N-barrier maximum principle for degenerate elliptic systems and its application
NN-BARRIER MAXIMUM PRINCIPLE FOR DEGENERATE ELLIPTIC SYSTEMSAND ITS APPLICATION
LI-CHANG HUNG, HSIAO-FENG LIU, AND CHIUN-CHUAN CHEN
Abstract.
In this paper, we prove the N-barrier maximum principle, which extends the result in [5] fromlinear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems ofporous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of thesolutions 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 coexistenceproblem in ecology, we show the nonexistence of waves in a three-species degenerate elliptic systems. Introduction and main results
The main perspective of the paper is to establish the
N-barrier maximum principle (NBMP, see[5, 7]) fordegenerate elliptic systems. To be more precise, we study d i ( u mi ) xx + θ ( u i ) x + u l i i f i ( u , u , · · · , u n ) = 0 , x ∈ R , i = 1 , , · · · , n, (1.1)where u i = u i ( x ), d i , l i > θ ∈ R , and f i ( u , u , · · · , u n ) ∈ C ( R + × R + × · · · × R + ) for i = 1 , , · · · , n .The NBMP for the linear diffusion case m = 1 has been presented in [5, 7]. In this sequel we will deal withthe nonlinear diffusion case m > x = ±∞ :( u , u , · · · , u n )( −∞ ) = e − , ( u , u , · · · , u n )( ∞ ) = e + , (1.2)where e − , e + ∈ (cid:110) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) u l i i f i ( u , u , · · · , u n ) = 0 ( i = 1 , , · · · , n ) , u , u , · · · , u n ≥ (cid:111) (1.3)are the equilibria of (1.1) which connect the solution ( u , u , · · · , u n )( x ) at x = −∞ and x = ∞ . This leadsto the boundary value problem of (1.1) and (1.2): (BVP) d i ( u mi ) xx + θ ( u i ) x + u l i i f i ( u , u , · · · , u n ) = 0 , x ∈ R , i = 1 , , · · · , n, ( u , u , · · · , u n )( −∞ ) = e − , ( u , u , · · · , u n )( ∞ ) = e + . Throughout, we assume, unless otherwise stated, that the following hypothesis on f i ( u , u , · · · , u n ) issatisfied:[ H ] For i = 1 , , · · · , n , there exist ¯ u i > ¯ u i > f i ( u , u , · · · , u n ) ≥ u , u , · · · , u n ) ∈ ¯ R ; f i ( u , u , · · · , u n ) ≤ u , u , · · · , u n ) ∈ ¯ R , where ¯ R = (cid:26) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) n (cid:88) i =1 u i ¯ u i ≤ , u , u , · · · , u n > (cid:27) ;¯ R = (cid:26) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) n (cid:88) i =1 u i ¯ u i ≥ , u , u , · · · , u n ≥ (cid:27) . Mathematics Subject Classification.
Primary 35B50; Secondary 35C07, 35K57.
Key words and phrases.
Maximum principle; traveling wave solutions;degenerate elliptic systems; reaction-diffusion equations; Lotka-Volterra. a r X i v : . [ m a t h . A P ] S e p lso, we denote by χ the characteristic function: χ = , if e + = (0 , · · · ,
0) or e − = (0 , · · · , , , otherwise . (1.4)The main contribution of the N-barrier maximum principle is that it provides rather generic a priori upperand lower bounds for the linear combination of the components of a vector-valued solution which hold for awide class of reaction terms and boundary conditions. In particular, the key ingredient in the poof relies onthe delicate construction of an appropriate N-barrier which allows us to establish the a priori estimates bycontradiction. Theorem 1.1 ( NBMP for m = 1 , [5, 7]) . Assume that [ H ] holds. Given any set of α i > i = 1 , , · · · , n ) ,suppose that ( u ( x ) , u ( x ) , · · · , u n ( x )) is a nonnegative C solution to (BVP) with m = 1 . Then ¯ λ ≤ n (cid:88) i =1 α i u i ( x ) ≤ ¯ λ, x ∈ R , (1.5) where ¯ λ = (cid:16) max ≤ i ≤ n α i ¯ u i (cid:17)(cid:16) max ≤ i ≤ n d i (cid:17) min ≤ i ≤ n d i , (1.6)¯ λ = (cid:16) min ≤ i ≤ n α i ¯ u i (cid:17)(cid:16) min ≤ i ≤ n d i (cid:17) max ≤ i ≤ n d i χ, (1.7) with χ given by (1.4) . (BVP) arises from the study of traveling waves in the Shigesada-Kawasaki-Teramoto (SKT) model (SKT) u t = ∆ (cid:0) u ( d + ρ u + ρ v ) (cid:1) + u ( σ − c u − c v ) , y ∈ Ω , t > ,v t = ∆ (cid:0) v ( d + ρ u + ρ v ) (cid:1) + v ( σ − c u − c v ) , y ∈ Ω , t > , which was proposed by Shigesada, Kawasaki and Teramoto ([36]) in 1979 to study the spatial segregationproblem 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 ∆ u and d ∆ v come from the randommovements of individual species with diffusion rates d , d >
