform bound in Lipschitz norm. Without loss of generality, we set in system (1), and consider the time-independent case:
Throughout this paper, we assume that the function and is positive somewhere in . Our main result is as follows.
Theorem 1. Let be non-negative solutions of (2). Then for every compact set there exist independent of k such that
Note that the study of strong-competition limits in corresponding elliptic or parabolic systems is of interest not only for questions of spatial segregation and coexistence, in population dynamics, as here and in  - , but also is key to the understanding of phase separation of Gross-Pitaevskii systems of modeling Bose-Einstein condensates, see  -  and reference therein.
The uniform Hölder regularity in related problems have been studied by many authors, see    , for the elliptic case,   for the parabolic case, and    for the fractional diffusion case. Concerning the uniform Lipschitz boundedness, some results have already been observed in literature. For the case of two components without advection and reaction terms, Conti, Terracini and Verzini in  proved that if are non-negative solutions of
with and traces , then is uniformly bounded in the Lipschitz norm. By using Kato’s inequality, Wang and Zhang  generalized the result to arbitrary number of components (possibly with suitable reaction terms). In  Berestycki, Lin, Wei and Zhao deal with the Gross-Pitaevskii system in dimension , they proved that if are uniformly bounded solutions of
with uniformly bounded coefficients , , then and are uniformly bounded in the Lipschitz norm. In the recent paper , Soave and Zilio extended the result of    to the case of arbitrary number of components and general reaction terms. The approach here follows the mainstream of , based upon the blow-up technique and the almost monotonicity formula by Caffarelli-Jerison-Kenig.
The rest of the paper is organized as follows: Section 2 is devoted to giving some prior estimates. Section 3 deals with the blow-up analysis. In Section 4, we prove the uniform bound in the Lipschitz norm.
2. Some Preliminary Results
In this section, we will derive some basic estimates. As in  , if we let , then system (2) is equivalent to
We start with the following observation of system (3).
Lemma 2. Let , and suppose that is a non-negative solution of (2). Then for all ,
Proof. We prove the estimate for and ; that for and follows similarly. Let denote a point where . Assume by contradiction that
Since , then by the Hopf lemma . Hence, we have , and . It then follows that
which is a contradiction. Hence, for all ,
and for all ,
This completes the proof of Lemma 2.
In the blow up procedure, we need the following lemma, which extends the result in , Lemma 4.4.
Lemma 3. Let be the open ball in . Assume that satisfying
where and , H are two positive constant. Then for every ,
where C is a positive constant depending only on , R and .
Proof. The proof is inspired by Conti et al. . Let and consider the following problem:
We claim that:
1) for ;
2) for ;
3) for , where .
To prove (1), we observe that is defined on and that , on . Indeed, if not, is positive on and , then ; On the other hand, since
then is strictly increasing on . Hence, , a contradiction. Since is positive, we have . Then using the initial conditions and comparison arguments, for , and thus (2) follows. Finally, we define . Then and
since , . Using again comparison arguments, we obtain
which gives (3).
Now let be the solution of
Clearly satisfies the assumptions in (4) for a suitable , so . Recall that , thus we have
If we let , then by construction we have that v is a radially symmetric function with in , on , and hence, by maximum principle, in . Moreover, since is an increasing function, if we prove that , then we will obtain the required bound for and the proof of the lemma will be concluded. Using (3) and choosing , , we obtain
Substituting in the inequality in (2), we finally have
then setting , provides the desired inequality.
3. Asymptotic of the Blow up Sequence
We deduce from Section 2 that the solutions of system (2) is uniform bounded in . For any compact set , we are aim to show that the Lipschitz semi-norm of solutions to system (2) is bounded in K, uniformly in k. To begin with, let be a cut-off function such that , in K and supp , we want to show that there exist a constant independent of k such that,
from which the desired result follows. Inspired from the work of Soave and Zilio in , we assume by contradiction that, up to a subsequence, it holds
Without loss of generality, we may assume that the supremum is achieved by at a point , that is
Now we introduce two blow-up sequences
where . We choose the scaling factor in such a way that
Note that, since , we have as . Furthermore, if is a solution to (2), then satisfies
The following lemma focuses on some preliminary properties of the blow up sequences.
