On the Uniqueness of the Solution to a Strongly Competing System
aa r X i v : . [ m a t h . A P ] M a r ON THE UNIQUENESS OF THE LIMITING SOLUTION TO ASTRONGLY COMPETING SYSTEM
AVETIK ARAKELYAN AND FARID BOZORGNIA*
Abstract.
This work is devoted to prove uniqueness result for the positive solution to astrongly competing system of Lotka-Volterra type in the limiting configuration, when thecompetition rate tends to infinity. Based on properties of limiting solution an alternativeproof to show uniqueness is given. Introduction
Let Ω ⊂ R d be an open, bounded, and connected domain with smooth boundary. Wetake m to be an integer number. The aim of this paper is to investigate the uniqueness ofsolution for a competition-diffusion system of Lotka-Volterra type, with Dirichlet boundaryconditions as the competition rate tends to infinity. This model of strongly competingsystems have been extensively studied from different point of views, see [3, 5, 7, 6, 8, 9]and references therein.The model describes the steady state of m competing species coexisting in the same areaΩ . Let u i ( x ) denote the population density of the i th component. The following systemshows the steady state of interaction between m components ∆ u εi = ε u εi P j = i u εj ( x ) in Ω ,u εi ≥ , i = 1 , · · · , m in Ω ,u εi ( x ) = φ i ( x ) , i = 1 , · · · , m on ∂ Ω . (1.1)Here φ i are non-negative C functions with disjoint supports that is, φ i · φ j = 0 , almosteverywhere on the boundary, and the term ε is the competition rate.This model is also called adjacent segregation, modeling when particles annihilate eachother on contact. The system (1.1) has been generalized for nonlinear diffusion or longsegregation, where species interact at a distance from each other see [4]. Also in [10] the Mathematics Subject Classification.
Key words and phrases.
Spatial segregation, Free boundary problems, Maximum principle.A. Arakelyan was partially supported by State Committee of Science MES RA, in frame of the researchproject No. 16YR-1A017 .*The corresponding author, F. Bozorgnia was supported by the FCT post-doctoral fellowshipSFRH/BPD/33962/2009 .
1N THE UNIQUENESS OF THE LIMITING SOLUTION TO A STRONGLY COMPETING SYSTEM 2 generalization of this problem has been considered for the extremal Pucci operator. Thenumerical treatment of the limiting case in system (1.1) is given in [2].The limiting configuration (solution) of (1.1) as ε tends to zero, is related to a freeboundary problem and the densities u i satisfy the system of differential inequalities. Theuniqueness of limiting solution is proven for the cases m = 2 in [5] and m = 3 in planardomain, see [7]. Later in [11] these uniqueness results have been generalized to arbitrarydimension and arbitrary number of species.In this work we give original proof for uniqueness of the limiting configuration forarbitrary m competing densities by employing properties of limiting solution, which isdifferent approach and straightforward.The outline of the paper is as follows: We state the problem in Section 2 and providemathematical background and known results, which will be used in our proof. In Section3 we prove the uniqueness of the system (1.1) in the limiting case as ε tends to zero.2. Known Results and Mathematical Background
In this section we mention some of known results for the solutions of the system (1.1),which play an important role in our study. Namely, we recall some estimates and com-pactness properties.To start with, for each fixed ε, the system (1.1) has a unique solution, see [11]. Theauthors in [11] use the sub- and sup-solution method for nonlinear elliptic systems toconstruct iterative monotone sequences which leads to the uniqueness in case of system(1.1).Let U ε = ( u ε , · · · , u εm ) be the unique solution of the system (1.1) for fixed ε. Then u εi for i = 1 , · · · , m, satisfies the following differential inequality:(1.2) − ∆ u εi ≤ . Define b u εi as b u εi := u εi − X j = i u εj , then it is easy to verify the following property(1.3) − ∆ b u εi = X j = i X h = j u εj u εh ≥ . By constructing of sub and super solution