0. Meanwhile, the terms ∆ (cid:0) u ( ρ u + ρ v ) (cid:1) and ∆ (cid:0) v ( ρ u + ρ v ) (cid:1) include the self-diffusion and cross-diffusion due to the directed movements ofthe individuals toward favorable habitats. The coefficients ρ and ρ are referred to as the self-diffusionrates, while ρ and ρ are the cross-diffusion rates. In addition, the coefficients σ i , c ii ( i = 1 , c ij ( i, j = 1 , i (cid:54) = j ) are the intrinsic growth rates, the intra-specific competition rates, and theinter-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 indicateswhich species is stronger and can survive. Inserting (1.8) into (SKT) with Ω = R leads to (SKT-tw) (cid:0) u ( d + ρ u + ρ v ) (cid:1) xx + θ u x + u ( σ − c u − c v ) = 0 , x ∈ R , (cid:0) v ( d + ρ u + ρ v ) (cid:1) xx + θ v x + v ( σ − c u − c v ) = 0 , x ∈ R . hen the self-diffusion and the cross-diffusion effects are neglected or ρ = ρ = ρ = ρ = 0, (SKT) with Ω = R and (SKT-tw) reduce respectively to (LV) u t = d ∆ u + u ( σ − c u − c v ) , y ∈ R , t > ,v t = d ∆ v + v ( σ − c u − c v ) , y ∈ R , t > , and (LV-tw) d u xx + θ u x + u ( σ − c u − c v ) = 0 , x ∈ R ,d v xx + θ v x + v ( σ − c u − c v ) = 0 , x ∈ R , 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 habitatof the two competing species u and v is resource-limited, the investigation of the total mass or the totaldensity 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. J = H ln( s ) , (1.9)where H = − s (cid:88) i =1 ι i · ln( ι i ) , (1.10) s is the total number of species, and ι i is the proportion of the i -th species determined by dividing thenumber of the i -th species species by the total number of all species. The species evenness index J for (LV-tw) is given by J = − u + v ) (cid:18) u ln (cid:16) uu + v (cid:17) + v ln (cid:16) vu + v (cid:17)(cid:19) . (1.11)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 = d , upper and lower bounds of u ( x ) + v ( x ) are given in [6] by an approach based on the ellipticmaximum principle. For the case of d (cid:54) = d , an affirmative answer to an even more general problem ofestimating α u + β v , where α, β > m = n = 2 and l i = 1 ( i = 1 , , · · · , n ) when d = d = ρ = ρ = 0 in (SKT-tw) . We therefore, address the following problem. Q : Under [ H ] , establish the NBMP for (BVP) , i.e. find nontrivial lower and upper bounds (dependingon the coefficients in (BVP) ) of n (cid:88) i =1 α i u i ( x ) , where α i > i = 1 , , · · · , n ) are arbitrary positive constants. Our main result is that (BVP) enjoys the following N-barrier maximum principle, which gives an affir-mative answer to Q . Indeed, we have Theorem 1.2 ( NBMP for (BVP) ) . Assume that [ H ] holds. Given any set of α i > i = 1 , , · · · , n ) ,suppose that ( u ( x ) , u ( x ) , · · · , u n ( x )) is a nonnegative C solution to (BVP) with m > . Then ¯ λ ≤ n (cid:88) i =1 α i u i ( x ) ≤ ¯ λ, x ∈ R , (1.12) where ¯ λ = m (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) m − (cid:18) max ≤ i ≤ n d i α m − i (cid:19)(cid:32) max ≤ i ≤ n α i d i ¯ u mi (cid:33) , (1.13) λ = m (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) − m (cid:18) min ≤ i ≤ n α m − i d i (cid:19) χ, (1.14) with χ given by (1.4) . We note that except the case in which either e + = (0 , · · · ,
0) or e − = (0 , · · · , ±∞ do not play any role in determining the upper and lower bounds given by Theorem 1.2. For either e + = (0 , · · · ,
0) or e − = (0 , · · · , m = n = 2, l i = 1 and f i ( u , u ) = u i ( σ i − c i u − c i u ) ( i = 1 , (BVP) becomes (NDC-tw) d ( u ) xx + θ ( u ) x + u ( σ − c u − c u ) = 0 , x ∈ R ,d ( u ) xx + θ ( u ) x + u ( σ − c u − c u ) = 0 , x ∈ R , ( u , u )( −∞ ) = e − , ( u , u )(+ ∞ ) = e + , where e − , e + ∈ (cid:26) ( u , u ) (cid:12)(cid:12)(cid:12) u i ( σ i − c i u − c i u ) = 0 ( i = 1 , , u , u ≥ (cid:27) . (1.15)The degenerate elliptic system (NDC-tw) arises from the study of traveling waves in (SKT) without thepresence of diffusion and cross-diffusion, and Ω replaced by R , i.e. (NDC) ( u ) t = d ( u ) yy + u ( σ − c u − c u ) , y ∈ R , t > , ( u ) t = d ( u ) yy + u ( σ − c u − c u ) , y ∈ R , t > . The nonlinear diffusion-competition system (NDC) has been studied, for example in [13]. Under suitablerestrictions on the coefficients, explicit spatially periodic stationary solutions to (NDC) can be found. Inaddition, 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 isthe following NBMP for (NDC-tw) . Corollary 1.3 ( NBMP for NDC-tw ) . Assume that ( u ( x ) , v ( x )) is a nonnegative C solution to (NDC-tw) . For any set of α i > i = 1 , , we have ¯ λ ≤ α u ( x ) + α u ( x ) ≤ ¯ λ, x ∈ R , (1.16) where ¯ λ = (cid:18) α d + α d (cid:19)(cid:115) max (cid:18) d α , d α (cid:19) max (cid:18) α d ¯ u , α d ¯ u (cid:19) , (1.17)¯ λ = d d ¯ u ¯ u min (cid:16) α d , α d (cid:17) (cid:114) α α ( α d ¯ u + α d ¯ u ) ( α d + α d ) χ, (1.18) with χ given by (1.4) and ¯ u = max (cid:18) σ c , σ c (cid:19) , ¯ u = max (cid:18) σ c , σ c (cid:19) , (1.19)¯ u = min (cid:18) σ c , σ c (cid:19) , ¯ u = min (cid:18) σ c , σ c (cid:19) , Proof.
