Uniform Hölder Bounds for Competition Systems with Strong Interaction on a Subdomain ()
1. Introduction
A central problem in population ecology is the understanding of spatial behavior of interacting species, in particular in the case when the interactions are large and of competitive type. Spatial segregation may occur when two or more species interact in a highly competitive way. Such phenomenon has been studied using competition models (or its parabolic case) with positive parameter
:
(1.1)
Here Ω is a smooth bounded domain in
,
,
denotes the density of the i-th population, whose internal dynamics is prescribed by
,
, and
is the number of the species. The positive constant
is the interspecific competition rate between the population
and
, which is possibly symmetric.
In the study of system (1.1), we are mostly concerned with the asymptotic behavior of the solutions as the parameter
. It turns out that uniformly bounded solutions
of system (1.1) converge, as
, to a limiting configuration in some weak sense,
, the limit satisfies
for
, which is called the spatial segregation (cf. [1] ). Segregation systems arise in different applicative contests, from biological models for competing species to the phase segregation phenomenon in Bose-Einstein condensation of the form:
(1.2)
In recent years, people show a lot of interests in segregation phenomenon, and abroad literature is present: starting from [2] - [9] , in a series of recent papers [10] - [22] , also in the fractional diffusion case [23] [24] [25] [26] . Among the others, the following results are known: the uniform Hölder bounds [7] [12] [15] [24] ) and the optimal Lipschitz bound [16] , the Lipschitz regularity of the limiting profiles and the regularity of the free boundaries, which is defined as the nodal set
of the singular limit. It is proved that the free boundary consists of two parts: a regular set, which is
locally smooth hypersurface, and a singular set of Hausdorff dimension less then
, see [11] [17] , for the nondivergence system, [10] [14] [17] for the variational one. Further information about the structure of the singular set has been provided in [27] .
Among the models proposed so far, the species compete strongly on the whole of Ω. However, in some heterogeneous environment, species may compete to some extent in the whole of a region Ω, but compete strongly on a subdomain A. To analysis the corresponding spatial segregation phenomenon governed by strong competition on A, Crooks and Dancer [28] proposed the following k-dependent system:
(1.3)
where k is again a positive competition parameter, u and v denote the densities of two species, the self-interaction functions f and g are assumed to be continuously differentiable and such that
and
for large y. A is a nonempty open subset of Ω with smooth boundary such that
. The parameters r and s are assumed to nonnegative, and
is the characteristic function on A.
Due to the presence of the characteristic function
in (1.3), we cannot expect classical solutions in general. By a k-dependent solution of (1.3), we will mean a pair of functions
such that
,
, and satisfy (1.3) almost everywhere. The asymptotic behavior of solutions to system (1.3) has been investigated in [28] , where it is proved uniform convergence of
to a limiting profile
, u and v segregate on
but not necessarily on Ω\A. The limit problem is a system on Ω\A and a scalar equation on A. The objective of this paper is to improve the convergence result of [28] , we shall establish the uniform Hölder bounds for solutions to system (1.3). To begin with, we define
(1.4)
Due to the apparent of subdomain, we can not expect boundedness for every Hölder exponent. In fact we have the following.
Theorem 1.1. Let
be nonnegative solutions of (1.3), and
be defined in (4). Assume that for every k, there exists
, independent of k, such that
Then for every
, there exists
, independent of k, such that
Notations Throughout the paper, we denote by
the open ball with center
and radius
. If
, we simply denote by
. We assume that any
can be written as
, with
and
. In this way, we denote by
. For any
we write
and
. We also denote by
and
the tangential derivative and the radial derivative of u, respectively.
The proof of Theorem 1.1 mainly follows the blow up method, developed by Terracini and her coauthors in [7] [15] . This method is a blow up analysis and need us to establish some Liouville type results, which can be achieved by some monotonicity formulas of Alt-Caffarelli-Friedman type. Compared with [7] [15] , the segregation occurs only in the subdomain
, and we lack the essential information both of the location of A and the boundary conditions on
. In the blow up procedure, the entire solutions may segregate only on the half space. Thus the Liouville type theorems established in [7] are no longer valid in the current situation. To attack this problem, new ACF type monotonicity formulas and corresponding Liouville type theorems are needed.
The rest of this paper is organized as follows: in Section 2, we establish a monotonicity formula of ACF type, and by utilizing this monotonicity formula, we prove a Liouville type theorem for entire solutions to a semilinear system. In Section 3, we perform the blow up procedure and complete the proof of Theorem 1.1.
2. Liouville-Type Results
In this section, we prove some nonexistence result in
. The main tools will be the monotonicity formula by Alt, Caffarelli, Friedman originally stated in [29] , as well as some generalizations made by Conti, Terracini, Verzini [7] , Dancer, Wang, Zhang [12] , and Terracini, Verzini, Zilio [23] [24] . The validity of ACF type formula depends on optimal partition problems involving spectral properties of the domain. In the current situation, the spectral problem we consider involves a pair of functions defined on
with disjoint support on
. In this way we are lead to consider the following optimal partition problem. Let E be an open subset of
, and we define the first eigenvalue associated to E as
Here
stands for the tangential gradient of u on E.