to the system (1.1), we can show that ∂u εi ∂n isbounded on ∂ Ω (independent of ε ). Then multiplying the inequality − ∆ u εi ≤ u εi andintegrating by part yields that u εi is bounded in H (Ω) for each ε .The above discussion show that the solution of the system (1.1) belongs to the followingclass F , see Lemma 2 . F = (cid:8) ( u , · · · , u m ) ∈ ( H (Ω)) m : u i ≥ , − ∆ u i ≤ , − ∆ b u i ≥ , u i = φ i on ∂ Ω (cid:9) , N THE UNIQUENESS OF THE LIMITING SOLUTION TO A STRONGLY COMPETING SYSTEM 3 where as in system(1.1) the boundary data φ i ∈ C ( ∂ Ω) , nonnegative functions and φ i · φ j = 0 , almost everywhere on the boundary.The following result in [3, 5] shows the asymptotic behavior of the system as ε → . Let U ε = ( u ε , · · · , u εm ) be the solution of system (1.1) for a fixed ε . If ε tends to zero, thenthere exists U = ( u , · · · , u m ) ∈ ( H (Ω)) m such that for all i = 1 , · · · , m :(1) up to a subsequences, u εi → u i strongly in H (Ω),(2) u i · u j = 0 if i = j a.e in Ω,(3) ∆ u i = 0 in the set { u i > } ,(4) Let x belongs to the common interface of two components u i and u j , thenlim y → x ∇ u i ( y ) = − lim y → x ∇ u j ( y ) . From above the limiting solution, as ε tends to zero, belongs to the following class: S = { ( u , · · · , u m ) ∈ F : u i · u j = 0 for i = j } . Note that the inequalities in (1.2) and (1.3) hold as ε tends to zero. Also − ∆ b u i = 0 on { x ∈ Ω : u i ( x ) > } . In this part we briefly review the known results about uniqueness of the limiting con-figuration of the system (1.1). In particular, for the case m = 2 , the limiting solution andthe rate of convergence are given (see Theorem 2 . Theorem 2.1.
Let W be harmonic in Ω with the boundary data φ − φ . Let u = W + , u = − W − , then the pair ( u , u ) is the limit configuration of any sequences ( u ε , u ε ) and: k u εi − u i k H (Ω) ≤ C · ε / as ε → , i = 1 , . For the case m = 3 , the uniqueness of the limiting configuration, as ε tends to zero, isshown in [7] on a planar domain, with appropriate boundary conditions. More precisely,the authors prove that the limiting configuration of the following system ∆ u εi = u εi ( x ) ε P j = i u εj ( x ) in Ω ,u εi ( x ) = φ i ( x ) on ∂ Ω ,i = 1 , , , minimizes the energy E ( u , u , u ) = Z Ω 3 X i =1 |∇ u i | dx, among all segregated states u i · u j = 0 , a.e. with the same boundary conditions. Remark . The system (1.1) is not in a variational form. In [6] for a class of segregationstates governed by a variational principle the proof of existence and uniqueness are shownunder some suitable conditions.
N THE UNIQUENESS OF THE LIMITING SOLUTION TO A STRONGLY COMPETING SYSTEM 4
In [11] the uniqueness of the limiting configuration and least energy property are gener-alized to arbitrary dimension and for arbitrary number of components. Following notationsin [11], let P denote the metric space { ( u , u , · · · , u m ) ∈ R m : u i ≥ , u i u j = 0 for i = j } . The authors in [11] show that the solution of the limiting problem ( u , · · · , u m ) ∈ S is aharmonic map into the space P . The harmonic map is the critical point (in weak sense)of the following energy functional Z Ω m X i =1 |∇ u i | dx, among all nonnegative segregated states u i · u j = 0 , a.e. with the same boundary condi-tions, see Theorem 1.6 in [11].Their proof is based on computing the derivative of the energy functional with respectto the geodesic homotopy between u and a comparison to an energy minimizing map v with same boundary values. This demands some procedures to avoid singularity of freeboundary. Unlike their approach, our proof is more direct and based on properties oflimiting solutions and doesn’t require results from regularity theory or harmonic maps.3. Uniqueness
In this section we prove the uniqueness for the limiting case as ε tends to zero. Ourapproach is motivated from the recent work related to the numerical analysis of a certainclass of the spatial segregation of reaction-diffusion systems (see [1]). We heavily use thefollowing notation: b w i ( x ) := w i ( x ) − X p = i w p ( x ) , for every 1 ≤ i ≤ m. Lemma 3.1.