We apply Theorem 1.2 to prove Corollary 1.3. Due to (1.19), it can be verified that [ H ] is satisfied.Indeed, we have ¯ R = (cid:40) ( u , u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 u i min j =1 , σ j c ji ≤ , u , u ≥ (cid:41) ; R = (cid:40) ( u , u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 u i max j =1 , σ j c ji ≥ , u , u ≥ (cid:41) . Since min j =1 , σ j c ji (cid:16) max j =1 , σ j c ji , respectively (cid:17) is the smallest (largest, respectively) u i -intercept of the two planes σ i − c i u − c i u = 0 ( i = 1 , σ i − c i u − c i u ≥ u , u ) ∈ ¯ R ; σ i − c i u − c i u ≤ u , u ) ∈ ¯ R , for each i = 1 ,
2. The desired result follows from Theorem 1.2. (cid:3)
As an interesting application of the linear diffusion NBMP (Theorem 1.1), we investigate the situationwhere 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 competingspecies 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: d u xx + θ u x + u ( σ − c u − c v − c w ) = 0 , x ∈ R ,d v xx + θ v x + v ( σ − c u − c v − c w ) = 0 , x ∈ R ,d w xx + θ w x + w ( σ − c u − c v − c w ) = 0 , x ∈ R . (1.20)Clearly, (1.1) includes (1.20) as a special case. For (1.20), existence of solutions with profiles of one-humpwaves coupled with the boundary conditions( u, v, w )( −∞ ) = (cid:16) σ c , , (cid:17) , ( u, v, w )( ∞ ) = (cid:16) , σ c , (cid:17) . (1.21)is investigated under certain assumptions on the parameters by finding exact solutions ([8, 6]) and using thenumerical tracking method AUTO ([8]). A one-hump wave is referred to as a traveling wave consisting ofa forward front v , a backward front u , and a pulse w in the middle. On the other hand, nonexistence ofsolutions for the problem (1.20), (1.21) is established by means of the NBMP (Theorem 1.1) as well as theelliptic maximum principle under certain conditions ([6, 5]).Recently, new dynamical patterns exhibited by the solutions of the Lotka-Volterra system of three com-peting species have been found in [26], where traveling wave solutions of the three species (i.e. solutionsof (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 threespecies 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 proportionalto the gradient of the population density. In some situations, however, evidences from field studies haveshown the inadequacy of this model. Due to population pressure, the phenomenon that species tend to avoidcrowded can be characterized by the population flux which depends on both the population density and itsgradient ([27, 35, 39]). Gurney and Nisbet considered the nonlinear diffusion effect described above, andproposed the following the model ([14, 15]) u t = ( u u x ) x + u ( u − , (1.22) here the population flux is proportional to u and u x . Based on porous medium version of the Fisherequation (1.22) ([28, 34, 33]), (1.20) becomes d ( u ) xx + θ u x + u ( σ − c u − c v − c w ) = 0 , x ∈ R ,d ( v ) xx + θ v x + v ( σ − c u − c v − c w ) = 0 , x ∈ R ,d ( w ) xx + θ w x + w ( σ − c u − c v − c w ) = 0 , x ∈ R , (1.23)For the existence of solutions of the problem (1.23), (1.21), it seems as far as we know, not available in theliterature. As a starting point to study this problem, we instead find the conditions on the parameters underwhich the solutions do not exist. With the aid of the NBMP for the problem (1.23), (1.21), this can beachieved as the following nonexistence result shows. Theorem 1.4 ( Nonexistence of three species waves ) . Under either (i) or (ii), (1.23) admits no positivesolution ( u ( x ) , v ( x ) , w ( x )) with u ( x ) , v ( x ) , w ( x ) (cid:54)≡ constant. (i) Let φ = σ − c σ c − and φ = σ − c σ c − . Assume that the following hypotheses hold: [ H0 ] ( u, v )( ±∞ ) (cid:54) = (0 , ; [ H1 ] max x ∈ R w ( x ) = w ( x ) for some x ∈ R ; [ H2 ] φ , φ > ; [ H3 ] λ ∗ := d d ¯ u ∗ ¯ v ∗ min (cid:18) c d , c d (cid:19) (cid:114) c c ( c d ¯ u ∗ + c d ¯ v ∗ ) ( c d + c d ) ≥ σ , where ¯ u ∗ = min (cid:18) φ c , φ c (cid:19) , ¯ v ∗ = min (cid:18) φ c , φ c (cid:19) . (1.24)(ii) Assume that the following hypotheses hold: [ H4 ] min x ∈ R w ( x ) = w ( x ) for some x ∈ R ; [ H5 ] λ ∗ := (cid:18) c d + c d (cid:19)(cid:115) max (cid:18) d c , d c (cid:19) max (cid:16) c d ¯ u ∗ , c d ¯ v ∗ (cid:17) < σ , where ¯ u ∗ = max (cid:18) σ c , σ c (cid:19) , ¯ v ∗ = max (cid:18) σ c , σ c (cid:19) . (1.25)[ H6 ] w ( ±∞ ) := w ±∞ , where either w −∞ < c ( σ − λ ∗ ) or w + ∞ < c ( σ − λ ∗ ) . 