N-barrier maximum principle for degenerate elliptic systems and its application

In this paper, we prove the N-barrier maximum principle, which extends the result in [5] from linear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems of porous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of the solutions to the above-mentioned degenerate nonlinear diffusion equations including the Shigesada-Kawasaki-Teramoto model as a special case. As an application of the N-barrier maximum principle to a coexistence problem in ecology, we show the nonexistence of waves in a three-species degenerate elliptic systems.

( 1.4) The main contribution of the N-barrier maximum principle is that it provides rather generic a priori upper and lower bounds for the linear combination of the components of a vector-valued solution which hold for a wide class of reaction terms and boundary conditions. In particular, the key ingredient in the poof relies on the delicate construction of an appropriate N-barrier which allows us to establish the a priori estimates by contradiction.
Theorem 1.1 (NBMP for m = 1, [5,7]). Assume that [H] holds. Given any set of α i > 0 (i = 1, 2, · · · , n), suppose that (u 1 (x), u 2 (x), · · · , u n (x)) is a nonnegative C 2 solution to (BVP) with m = 1. Then which was proposed by Shigesada, Kawasaki and Teramoto ( [36]) in 1979 to study the spatial segregation problem for two competing species. Here u(y, t) and v(y, t) stand for the density of the two species u and v, respectively, and Ω ⊆ R n is the habitat of the two species. d 1 ∆u and d 2 ∆v come from the random movements of individual species with diffusion rates d 1 , d 2 > 0. Meanwhile, the terms ∆ u (ρ 11 u + ρ 12 v) and ∆ v (ρ 21 u + ρ 22 v) include the self-diffusion and cross-diffusion due to the directed movements of the individuals toward favorable habitats. The coefficients ρ 11 and ρ 22 are referred to as the self-diffusion rates, while ρ 12 and ρ 21 are the cross-diffusion rates. In addition, the coefficients σ i , c ii (i = 1, 2), and c ij (i, j = 1, 2 with i = j) are the intrinsic growth rates, the intra-specific competition rates, and the inter-specific competition rates, which are all assumed to be positive, respectively. To tackle the problem as to which species will survive in a competitive system is of importance in ecology. To this end, we consider traveling wave solutions, which are solutions of the form (u(y, t), v(y, t)) = (u(x), v(x)), x = y − θ t, (1.8) where x ∈ R and θ ∈ R is the propagation speed of the traveling wave. Ecologically, the sign of θ indicates which species is stronger and can survive. Inserting (1.8) into (SKT) with Ω = R leads to When the self-diffusion and the cross-diffusion effects are neglected or ρ 11 = ρ 12 = ρ 21 = ρ 22 = 0, (SKT) with Ω = R and (SKT-tw) reduce respectively to where (LV) is the celebrated Lotka-Volterra competition-diffusion system of two species and the NBMP for (LV-tw) has been established by applying Theorem 1.1 for (LV-tw) ( [5]). We illustrate our motivation for establishing Theorem 1.1 for (LV-tw) as follows. When the habitat of the two competing species u and v is resource-limited, the investigation of the total mass or the total density of the two species v and v is essential. This gives rise to the problem of estimating the total density u(x) + v(x) in (LV-tw). In addition, another issue which motivates us to study the estimate of u(x) + v(x) is the measurement of the species evenness index J for (LV-tw). J is defined via Shannon's diversity index H ( [3,11,30,37]), i.e.
s is the total number of species, and ι i is the proportion of the i-th species determined by dividing the number of the i-th species species by the total number of all species. The species evenness index J for (LV-tw) is given by We see u(x) + v(x) is involved in the calculation of J . Another problem we are concerned with is the parameter dependence on the estimate of u(x) + v(x). When d 1 = d 2 , upper and lower bounds of u(x) + v(x) are given in [6] by an approach based on the elliptic maximum principle. For the case of d 1 = d 2 , an affirmative answer to an even more general problem of estimating α u + β v, where α, β > 0 are arbitrary constants, is given by means of Theorem 1.1.