Let two elements ( u , · · · , u m ) and ( v , · · · , v m ) belong to S . Then thefollowing equation for each ≤ i ≤ m holds: max Ω ( b u i ( x ) − b v i ( x )) = max { u i ( x ) ≤ v i ( x ) } ( b u i ( x ) − b v i ( x )) . Proof.
We argue by contradiction. Let there exists some i such that(1.4) max Ω ( b u i − b v i ) = max { u i >v i } ( b u i − b v i ) > max { u i ≤ v i } ( b u i − b v i ) . Assume D = { x ∈ Ω : u i ( x ) > v i ( x ) } , then in D we have(1.5) (cid:26) − ∆ b u i ( x ) = 0 , − ∆ b v i ( x ) ≥ , N THE UNIQUENESS OF THE LIMITING SOLUTION TO A STRONGLY COMPETING SYSTEM 5 which implies that ∆( b u i ( x ) − b v i ( x )) ≥ . The weak maximum principle yieldsmax D ( b u i − b v i ) ≤ max ∂D ( b u i − b v i ) ≤ max { u i = v i } ( b u i − b v i ) , which is inconsistent with our assumption (1.4). It is clear that we can interchange therole of b u i and b v i . Thus, we also havemax Ω ( b v i ( x ) − b u i ( x )) = max { v i ( x ) ≤ u i ( x ) } ( b v i ( x ) − b u i ( x )) , for all 1 ≤ i ≤ m. (cid:3) In view of Lemma 3.1 we define the following quantities P := max ≤ i ≤ m (cid:18) max Ω ( b u i ( x ) − b v i ( x )) (cid:19) = max ≤ i ≤ m (cid:18) max { u i ≤ v i } ( b u i ( x ) − b v i ( x )) (cid:19) ,Q := max ≤ i ≤ m (cid:18) max Ω ( b v i ( x ) − b u i ( x )) (cid:19) = max ≤ i ≤ m (cid:18) max { v i ≤ u i } ( b v i ( x ) − b u i ( x )) (cid:19) . Lemma 3.2.
Let two elements ( u , · · · , u m ) and ( v , · · · , v m ) belong to S . We set P and Q as defined above. If P > is attained for some index ≤ i ≤ m, then we have P = Q > . Moreover, there exist another index j = i and a point x ∈ Ω , such that: P = Q = max { u i ≤ v i } ( b u i − b v i ) = max { u i = v i =0 } ( b u i − b v i ) = v j ( x ) − u j ( x ) . Proof.
Let the maximum
P > i component. According tothe previous lemma, we know that ( b u i ( x ) − b v i ( x )) attains its maximum on the set { u i ( x ) ≤ v i ( x ) } . Let that maximum point be x ∗ ∈ { u i ( x ) ≤ v i ( x ) } . It is easy to see that b u i ( x ∗ ) − b v i ( x ∗ ) = P > , implies u i ( x ∗ ) = v i ( x ∗ ) = 0 . Indeed, if u i ( x ∗ ) = v i ( x ∗ ) > , then in the light of disjointness property of the components of u i and v i we get P = b u i ( x ∗ ) − b v i ( x ∗ ) = u i ( x ∗ ) − v i ( x ∗ ) = 0 which is a contradiction. If u i ( x ∗ ) < v i ( x ∗ ) , then again due to the disjointness of the densities u i , v i , we have0 < P = b u i ( x ∗ ) − b v i ( x ∗ ) = b u i ( x ∗ ) − v i ( x ∗ ) ≤ u i ( x ∗ ) − v i ( x ∗ ) < . This again leads to a contradiction. Therefore u i ( x ∗ ) = v i ( x ∗ ) = 0 . Now assume by contradiction that Q ≤ . Then by definition of Q we should have b v j ( x ) ≤ b u j ( x ) , ∀ x ∈ Ω , j = 1 , · · · , m. This apparently yields v j ( x ) ≤ u j ( x ) , ∀ x ∈ Ω , j = 1 , · · · , m. N THE UNIQUENESS OF THE LIMITING SOLUTION TO A STRONGLY COMPETING SYSTEM 6
Let D i = { u i ( x ) = v i ( x ) = 0 } , then we have0 < P = max D i ( b u i ( x ) − b v i ( x )) = max D i X j = i ( v j ( x ) − u j ( x )) ≤ . This contradiction implies that
Q >