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 ecologythat when the intrinsic growth rate σ 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-mediatedcoexistence cannot occur in such a circumstance. On the other hand, [ H6 ] is satisfied when the boundaryconditions are ( u, v, w )( −∞ ) = (cid:16) σ c , , (cid:17) , ( u, v, w )( ∞ ) = (cid:16) , ˜ v, ˜ w (cid:17) , (1.26)where v = ˜ v , w = ˜ w solves σ − c v − c w = 0 , σ − c v − c w = 0 (1.27)or ˜ v = c σ − c σ c c − c c , ˜ w = c σ − c σ c c − c c , (1.28)whenever the coexistence state (˜ v, ˜ w ) exists. [ H4 ] is an extra hypothesis on the profile of the wave. Asa 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 cannotoccur when the intrinsic growth rate σ 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. Asan application of Theorem 1.2, we show in Section 3 the nonexistence result of three species in Theorem 1.4. n Section 4, we propose some open problems concerning the NBMP. Finally, some exact traveling wavesolutions and the solutions of a system of algebraic equations needed in the proof of Theorem 1.2 are givenin the Appendix (Section 5). 2. Proof of Theorem 1.2
Proposition 1 ( Lower bound in NBMP ) . Suppose that u i ( x ) ∈ C ( R ) with u i ( x ) ≥ i = 1 , , · · · , n ) and satisfy the following differential inequalities and asymptotic behavior: (BVP-u) d i ( u mi ) xx + θ ( u i ) x + u l i i f i ( u , u , · · · , u n ) ≤ , x ∈ R , i = 1 , , · · · , n, ( u , u , · · · , u n )( −∞ ) = e − , ( u , u , · · · , u n )( ∞ ) = e + , where e − and e + are given by (1.3) . If the hypothesis [ H ] For i = 1 , , · · · , n , there exist ¯ u i > such that f i ( u , u , · · · , u n ) ≥ whenever ( u , u , · · · , u n ) ∈ ¯ R , where ¯ R is as defined in [ H ] holds, then we have for any α i > i = 1 , , ..., n ) n (cid:88) i =1 α i u i ( x ) ≥ m (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) − m (cid:18) min ≤ i ≤ n α m − i d i (cid:19) χ (2.1) where χ is defined as in (1.4) .Proof. For the case where e + = (0 , · · · ,
0) or e − = (0 , · · · , n (cid:88) i =1 α i u i ( x ) is 0. Itsuffices to show (2.1) for the case e + (cid:54) = (0 , ...,
0) and e − (cid:54) = (0 , ..., p ( x ) = n (cid:88) i =1 α i u i ( x ); (2.2) q ( x ) = n (cid:88) i =1 α i d i u mi ( x ) . (2.3)Adding the n equations in (BVP-u) , we obtain a single equation involving p ( x ) and q ( x ) d q ( x ) dx + θ dp ( x ) dx + F ( u ( x ) , u ( x ) , · · · , u n ( x )) ≤ , x ∈ R , (2.4)where F ( u , u , · · · , u n ) := n (cid:88) i =1 α i u l i i f i ( u , u , · · · , u n ). First of all, we show how to construct the N-barrier .Determining an appropriate N-barrier is crucial in establishing (2.1). The construction of the N-barrierconsists of determining the positive parameters λ , λ , η and η such that the two hyper-ellipsoids n (cid:88) i =1 α i d i u mi = λ and n (cid:88) i =1 α i d i u mi = λ , and the two hyperplanes n (cid:88) i =1 α i u i = η and n (cid:88) i =1 α i u i = η satisfy the relationship P η ⊂ Q λ ⊂ P η ⊂ Q λ ⊂ ¯ R , (2.5)where P η = (cid:110) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) n (cid:88) i =1 α i u i ≤ η, u , u , · · · , u n ≥ (cid:111) ; (2.6) Q λ = (cid:110) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) n (cid:88) i =1 α i d i u mi ≤ λ, u , u , · · · , u n ≥ (cid:111) . (2.7) he hyper-ellipsoids n (cid:88) i =1 α i d i u mi = λ and n (cid:88) i =1 α i d i u mi = λ , and the hyperplane n (cid:88) i =1 α i u i = η form theN-barrier; it turns out that the hyperplane n (cid:88) i =1 α i u i = η determines a lower bound of p ( x ). We follow thethree steps below to construct the N-barrier:(1) Let the hyperplane n (cid:88) i =1 u i ¯ u i = 1 be tangent to the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ at ( u , u , · · · , u n )with u , u , · · · , u n > Q λ ⊂ ¯ R . This leads to the following equations: α i d i u m − i ¯ u i = α j d j u m − j ¯ u j , i, j = 1 , , · · · , n ; (2.8) n (cid:88) i =1 u i ¯ u i = 1; (2.9) n (cid:88) i =1 α i d i u mi = λ . (2.10)By Lemma 5.1 (see Section 5), λ is determined by λ = (cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − m . (2.11)(2) Setting η = m (cid:115) λ min ≤ i ≤ n α m − i d i , (2.12)the hyperplane n (cid:88) i =1 α i u i = η has the n