To illustrate Theorem 1.2, we present an example. Suppose that m = n = 2, l i = 1 and The degenerate elliptic system (NDC-tw) arises from the study of traveling waves in (SKT) without the presence of diffusion and cross-diffusion, and Ω replaced by R, i.e. (NDC) The nonlinear diffusion-competition system (NDC) has been studied, for example in [13]. Under suitable restrictions on the coefficients, explicit spatially periodic stationary solutions to (NDC) can be found. In addition, for appropriate diffusion coefficients the existence of an explicit, unbounded traveling wave to (NDC) is proved under either strong or weak competition. An immediate consequence of Theorem 1.2 is the following NBMP for (NDC-tw). Corollary 1.3 (NBMP for NDC-tw). Assume that (u(x), v(x)) is a nonnegative C 2 solution to (NDCtw). For any set of α i > 0 (i = 1, 2), we havē with χ given by (1.4) andū σ j c ji , respectively is the smallest (largest, respectively) u i -intercept of the two planes for each i = 1, 2. The desired result follows from Theorem 1.2.
As an interesting application of the linear diffusion NBMP (Theorem 1.1), we investigate the situation where one exotic competing species (say, w) invades the ecological system of two native species (say, u and v) that are competing in the absence of w. A problem related to competitive exclusion ( [2,18,19,21,25,38]) or competitor-mediated coexistence ( [4,22,26]) then arises. The Lotka-Volterra system of three competing species is usually used to model this situation ( [1,10,12,16,17,23,24,26,29,9,40]). Under this situation, the traveling wave solution (u(x), v(x), w(x)) satisfies the following system: is investigated under certain assumptions on the parameters by finding exact solutions ( [8,6]) and using the numerical tracking method AUTO ( [8]). A one-hump wave is referred to as a traveling wave consisting of a forward front v, a backward front u, and a pulse w in the middle. On the other hand, nonexistence of solutions for the problem (1.20), (1.21) is established by means of the NBMP (Theorem 1.1) as well as the elliptic maximum principle under certain conditions ( [6,5]). Recently, new dynamical patterns exhibited by the solutions of the Lotka-Volterra system of three competing species have been found in [26], where traveling wave solutions of the three species (i.e. solutions of (1.20) are used as building blocks (1.20) to generate dynamical patterns in which three species coexist. This numerical evidence demonstrates (indicates) from the viewpoint of dynamical coexistence of the three species the great importance of the one-hump waves in the problem (1.20), (1.21).
The linear diffusion terms in (1.20) are based on Fick's law in which the population flux is proportional to the gradient of the population density. In some situations, however, evidences from field studies have shown the inadequacy of this model. Due to population pressure, the phenomenon that species tend to avoid crowded can be characterized by the population flux which depends on both the population density and its gradient ( [27,35,39]). Gurney and Nisbet considered the nonlinear diffusion effect described above, and proposed the following the model ( [14,15]) Assume that the following hypotheses hold: (ii) Assume that the following hypotheses hold: We note that when the boundary conditions are imposed at x = ±∞ like (1.21), hypotheses [H0] and [H1] are simultaneously satisfied. Roughly speaking, (i) of Theorem 1.4 says from the viewpoint of ecology that when the intrinsic growth rate σ 3 of w is sufficiently small ( i.e. [H3]), the three species u, v and w cannot coexist in the ecological system modeled by (1.23), (1.21). In other words, competitor-mediated coexistence cannot occur in such a circumstance. On the other hand, [H6] is satisfied when the boundary conditions are where v =ṽ, w =w solves whenever the coexistence state (ṽ,w) exists.
[H4] is an extra hypothesis on the profile of the wave. As a consequence, (ii) of Theorem 1.4 asserts that under certain conditions on the boundary conditions (i.e.
[H6]) and on the profile of the wave (i.e. [H4]), coexistence among the three species u, v and w cannot occur when the intrinsic growth rate σ 3 of w is sufficiently large (i.e. [H5]). The remainder of this paper is organized as follows. Section 2 is devoted to the proof of Theorem 1.2. As an application of Theorem 1.2, we show in Section 3 the nonexistence result of three species in Theorem 1.4.
In Section 4, we propose some open problems concerning the NBMP. Finally, some exact traveling wave solutions and the solutions of a system of algebraic equations needed in the proof of Theorem 1.2 are given in the Appendix (Section 5).