0. By analogous proof, one can see that if P be non-positive then Q will be non-positive as well. Next, assume the maximum P is attained ata point x ∈ D i . Then, we get(1.6) 0 < P = b u i ( x ) − b v i ( x ) = ( u i ( x ) − v i ( x ))++ X j = i ( v j ( x ) − u j ( x )) = X j = i ( v j ( x ) − u j ( x )) . This shows that X j = i v j ( x ) = X j = i u j ( x ) + P > . Since ( v , · · · , v m ) ∈ S, then there exists j = i such that v j ( x ) > . This implies0 < P = b u i ( x ) − b v i ( x ) = v j ( x ) − X j = i u j ( x ) ≤ b v j ( x ) − b u j ( x ) ≤ Q. The same argument shows that Q ≤ P which yields P = Q . Hence, we can write P = v j ( x ) − X j = i u j ( x ) = b v j ( x ) − b u j ( x ) = Q. This gives us 2 P j = j u j ( x ) = 0 , and therefore u j ( x ) = 0 , ∀ j = j , which completes the last statement of the proof. (cid:3) We are ready to prove the uniqueness of a limiting configuration.
Theorem 3.3.
There exists a unique vector ( u , · · · , u m ) ∈ S, which satisfies the limitingsolution of (1.1).Proof. In order to show the uniqueness of the limiting configuration, we assume that twom-tuples ( u , · · · , u m ) and ( v , · · · , v m ) are the solutions of the system (1.1) as ε tendsto zero. These two solutions belong to the class S . For them we set P and Q as above.Then, we consider two cases P ≤ P > . If we assume that P ≤ Q ≤
0. This leads to0 ≤ − Q ≤ b u i ( x ) − b v i ( x ) ≤ P ≤ , for every 1 ≤ i ≤ m, and x ∈ Ω . This provides that b u i ( x ) = b v i ( x ) i = 1 , · · · , m, N THE UNIQUENESS OF THE LIMITING SOLUTION TO A STRONGLY COMPETING SYSTEM 7 which in turn implies u i ( x ) = v i ( x ) . Now, suppose
P > . We show that this case leads to a contradiction. Let the value P is attained for some i , then due to Lemma 3.2 there exist x ∈ Ω and j = i such that:0 < P = Q = b u i ( x ) − b v i ( y ) = max { u i = v i =0 } ( b u i ( x ) − b v i ( x )) = v j ( x ) − u j ( x ) . Let Γ be a fixed curve starting at x and ending on the boundary of Ω . Since Ω is con-nected, then one can always choose such a curve belonging to Ω . By the disjointness andsmoothness of v j and u j there exists a ball centered at x , and with radius r ( r dependson x ) which we denote it B r ( x ), such that v j ( x ) − u j ( x ) > B r ( x ) . This yields ∆( b v j ( x ) − b u j ( x )) ≥ B r ( x ) . The maximum principle implies thatmax B r ( x ) ( b v j ( x ) − b u j ( x )) = max ∂B r ( x ) ( b v j ( x ) − b u j ( x )) ≤ P. One the other hand, in view of Lemma 3.2 we have b v j ( x ) − b u j ( x ) = v j ( x ) − u j ( x ) = P, which implies that P is attained at the interior point x ∈ B r ( x ) . Thus, b v j ( x ) − b u j ( x ) ≡ P > B r ( x ) . Next let x ∈ Γ ∩ ∂B r ( x ) . We get b v j ( x ) − b u j ( x ) = P > , which leads to v j ( x ) ≥ u j ( x ) . We proceed as follows: If v j ( x ) > u j ( x ) , then as above v j ( x ) > u j ( x ) in B r ( x ) . Thisin turn implies ∆( b v j ( x ) − b u j ( x )) ≥ B r ( x ) . Again following the maximum principle and recalling that b v j ( x ) − b u j ( x ) = P we con-clude that b v j ( x ) − b u j ( x ) = P > B r ( x ) . If v j ( x ) = u j ( x ) , then clearly the only possibility is v j ( x ) = u j ( x ) = 0 . Thus,0 < P = b v j ( x ) − b u j ( x ) = X j = j ( u j ( x ) − v j ( x )) . Following the lines of the proof of Lemma 3.2, we find some k = j , such that P = u k ( x ) − v k ( x ) = b u k ( x ) − b v k ( x ) . It is easy to see that there exists a ball B r ( x ) (without loss of generality one keeps thesame notation) ∆( b u k ( x ) − b v k ( x )) ≥ B r ( x ) . N THE UNIQUENESS OF THE LIMITING SOLUTION TO A STRONGLY COMPETING SYSTEM 8