Lemma 4. In the previous blow-up setting, the following assertions hold:
1) , , uniformly in as , in particular,
2) we have
uniformly in all as ;
3) the sequence and have uniformly bounded Lip-seminorm:
furthermore and as ;
4) there exist , globally Lipschitz continuous in with Lipschitz constant equal to 1, such that up to a subsequence:
5) there holds in as , and for any there exist , independent of k, such that
If , then as . Moreover the limit w, z satisfies
Proof. 1) Since , then for every ,
Note also that is positive somewhere in , thus there exists a positive constant , such that .
2) By Lemma 2 and the definitions of and , we have
Similarly, we have .
The uniform bound on the Lipschitz seminorm of , and the fact that , are direct consequence of the definitions. Moreover
as , then (3) holds.
4) We will only prove the estimate of and , that of and are similar. For any fixed , we may let k sufficient large such that . The sequence has a uniformly Lipschitz seminorm in , and is uniformly bounded in 0. Hence by the Ascoli-Arzelà theorem, it is uniformly convergent (up to a subsequence) to some having Lipschitz-seminorm bounded by 1. To complete the proof, we shall show that as in . To this aim, it is sufficient to observe that for any compact ,
where l denotes the Lipschitz constant of , C is the uniformly boundedness of . Since and K is compact, the desired result follows.
5) To prove (7), it is sufficient to test the equation for against a smooth cut-off function such that in and in , we obtain:
By the uniform boundedness of in compact sets and the fact that , there exists a constant independent of k, such that
Testing the equation for against , we also deduce that
where C is a positive constant independent of k. This implies that, up to a subsequence,
To prove the strong convergence, we test the equation for against , and recalling that uniformly in , we deduce that as ,
From this we can pass from the weak convergence to the strong one.
To prove (8), we note that
By strong convergence and (1), above equation can be passing to the limit. So up to a subsequence, we have in the distribute sense that
Since , we have in , and thus
Similarly, the result holds for z. This completes the proof of Lemma 4.
Lemma 5. The limit function is not constant. In particular, w is neither trivial nor constant.
Proof. We divide the proof according to properties of .
Case 1. ( ) is bounded. The equation for can be rewrite as:
Since is uniformly bounded in any compact set of , by standard regularity theory for elliptic equations, we deduce that for every compact there exist independent of k such that . This implies that, up to a subsequence
So that in particular , and cannot be a vector of constant functions.
Case 2. . By Lemma 4 (5) we infer that in , and the choice of implies that , so there are only two possibilities: either , or .
Assume at first that , then , and by continuity of it results that in an open neighbourhood of 0. Moreover, there exists , such that
for sufficient large k. Thanks to Lemma 4 (4), we have
as , for every . Thus, whenever k is sufficiently large, in . As a consequence, if we Let , then satisfies
By Lemma 3,
Hence for every ,
Note that . By standard regularity theory for elliptic equations, we have . Note also that and (by Lemma 4), we then deduce that
This implies that up to a subsequence in . In particular , in contradiction with the fact that in a neighbourhood of 0. Thus, the case is impossible, therefore . As a consequence the same argument described above provides in . If we let , then satisfies
By Lemma 3 again,
By the uniform boundedness of the sequence in , we infer that,
And hence up to a subsequence in . In particular, by Lemma 4 (3) we have
which completes the proof.
Lemma 6. There exist such that .
Proof. Let us assume by contradiction that there exists a subsequence . Reasoning as in the previous lemma, the limiting function satisfies
and , thus thanks to the Liouville theorem, are constant. This contradicts the fact that .
We conclude this section by summing up what we proved so far in the following statement.