Proof of Theorem 1.2
and satisfy the following differential inequalities and asymptotic behavior: where e − and e + are given by (1.3). If the hypothesis [H] For i = 1, 2, · · · , n, there existū i > 0 such that whereR is as defined in [H] holds, then we have for any Proof. For the case where e + = (0, · · · , 0) or e − = (0, · · · , 0), a trivial lower bound of 1) for the case e + = (0, ..., 0) and e − = (0, ..., 0). To this end, we let Adding the n equations in (BVP-u), we obtain a single equation involving p(x) and q(x) First of all, we show how to construct the N-barrier.
(1) Since the line ū u in the first quadrant of the uv-plane, this leads to the following equations:

31)
By Lemma 5.1 (see Section 5), λ 1 is given by (2.32) (2) The u-coordinate of the u-intercept and the v-coordinate of the v-intercept of the ellipse α d 1 u 2 + β d 2 v 2 = λ 1 are λ 1 α d 1 and λ 1 β d 2 , respectively; the u-coordinate of the u-intercept and the vcoordinate of the line η 1 = α u + β v are η 1 α and η 1 β , respectively. Because of (2.37) This means that when min α d 1 , β d 2 = α d 1 , the ellipse α d 1 u 2 +β d 2 v 2 = λ 1 and the line η 1 = α u+β v possesess the same u-coordinate of the u-intercept, i.e. λ 1 α d 1 = η 1 α ; meanwhile, the inequality indicates that the v-coordinate of the v-intercept of the line η 1 = α u + β v is not larger than that of the v-intercept of the ellipse α d 1 u 2 + β d 2 v 2 = λ 1 . A similar conclusion can be drawn for the case of min (3) The fact that the line η 1 = α u + β v is tangent to the ellipse α d 1 u 2 + β d 2 v 2 = λ 2 at (u, v) in the first quadrant of the uv-plane yields the following equations: Employing Lemma 5.2 in Section 5, we obtain (2.41) The above three steps complete the construction of the N-barrier. Finally, we determine the line η 2 = α u+β v by setting such that, as in step (ii), the line η 2 = α u + β v lies entirely below the ellipse α d 1 u 2 + β d 2 v 2 = λ 2 in the first quadrant of the uv-plane. Combining (2.32), (2.33), (2.41) and (2.42), we arrive at The lower bound η 2 coincides with that given in Corollary 1.3.
It follows immediately from step (ii) that there are two conditions: We show the N-barrier for each condition in Figure 1: the N-barrier for the case min is shown in Figure 1(a), while the one for the case min α d 1 , β d 2 = β d 2 is shown in Figure 1(b). We note that through the example of Figure 1 in which the N-barrier for the lower dimensional problem (2.25) is constructed, the N-barrier in the hyper-space in the proof of Proposition 1 become immediate.