In view of the maximum principle and above steps we obtain: b u k ( x ) − b v k ( x ) = P > B r ( x ) . Then we take x ∈ Γ ∩ ∂B r ( x ) such that x stands between the points x and x alongthe given curve Γ . According to the previous arguments for the point x we will find anindex l ∈ { , · · · , m } and corresponding ball B r ( x ) , such that | b u l ( x ) − b v l ( x ) | = P in B r ( x ) . We continue this way and obtain a sequence of points x n along the curve Γ, which aregetting closer to the boundary of Ω . Since for all j = 1 , · · · , m and x ∈ ∂ Ω we have b u j ( x ) − b v j ( x ) = b v j ( x ) − b u j ( x ) = 0 , then obviously after finite steps N we find the point x N , which will be very close to the ∂ Ω and for all j = 1 , · · · , m | b u j ( x N ) − b v j ( x N ) | < P/ . On the other hand, according to our construction for the point x N , there exists an index1 ≤ j N ≤ m such that | b u j N ( x N ) − b v j N ( x N ) | = P, which is a contradiction. This completes the proof of the uniqueness. (cid:3) References [1]
Arakelyan, A., and Barkhudaryan, R.
A numerical approach for a general class of the spatialsegregation of reaction–diffusion systems arising in population dynamics.
Computers & Mathematicswith Applications 72 , 11 (2016), 2823–2838.[2]
Bozorgnia, F.
Numerical algorithm for spatial segregation of competitive systems.
SIAM Journalon Scientific Computing 31 , 5 (2009), 3946–3958.[3]
Caffarelli, L. A., and Lin, F.-H.
Singularly perturbed elliptic systems and multi-valued harmonicfunctions with free boundaries.
Journal of the American Mathematical Society 21 , 3 (2008), 847–862.[4]
Caffarelli, L. A., Quitalo, V., and Patrizi, S.
On a long range segregation model. arXiv preprintarXiv:1505.05433 (2015).[5]
Conti, M., Terracini, S., and Verzini, G.
Asymptotic estimates for the spatial segregation ofcompetitive systems.
Advances in Mathematics 195 , 2 (2005), 524–560.[6]
Conti, M., Terracini, S., and Verzini, G.
A variational problem for the spatial segregation ofreaction-diffusion systems.
Indiana University Mathematics Journal 54 , 3 (2005), 779–815.[7]
Conti, M., Terracini, S., and Verzini, G.
Uniqueness and least energy property for solutions tostrongly competing systems.
Interfaces and Free Boundaries 8 , 4 (2006), 437–446.[8]
Dancer, E. N., and Du, Y. H.
Competing species equations with diffusion, large interactions, andjumping nonlinearities.
Journal of Differential Equations 114 , 2 (1994), 434–475.[9]
Ei, S.-I., and Yanagida, E.
Dynamics of interfaces in competition-diffusion systems.
SIAM Journalon Applied Mathematics 54 , 5 (1994), 1355–1373.[10]
Quitalo, V.
A free boundary problem arising from segregation of populations with high competition.
Archive for Rational Mechanics and Analysis 210 , 3 (2013), 857–908.[11]
Wang, K., and Zhang, Z.
Some new results in competing systems with many species. In
Annalesde l’Institut Henri Poincare (C) Non Linear Analysis (2010), vol. 27, Elsevier, pp. 739–761.
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Institute of Mathematics, NAS of Armenia, 0019 Yerevan, Armenia
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