intercepts (cid:16) η α , , · · · , (cid:17) , (cid:16) , η α , , · · · , (cid:17) , · · · , and (cid:16) , , · · · , , η α n (cid:17) and the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ has the n intercepts (cid:18) m (cid:114) λ α d , , · · · , (cid:19) , (cid:18) , m (cid:114) λ α d , , · · · , (cid:19) , · · · ,and (cid:18) , , · · · , , m (cid:114) λ α n d n (cid:19) . It is easy to verify that P η ⊂ Q λ since η α j ≤ m (cid:115) λ α j d j for j =1 , , · · · , n . Indeed, we have η α j = (cid:18) min ≤ i ≤ n λ α m − i d i (cid:19) m α j (2.13) ≤ (cid:18) λ α m − j d j α mj (cid:19) m = (cid:18) λ α j d j (cid:19) m . (3) Let the hyperplane n (cid:88) i =1 α i u i = η be tangent to the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ at ( u , u , · · · , u n )with u , u , · · · , u n > Q λ ⊂ P η . This leads to the following equations: d i u m − i = d j u m − j , i, j = 1 , , · · · , n ; (2.14) n (cid:88) i =1 α i u i = η ; (2.15) n (cid:88) i =1 α i d i u mi = λ . (2.16) mploying Lemma 5.2 in Section 5, we obtain λ = η m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) − m . (2.17)Steps (i) ∼ (iii) complete the construction of the N-barrier. As in step (ii), we determine η by η = m (cid:115) λ min ≤ i ≤ n α m − i d i (2.18)such that P η ⊂ Q λ . From (2.11), (2.12), (2.17) and (2.18), it follows immediately that η is given by η = λ m (cid:18) min ≤ i ≤ n α m − i d i (cid:19) m = η (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) − mm (cid:18) min ≤ i ≤ n α m − i d i (cid:19) m (2.19)= λ m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) − mm (cid:18) min ≤ i ≤ n α m − i d i (cid:19) m = (cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − mm (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) − mm (cid:18) min ≤ i ≤ n α m − i d i (cid:19) m . (The construction of the N-barrier for the simplified case m = n = 2 is illustrated in Remark 2.1, whichprovides an intuitive idea of the construction of the N-barrier in higher dimensional cases.)We claim that q ( x ) ≥ λ , x ∈ R . This proves (2.1), i.e q ( x ) ≥ η , x ∈ R since the α i > i =1 , , · · · , n ) are arbitrary and the relationship P η ⊂ Q λ holds. Now we prove the claim by contradiction.Suppose that, contrary to our claim, there exists z ∈ R such that q ( z ) < λ . Since u i ( x ) ∈ C ( R ) and( u , u , · · · , u n )( ±∞ ) = e ± , we may assume min x ∈ R q ( x ) = q ( z ). We denote respectively by z and z the firstpoints at which the solution ( u ( x ) , u ( x ) , · · · , u n ( x )) intersects the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ when x moves from z towards ∞ and −∞ . For the case where θ ≤
0, we integrate (2.4) with respect to x from z to z and obtain q (cid:48) ( z ) − q (cid:48) ( z ) + θ ( p ( z ) − p ( z )) + (cid:90) zz F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx ≤ . (2.20)On the other hand we have: • q (cid:48) ( z ) = 0 because of min x ∈ R q ( x ) = q ( z ); • q ( z ) = λ follows from the fact that z is on the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ . Since z is thefirst point for q ( x ) taking the value λ when x moves from z to −∞ , we conclude that q ( z + δ ) ≤ λ for z − z > δ > q (cid:48) ( z ) ≤ • p ( z ) < η since z is below the hyperplane n (cid:88) i =1 α i u i = η ; p ( z ) > η since z is above the hyperplane n (cid:88) i =1 α i u i = η ; • let F + = (cid:110) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) F ( u , u , · · · , u n ) > , u , u , · · · , u n ≥ (cid:111) . Due to the fact that( u ( z ) , u ( z ) , · · · , u n ( z )) is on the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ and ( u ( z ) , u ( z ) , · · · , u n ( z )) ∈ Q λ , ( u ( z ) , u ( z ) , · · · , u n ( z )), we have ( u ( z ) , u ( z ) , · · · , u n ( z )) ∈ ¯ R by (2.5). Because of [ H ] nd F ( u , u , · · · , u n ) = n (cid:88) i =1 α i u l i i f i ( u , u , · · · , u n ), it is easy to see that (cid:110) ( u ( x ) , u ( x ) , · · · , u n ( x )) (cid:12)(cid:12)(cid:12) z ≤ x ≤ z (cid:111) ⊂ ¯ R ⊂ F + . (2.21)Therefore we have (cid:90) zz F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx > q (cid:48) ( z ) − q (cid:48) ( z ) + θ ( p ( z ) − p ( z )) + (cid:90) zz F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx > , (2.22)which contradicts (2.20). Therefore when θ ≤ q ( x ) ≥ λ for x ∈ R . For the case where θ ≥