Remark 2.2 (N-barrier for upper bounds).
We illustrate the construction of the N-barrier in Proposition 2 for the case when m = n = 2. For consistency, we use the setting in Remark 2.1.
(i) Ellipse α d 1 u 2 + β d 2 v 2 = λ 1 We first determine the ellipse α d 1 u 2 + β d 2 v 2 = λ 1 by letting The u-coordinate of the u-intercept and the v-coordinate of the v-intercept of the ellipse α d 1 u 2 + β d 2 v 2 = λ 1 are λ 1 α d 1 and λ 1 β d 2 , respectively; the u-coordinate of the u-intercept and the vcoordinate of the line ū u + v v = 1 areū andv, respectively. It turns out that when max α d 1ū 2 , β d 2v 2 = α d 1ū 2 , we have This means that the ellipse α d 1 u 2 + β d 2 v 2 = λ 1 lies entirely above the line ū u in the first quadrant of the uv-plane, we have the following equations: Employing Lemma 5.2 in Section 5, we obtain (2.73) We note that the line η 1 = α u + β v lies entirely above the ellipse α d 1 u 2 + β d 2 v 2 = λ 1 in the first quadrant of the uv-plane.
The u-coordinate of the u-intercept and the v-coordinate of the v-intercept of the ellipse α d 1 u 2 + β d 2 v 2 = λ 2 are λ 2 α d 1 and λ 2 β d 2 , respectively; the u-coordinate of the u-intercept and the vcoordinate of the line η 1 = α u + β v are η 1 α and η 1 β , respectively. It follows that We see from the construction of the ellipse α d 1 u 2 +β d 2 v 2 = λ 2 that the ellipse α d 1 u 2 +β d 2 v 2 = λ 2 lies entirely above the line η 1 = α u + β v in the first quadrant of the uv-plane.
It is readily seen from that, depending on max α d 1ū 2 , β d 2v 2 and max We are now in the position to prove Theorem 1.2.
Proof of Theorem 1.2. In Propositions 1 and 2, we obtain a lower and upper bound for  (2.83) Letting α = 1 2 and β = 1 3 , it follows immediately that α u(x) + β v(x) = 30 tanh 2 x − 172 3 tanh x + 98 3 is monotonically decreasing in x. As a result, On the other hand, upper and lower bounds given by Corollary 1.3 turn out to be In this section, we prove Theorem 1.4 by contradiction.
Proof of Theorem 1.4. We first prove (i). Suppose to the contrary that there exists a solution (u(x), v(x), w(x)) to the problem (1.23). Due to [H1], we have w x (x 0 ) = 0 and w xx (x 0 ) ≤ 0. Since w(x) satisfies and (w 2 ) xx = 2 (w 2 x + w w xx ), we obtain     This lead to an upper bound of w(x), i.e.
By virtue of the inequality w(x) < σ 3 c 33 , the last two equations in (1.23) become 6) which contradicts (3.2). This completes the proof of (i). To prove (ii), an easy observation leads to since w(x) > 0, x ∈ R. Letting u 1 = u, u 2 = v and α 1 = c 31 , α 2 = c 32 , an upper bound of c 31 u(x) + c 32 v(x) given by Corollary 1.3 is whereū * andv * are defined in [H5]. It follows from the last inequality that On the other hand, [H4] leads to the fact that w x (x 0 ) = 0 and w xx (x 0 ) ≥ 0, and hence However, this is a contradiction with [H6]. We complete the proof of (ii).

Concluding Remarks
In this paper, we have shown the NBMP for (BVP) with m > 1, and apply it the establish the nonexistence of three species waves in (1.23) under certain conditions. In particular, the upper and lower bounds given by the NBMP are verified by using exact solutions.
The N-barrier method is still under investigation, and there is a number of open problems concerning NBMP. We point out some of them for further study: • NBMP for periodic solutions: As we can see from [13], (NDC) admits periodic stationary solutions under certain conditions on the parameters. Motivated by this work, we show in Theorem 5.4 (see Section 5.3) that for the three-specie case (1.23) also admits periodic solutions under certain conditions on the parameters. The question is how to correct the N-barrier method adapted for periodic solutions? • NBMP for multi-dimensional equations: The N-barrier method has not yet been applied to multidimensional equations since there is still a lack of systematic formulation of the method in the multidimensional case. The difficulty is to construct appropriate N-barriers corresponding to operator like ∆u, ∇u, ∆(u 2 ) etc.. • NBMP for strongly-coupled equations: The N-barrier method developed to study (1.1) can also be applied to a wide class of elliptic systems, for instance, the system (SKT-tw) in which diffusion, self-diffusion, and cross-diffusion are strongly coupled.
These are left as the future work. , i, j = 1, 2, · · · , n; Proof. Due to (5.1), we may assume for some K > 0. It follows immediately from (5.2) that K is determined by and hence Therefore, Λ is given by we have Proof. Lemma 5.2 follows from lettingū i = 1 α i in Lemma 5.1.

5.2.
Exact solutions using Tanh method. Enlightened by the works of [20,8,31,32], our idea is to look for a monotone solution with a hyperbolic tangent profile. We make the following ansätz for solving (2.82): where k 1 and k 2 are positive constants to be determined. Since the derivative of tanh x is expressible in terms of itself, i.e. d dx tanh x = 1 − tanh 2 x, we see that the nth derivative of a polynomial in tanh x with any order is also a a polynomial in tanh x. Inserting ansätz (5.12) into (2.82), this fact enables us to get The result obtained is summarized in the following Theorem 5.3. System (2.82) has a solution of the form (5.12) provided that (5.38) holds.