Uniform Lipschitz Bound for a Competition Diffusion Advection System with Strong Competition ()
1. Introduction
In this paper, we consider the following competition-diffusion-advection system
(1)
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 [1].
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 [2]. For the case when
,
, Cantrell et al. [3] [4] 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. [3] and Chen et al. [5] 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. [1] 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 [6] 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 [6], proving the uniform bound in Lipschitz norm. Without loss of generality, we set
in system (1), and consider the time-independent case:
(2)
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 [7] - [14], but also is key to the understanding of phase separation of Gross-Pitaevskii systems of modeling Bose-Einstein condensates, see [15] - [24] and reference therein.
The uniform Hölder regularity in related problems have been studied by many authors, see [10] [11] [15] [25], for the elliptic case, [15] [18] for the parabolic case, and [26] [27] [28] 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 [10] 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 [14] generalized the result to arbitrary number of components (possibly with suitable reaction terms). In [29] 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 [30], Soave and Zilio extended the result of [10] [14] [29] to the case of arbitrary number of components and general reaction terms. The approach here follows the mainstream of [30], 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 [1] [4], if we let
,
then system (2) is equivalent to
(3)
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
,
Furthermore,
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 [11], 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. [11]. Let
and consider the following problem:
(4)
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
Furthermore,
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
that gives
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,
(5)
from which the desired result follows. Inspired from the work of Soave and Zilio in [30], 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
(6)
where
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
(7)
If
, then
as
. Moreover the limit w, z satisfies
(8)
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
Therefore,
Notice that
,
,
. Hence in the sense of distributions that
, and in particular
(9)
for k sufficiently large.
Lemma 9. There exist a constant
independent of k such that for any
and
,
(10)
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 [31].
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 [29], 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.
Acknowledgements
We thank the Editor and the referee for their comments. The work is partially supported by PRC grant NSFC 11601224.