0, integrating(2.4) with respect to x from z to z yields q (cid:48) ( z ) − q (cid:48) ( z ) + θ ( p ( z ) − p ( z )) + (cid:90) z z F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx ≤ . (2.23)In a similar manner, it can be shown that q (cid:48) ( z ) ≥ q (cid:48) ( z ) = 0, p ( z ) > η , p ( z ) < η , and (cid:90) z z F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx > . (2.24)These together contradict (2.23). Consequently, (2.1) is proved and the proof is completed. (cid:3) Remark 2.1 ( N-barrier for lower bounds ) . When σ = σ = c = c = 1 , c = a , and c = a in (NDC-tw) with the asymptotic behavior e − = (1 , and e + = (0 , , we are led to the problem d ( u ) xx + θ u x + u (1 − u − a v ) = 0 , x ∈ R ,d ( v ) xx + θ v x + v (1 − a u − v ) = 0 , x ∈ R , ( u, v )( −∞ ) = (1 , , ( u, v )(+ ∞ ) = (0 , . (2.25) To satisfy the hypothesis [ H ] , we let as in the proof of Corollary 1.3 ¯ u = min (cid:16) , a (cid:17) , (2.26)¯ v = min (cid:16) , a (cid:17) . (2.27) For simplicity, we shall always assume the bistable condition a , a > for (2.25) . This gives ¯ u = 1 a and ¯ v = 1 a . We readily verify that under a , a > , the quadratic curve F ( u, v ) := α u (1 − u − a v ) + β v (1 − a u − v ) = 0 (2.28) in the first quadrant of the uv -plane is a hyperbola for any α, β > and it passes through the equilibria (0 , , (1 , , (1 , and (cid:16) a − a a − , a − a a − (cid:17) .We are now in the position to follow the three steps in the proof of Proposition 1 to construct the N-barrierfor the problem (2.25) . (1) Since the line u ¯ u + v ¯ v = 1 is tangent to the ellipse α d u + β d v = λ at ( u, v ) in the first quadrantof the uv -plane, this leads to the following equations: α d uβ d v = ¯ v ¯ u , (2.29) u ¯ u + v ¯ v = 1 , (2.30) α d u + β d v = λ . (2.31) y Lemma 5.1 (see Section 5), λ is given by λ = 11 α d ¯ u + 1 β d ¯ v = α β d d ¯ u ¯ v α d ¯ u + β d ¯ v . (2.32)(2) The u -coordinate of the u -intercept and the v -coordinate of the v -intercept of the ellipse α d u + β d v = λ are (cid:114) λ α d and (cid:115) λ β d , respectively; the u -coordinate of the u -intercept and the v -coordinate of the line η = α u + β v are η α and η β , respectively. Because of η = (cid:114) λ min (cid:16) αd , βd (cid:17) , (2.33) • when min (cid:16) αd , βd (cid:17) = αd , we clearly have η α = 1 α (cid:114) λ αd = (cid:114) λ α d , (2.34) η β = 1 β (cid:114) λ αd ≤ √ λ β (cid:114) βd = (cid:115) λ β d ; (2.35) • when min (cid:16) αd , βd (cid:17) = βd , we clearly have η α = 1 α (cid:114) λ βd ≤ √ λ α (cid:114) αd = (cid:114) λ α d , (2.36) η β = 1 β (cid:114) λ βd = (cid:115) λ β d . (2.37) This means that when min (cid:16) αd , βd (cid:17) = αd , the ellipse α d u + β d v = λ and the line η = α u + β v possesess the same u -coordinate of the u -intercept, i.e. (cid:114) λ α d = η α ; meanwhile, the inequality η β ≤ (cid:115) λ β d indicates that the v -coordinate of the v -intercept of the line η = α u + β v is not largerthan that of the v -intercept of the ellipse α d u + β d v = λ . A similar conclusion can be drawnfor the case of min (cid:16) αd , βd (cid:17) = βd . (3) The fact that the line η = α u + β v is tangent to the ellipse α d u + β d v = λ at ( u, v ) in thefirst quadrant of the uv -plane yields the following equations: α d uβ d v = αβ , (2.38) α u + β v = η , (2.39) α d u + β d v = λ . (2.40) Employing Lemma 5.2 in Section 5, we obtain λ = η αd + βd = η d d α d + β d . (2.41) he above three steps complete the construction of the N-barrier. Finally, we determine the line η = α u + β v by setting η = (cid:114) λ min (cid:16) αd , βd (cid:17) (2.42) such that, as in step (ii), the line η = α u + β v lies entirely below the ellipse α d u + β d v = λ in thefirst quadrant of the uv -plane. Combining (2.32) , (2.33) , (2.41) and (2.42) , we arrive at η = (cid:114) λ min (cid:16) αd , βd (cid:17) = η (cid:115) min (cid:16) αd , βd (cid:17) d d α d + β d (2.43)= (cid:114) λ min (cid:16) αd , βd (cid:17) (cid:115) min (cid:16) αd , βd (cid:17) d d α d + β d = min (cid:16) αd , βd (cid:17) (cid:115) α β d d ¯ u ¯ v α d ¯ u + β d ¯ v d d α d + β d = d d ¯ u ¯ v min (cid:16) αd , βd (cid:17) (cid:115) α β ( α d ¯ u + β d ¯ v ) ( α d + β d )= α β ¯ u ¯ v min (cid:16) d α , d β (cid:17) (cid:115) α β ( α d ¯ u + β d ¯ v ) ( α d + β d ) . The lower bound η coincides with