Proposition 7. Under the previous notations, we have
1) Up to a subsequence
is non-trivial and non-constant, and in particular ;
2) There exist such that ;
3) If ( ) is bounded, then
where as ;
4) If , then both w and z are subharmonic in , and
4. Uniform Lipschitz Bounds with Respect to k
This section is devoted to the study of the Lipschitz uniform continuty of the system (2). In Section 3, we have proved that the limit is non-trivial and non-constant, and in particular (Proposition 7). In what follows, we will show that one of the components of is identically zero and the other is a constant, which bring us to a contradiction.
For any given functions, we let
we shall make use of the celebrated almost monotonicity formula of Callarelli-Jerison-Kening, which we recall here in its original formulation.
Theorem 8. (Callarelli-Jerison-Kening almost monotonicity). Suppose u, v are non-negative, continuous functions on the unit ball . Suppose that and in the sense of distributions and that for all . Then there exist a constant C depending only on dimension such that for every :
Moreover, if u and v satisfy the same assumptions also in the ball , then there exist a dimensional constant such that
Now we consider the following systems
Notice that , , . Hence in the sense of distributions that , and in particular
for k sufficiently large.
Lemma 9. There exist a constant independent of k such that for any and ,
Proof. By (9), it follows that the positive and negative part of fall under the assumptions of Theorem 8, and in particular
where is independent of k.
Corollary 1. Any blow-up limit is made of ordered functions, that is if
then either or , in .
Proof. Indeed, scaling properly of the estimate (10), we obtain for every and k large enough
as . The conclusion follows by strong convergence of the blow-up sequence and by the continuity of the blow-up limit.
In order to complete the proof of Theorem 1, we need the following classical result, for which we refer to Lemma 2 in .
Lemma 10. Let , and let be a solution of
if we assume u to be non-negative, then .
With the lemmas above, we can now complete the proof of uniform Lipschitz bounds.
Proof of Theorem 1. According to , we divided the proof in two steps.
Step 1. The case ( ) bounded. In this case by Proposition 7 the limiting function is a non-negative, non-trivial, non-constant and sublinear solution of
By Corollary 1, we evince that either in , or in . Without loss of generality, we suppose that and . Thus
Thanks to Lemma 10, we have . But then
Then by the classical Liouville theorem, we have w is a constant, which is in contradiction with the fact that w is non-trivial and non-constant.
Step 2. the case . In such a situation, . Notice that
that is . Then Corollary 1 implies that either , or . Without loss of generality, we suppose that , then the classical Liouville theorem shows that
since , Therefore .
We deduce that , and , this implies that w is a constant, similarly, a contradiction. This completes the proof of Theorem 1.
5. Conclusion and Further Works
The study of the asymptotic behavior of singular perturbed equations and systems of elliptic or parabolic type is very broad and active subject of research. In this paper, we study a competition-diffusion-advection system for two competing species in a spatially heterogenous environment. We prove the uniform Lipschitz bound for solutions of the system, which extends known quasi-optimal results and covers the optimal case for this problem. We remark that the existence of uniform Lipschitz bounds is relevant not only for a pure mathematical flavour. As already observed in , it is necessary to obtain, rigorous qualitative description of phase separation phenomena (the uniform Hölder bounds would not be sufficient for this purpose.)
Finally, we mention that there are many interesting problems for further study. Note that we established uniform Lipschitz bound for solutions to elliptic system (2), naturally to ask whether our results can be extended to the parabolic system (1)? Up to our knowledge, the optimal Lipschitz bound for parabolic setting is unknown even for the case when (without advection terms) in system (1). Moreover, in system (2) the advection rates and are fixed nonzero constants, what happens if and are k-dependent and are suitably large? In such situation, the regularity of the solutions remains a challenge, and it will be the object of a forthcoming paper.
We thank the Editor and the referee for their comments. The work is partially supported by PRC grant NSFC 11601224.
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