Lemma 2.1. Let
be as in (1.4). We define the nondecreasing function
as
and the admissible set
by
Then we have
Proof If
, it is obviously that
A symmetrization argument gives that the optimal domain is a connected arc. Moreover, the longer the arc is, the smaller the first eigenvalue is. Thus the sum
takes its minimum for two arcs
with
If we assume that the length of
is
, then the length of
is
, and the corresponding eigenfunctions are
Thus we have
and the equality holds if and only if
.
If
, according to the argument in [30] , we have
where
,
, and
is convex and decreasing:
Setting
,
, we then have
This completes the proof of Lemma 2.1.
In the following, we shall prove an ACF type monotonicity formula associated with the following system
(2.1)
where
,
is the characteristic function on T. As in [15] , we introduced an auxiliary function:
and denote
. In this setting, we note that
is bounded in
, vanishes in
and
for a.e. x.
Under the previous notations, we can prove the following monotonicity formula.
Theorem 2.2. Let
be positive solutions of (2.1) and let
be fixed. Then there exists
such that the function
is increasing for
.
Proof The proof is inspired by [15] . In order to simplify notations we shall denote
Then
. Let us first evaluate the derivative of
for
. A straightforward calculation leads to
(2.2)
By testing the equation for u in (2.1) with
on
, we obtain
Thus we can rewrite the term
in a different way
(2.3)
Now we define
where
. Then for every
, by Hölder inequality and Young’s inequality, there holds
Substituting in (2.3) we obtain
Now we choose
in such a way that
After some calculation, we obtain
where
is defined as
With this choice of
we have
Similarly, we also have
Substituting in (2.2) we obtain
Therefore it only remains to prove that there exists a
such that for every
there holds
(2.4)
To this aim we define the functions
as
Then a change of variables gives
Notice first of all that there exists a constant
such that
for r sufficiently large. Indeed assume by contradiction this is not true, then
, which implies
since u is subharmonic, and this contradicts the assumption
. The same result clear holds also for
.
Assume (2.4) does not hold, then there exists
such that
(2.5)
in particular,
are bounded. We define
Then there exists a constant
(independent of
) such that
which ensure the existence of
such that up to a subsequence, we have
in
. Moreover, since
We infer that
on
. Then Lemma 2.1 yields
that is in contradiction with (2.5).
As in [15] , we have a suitable monotonicity formula we are ready to prove a Liouville type result for solutions to system (2.1). To begin with, we recall a Liouville type result for harmonic functions.
Lemma 2.3. ( [15] ) Let u be a harmonic function in
such that for some
there holds
Then u is constant.
Theorem 2.4. Let
be nonnegative solutions of system (2.1). Assume that for some
there holds
(2.6)
Then one of the functions is identically zero and the other is a constant.
Proof We first note that, by (2.6) and Lemma 2.3, if one of the functions is identically zero or a positive constant, then the other must be a constant or 0 respectively. Hence we may assume by contradiction that neither u nor v is constant. Then by the maximum principle u and v are positive, and Theorem 2.2 ensures the existence of a constant
such that
(2.7)
for r sufficiently large. Let
be any smooth, radial, cut-off function with the following properties:
,
in
,
in
and
. Testing the equation
with the function
on
, we obtain
Consequently,
Testing the equality
with the function
on
, we obtain
which together with the previous inequality and the fact that
, gives
Now, recalling the definition of
and f and using assumption (2.6), we obtain
Similarly,
Thus we have
which contradicts with (2.7) for r large.
Remark 2.5. If
, then
compete in the whole
. In this case we have
, see [7] for detailed proof.
A similar nonexistence result is true when studying 2-tuple of subharmonic functions on
having disjoint supports on T
Corollary 2.6. Let
such that
and for some fixed
, there exists a constant
such that
then one of the functions is identically zero and the other is a constant.
3. The Uniform Hölder Bounds
In this section, we shall establish the uniform Hölder bounds for solutions to system (1.3). Note that the strong competition effect of the system only occurs in subdomain A, while in the other regions, the equation does not contain the strong competition parameter k, so the solutions
and
are uniformly bounded independently of k in
, for each
, and in
, for each
. Therefore, up to a subsequence,
converge strongly in
. In order to improve the uniform convergence result obtained in [28] , it suffices to establish the uniform
bounds on subdomain
. We now state the main results in this section.
Theorem 3.1. Let
be nonnegative solutions of system (1.3) uniformly bounded in
. Then for every
there exists
, independent of k, such that
for every
.