that given in Corollary 1.3.It follows immediately from step (ii) that there are two conditions: min (cid:16) αd , βd (cid:17) = αd and min (cid:16) αd , βd (cid:17) = βd . We show the N-barrier for each condition in Figure 1: the N-barrier for the case min (cid:16) αd , βd (cid:17) = αd is shown in Figure 1(a), while the one for the case min (cid:16) αd , βd (cid:17) = βd is shown in Figure 1(b). We notethat through the example of Figure 1 in which the N-barrier for the lower dimensional problem (2.25) isconstructed, the N-barrier in the hyper-space in the proof of Proposition 1 become immediate. Proposition 2 ( Upper bound in NBMP ) . Suppose that u i ( x ) ∈ C ( R ) with u i ( x ) ≥ i = 1 , , · · · , n ) and satisfy the following differential inequalities and asymptotic behavior: (BVP-l) d i ( u mi ) xx + θ ( u i ) x + u l i i f i ( u , u , · · · , u n ) ≥ , x ∈ R , i = 1 , , · · · , n, ( u , u , · · · , u n )( −∞ ) = e − , ( u , u , · · · , u n )( ∞ ) = e + , where e − and e + are given by (1.3) . If the hypothesis [ ¯H ] For i = 1 , , · · · , n , there exist ¯ u i > such that f i ( u , u , · · · , u n ) ≤ whenever ( u , u , · · · , u n ) ∈ ¯ R , where ¯ R is as defined in [ H ] holds, then we have for any α i > i = 1 , , · · · , n ) n (cid:88) i =1 α i u i ( x ) ≤ m (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) m − (cid:18) max ≤ i ≤ n d i α m − i (cid:19)(cid:32) max ≤ i ≤ n α i d i ¯ u mi (cid:33) (2.44) Proof.
We show by employing the N-barrier method as in the proof of Proposition 1 the upper bound givenby (2.44). The construction of an appropriate N-barrier is the main ingredient of our proof. To do this, let P η = (cid:110) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) n (cid:88) i =1 α i u i ≥ η, u , u , · · · , u n ≥ (cid:111) ; (2.45) Q λ = (cid:110) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) n (cid:88) i =1 α i d i u i ≥ λ, u , u , · · · , u n ≥ (cid:111) . (2.46) z z u v (a) min (cid:16) αd , βd (cid:17) = αd : d = 3, d = 4, a = 2, a = 3, α = 1, β = 2, u = 13 , v = 12 , λ = 27 , λ = 435 , η = (cid:114)
221 , η = 2 √
105 . z z z u v (b) min (cid:16) αd , βd (cid:17) = βd : d = 3, d = 4, a = 2, a = 3, α = 1, β = 1, u = 13 , v = 12 , λ = 14 , λ = 328 , η = 14 , η = (cid:114)
374 .
Figure 1.
Red line: 1 − u − a v = 0; blue line: 1 − a u − v = 0; green curve: F ( u, v ) := α u (1 − u − a v ) + β v (1 − a u − v ) = 0; brown line: uu + vv = 1, where u and v are given by(2.26) and (2.27) ; magenta ellipse (above): α d u + β d v = λ , where λ is given by (2.32);magenta ellipse (below): α d u + β d v = λ , where λ is given by (2.41); yellow line (above): α u + β v = η , where η is given by (2.33); yellow line (below): α u + β v = η , where η is given by(2.42); dashed orange curve: the solution ( u ( x ) , v ( x )); dotted line (above): u (cid:113) λ α d + v (cid:113) λ β d = 1;dotted line (below): u (cid:113) λ α d + v (cid:113) λ β d = 1. Recall (2.2) in the proof of Proposition 1. Adding the n equations in (BVP-l) , we obtain the equation d q ( x ) dx + θ dp ( x ) dx + F ( u ( x ) , u ( x ) , · · · , u n ( x )) ≥ , x ∈ R , (2.47)where F ( u , u , · · · , u n ) := n (cid:88) i =1 α i u l i i f i ( u , u , · · · , u n ).We determine the positive parameters λ , λ , η and η such that the two hyper-ellipsoids n (cid:88) i =1 α i d i u mi = λ , n (cid:88) i =1 α i d i u mi = λ , and the two hyperplanes n (cid:88) i =1 α i u i = η , n (cid:88) i =1 α i u i = η satisfy the relationship P η ⊃ Q λ ⊃ P η ⊃ Q λ ⊃ ¯ R . (2.48) he hyper-ellipsoids n (cid:88) i =1 α i d i u mi = λ , n (cid:88) i =1 α i d i u mi = λ , and the hyperplane n (cid:88) i =1 α i u i = η form the N-barrier and it turns out that the hyperplane n (cid:88) i =1 α i u i = η determines the upper bound in (2.44). We followthe three steps below to construct the N-barrier:(1) Setting λ = max ≤ i ≤ n α i d i ¯ u mi , (2.49)the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ has the n intercepts (cid:18) m (cid:114) λ α d , , · · · , (cid:19) , (cid:18) , m (cid:114) λ α d , , · · · , (cid:19) , · · · ,and (cid:18) , , · · · , , m (cid:114) λ α n d n (cid:19) and the hyperplane n (cid:88) i =1 u i ¯ u i = 1 has the n intercepts (¯ u , , · · · , , ¯ u , , · · · , · · · , and (0 , , · · · , , ¯ u n ). It is easy to verify that Q λ ⊃ ¯ R since ¯ u j ≤ m (cid:115) λ α j d j