The proof of Theorem 3.1 is inspired from the work of [15] . We assume by contradiction that, for some
, up to a subsequence, it holds
We can assume that
is achieved, say, by
at the pair
. That is
Let us define the rescaled functions
where
will be chosen later. By direct calculation, (
and
) satisfy the following system
where
. We note that
as
, and depending on the asymptotic behavior of the distance
, we have
, where T is either
or an half-space (when
or the limit is finite, respectively). We also observe that
Since
are uniformly bounded in
,
and
, by diect calculations it is easy to see that
(3.1)
(3.2)
In the following, we need to make different choices of the sequence
. Once
is chosen, we will use Ascoli-Arzelà’s Theorem to pass to the limit on compact sets. Now since the
’s are uniform α-Hölder continuous, it is suffices to show that
and
are bounded in k. To begin with, we need the following technical lemma, which is proved in [7] .
Lemma 3.2. [7] Let
satisfy that
where H is a positive constant, then for every
, it holds
where
is a constant, and only dependent on
.
Lemma 3.3. Let
as
be such that
(i)
for some
.
(ii)
.
Then
are uniformly bounded in k.
Proof We prove the estimate for
; that for
follows similarly. Assume by contradiction that
is unbounded. Let
and choose k sufficiently large such that
. Moreover since
and
, we have
Claim.
,
.
Indeed, note that
and by (3.1), we have
(3.3)
In order to simplify the notation, let
and for each compact set
, we choose a cut-off function
such that
on
,
on
. Then by testing (3.3) with
on K, we obtain
So, there exist two positive constants
such that
(3.4)
Since
, we have
Then (3.4) implies that
If we choose k sufficient large such that
, then
which implies the boundedness of
in K. Thus we can apply Lemma 3.1, which gives
and the claim can be easily seen.
Define
We have
, and it is Hölder continuous, and
(3.5)
where
. Moreover, by the claim
where
, as
, uniformly on any
.
From this equation, we can infer from
theory and Sobolev embeddig theory that
is uniformly Lipschitz continuous, that is, there exists
is a constant, independent with k such that
We have that, up to a subsequence,
, since
. We claim that
.
In fact, if
, then we obtain
as
, which is a contradiction. After passing to a subsequence,
converges to a continuous function
on compacts and satisfying
Moreover, (3.5) can be passed to the limit, which is
(3.6)
Thus we have
by Lemma 2.3, this contradicts (3.6). So
is uniformly bounded.
Lemma 3.4. Up to a subsequence, we have
.
Proof We assume by contradiction that there exists
such that
. Let
then we obtain
while k is sufficient large, so we can use Lemma 3.3 to conclude that
are uniformly bounded.
On the other hand, by the uniform Hölder continuity and the Ascoli-Arzelà theorem we have that, up to a subsequence, there exist
and
such that
uniformly on the compact set of
. Moreover, the choice of
implies that the equations of
and
are
(3.7)
From the first equation, we can also obtain a uniform Lipschitz estimate of
, and we also have
(3.8)
Let
in (3.7), we can obtain, up to a subsequence,
where
, or
. So we can use Theorem 2.4 and Remark 2.5 to conclude that one of
and
is identically zero and the other is a constant, which contradicts (3.8).
Now we come to the proof of Theorem 3.1.
The proof of Theorem 3.1. From Lemma 3.4, we must have
. Let
. With this choice, we know that all the assumptions of Lemma 3.3 are satisfied and hence
and
are uniformly bounded. Again by the uniform Hölder continuity and the Ascoli-Arzelà theorem we have that, up to a subsequence, there exist
and
such that
uniformly on the compact set of
. Note that
, (3.5) implies that
(3.9)
Moreover
and
satisfy the following inequalities
(3.10)
(3.11)
Let
in (3.10) and (3.11), we obtain
Now let
be a compact set, we can choose k sufficient large such that
. Let us choose a cut-off function
such that
and
on K. Multiplying (3.10) by
and integrating by parts, we obtain
Since
is uniformly Hölder continuous, it then form the boundedness of
that
is uniformly bounded on compact set K. Therefore the right hand side of previous inequality is uniformly bounded. Because
, we obtain
which yields
To sum up, we have
Thus we can infer from Corollary 2.6 that one of the limiting functions is identically zero and the other is a constant, which contradicts (3.9). The proof of Theorem 3.1 is complete.
4. Conclusion and Further Works
The study of the asymptotic behavior of singular perturbed equations and system of elliptic or parabolic type is very broad and subject of research. In this paper, We study the large-interaction limit of solutions to a singularly perturbed elliptic system modeling the steady states of two species u and v which compete to some extent throughout a domain Ω but compete strongly on a subdomain
. We improve the uniform convergence result of [28] , proving bounds in Hölder norms whenever
is a smooth bounded domain.
Finally, we mention that there many interesting problems for further study. Note that we prove the uniform Hölder bounds to a singularly perturbed elliptic system, naturally to ask whether this result can be extended to the corresponding parabolic system? Up to our knowledge, the uniform Hölder bounds for parabolic setting is unknown, and both the asymptotics and the qualitative properties of the limit segregated profiles remain a challenge, this will be the object of a forthcoming paper.
Founding
The work is partially supported by PRC grant NSFC 11601224.