for j = 1 , , · · · , n . Indeed, we have (cid:16) λ α j d j (cid:17) m = (cid:32) max ≤ i ≤ n α i d i ¯ u mi α j d j (cid:33) m (2.50) ≥ (cid:32) α j d j ¯ u mj α j d j (cid:33) m = ¯ u j . (2) Let the hyperplane n (cid:88) i =1 α i u i = η be tangent to the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ at ( u , u , · · · , u n )with u , u , · · · , u n > P η ⊃ Q λ . This leads to the following equations: d i u m − i = d j u m − j , i, j = 1 , , · · · , n ; (2.51) n (cid:88) i =1 α i u i = η ; (2.52) n (cid:88) i =1 α i d i u mi = λ . (2.53)Employing Lemma 5.2 in Section 5, we obtain η = λ m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) m − m . (2.54)(3) Setting λ = η m (cid:18) max ≤ i ≤ n d i α m − i (cid:19) , (2.55)the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ has the n intercepts (cid:16) m (cid:114) λ α d , , · · · , (cid:17) , (cid:16) , m (cid:114) λ α d , , · · · , (cid:17) , · · · ,and (cid:16) , , · · · , , m (cid:114) λ α n d n (cid:17) and the hyperplane n (cid:88) i =1 α i u i = η has the n intercepts (cid:16) η α , , · · · , (cid:17) , (cid:16) , η α , , · · · , (cid:17) , · · · , and (cid:16) , , · · · , , η α n (cid:17) . It is easy to verify that Q λ ⊃ P η since η α j ≤ (cid:16) λ α j d j (cid:17) m or j = 1 , , · · · , n . Indeed, we have (cid:16) λ α j d j (cid:17) m = η (cid:32) max ≤ i ≤ n d i α − mi α j d j (cid:33) m (2.56) ≥ η (cid:32) d j α − mj α j d j (cid:33) m = η α j . Steps (i) ∼ (iii) complete the construction of the N-barrier. As in step (ii), we determine η by lettingthe hyperplane n (cid:88) i =1 α i u i = η be tangent to the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ at ( u , u , · · · , u n ) with u , u , · · · , u n > P η ⊃ Q λ . This leads to the following equations: d i u m − i = d j u m − j , i, j = 1 , , · · · , n ; (2.57) n (cid:88) i =1 α i u i = η ; (2.58) n (cid:88) i =1 α i d i u mi = λ . (2.59)Employing Lemma 5.2 in Section 5 again, we obtain η = λ m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) m − m . (2.60)such that P η ⊂ Q λ . From (2.49), (2.54), (2.55) and (2.60), it follows immediately that η is given by η = λ m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) m − m = η (cid:18) max ≤ i ≤ n d i α m − i (cid:19) m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) m − m (2.61)= λ m (cid:18) max ≤ i ≤ n d i α m − i (cid:19) m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) m − m = (cid:18) max ≤ i ≤ n α i d i ¯ u mi (cid:19) m (cid:18) max ≤ i ≤ n d i α m − i (cid:19) m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) m − m . (An illustration of the N-barrier for m = n = 2 is given in Remark 2.2.)As the proof of Proposition 1, we claim by contradiction that q ( x ) ≤ λ for x ∈ R , from which (2.44) followssince the α i > i = 1 , , · · · , n ) are arbitrary and the relationship P η ⊃ Q λ holds. Suppose that, contraryto our claim, there exists z ∈ R such that q ( z ) > λ . Since u i ( x ) ∈ C ( R ) and ( u , u , · · · , u n )( ±∞ ) = e ± ,we may assume max x ∈ R q ( x ) = q ( z ). We denote respectively by z and z the first points at which the solution( u ( x ) , u ( x ) , · · · , u n ( x )) intersects the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ when x moves from z towards ∞ and −∞ . For the case where θ ≤
0, we integrate (2.47) with respect to x from z to z and obtain q (cid:48) ( z ) − q (cid:48) ( z ) + θ ( p ( z ) − p ( z )) + (cid:90) zz F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx ≥ . (2.62)On the other hand we have: • q (cid:48) ( z ) = 0 because of max x ∈ R q ( x ) = q ( z ); q ( z ) = λ follows from the fact that z is on the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ . Since z is thefirst point for q ( x ) taking the value λ when x moves from z to −∞ , we conclude that q ( z + δ ) ≥ λ for z − z > δ > q (cid:48) ( z ) ≥ • p ( z ) > η since z is above the hyperplane n (cid:88) i =1 α i u i = η ; p ( z ) < η since z is below the hyperplane n (cid:88) i =1 α i u i = η ; • let F − = (cid:110) ( u , u , · · · , u n ) (cid:12)(cid:12)(cid:12) F ( u , u , · · · , u n ) < , u , u , · · · , u n ≥ (cid:111) . Due to the fact that( u ( z ) , u ( z ) , · · · , u n ( z )) is on the hyper-ellipsoid n (cid:88) i =1 α i d i u mi = λ and ( u ( z ) , u ( z ) , · · · , u n ( z )) ∈ Q λ , ( u ( z ) , u ( z ) , · · · , u n ( z )), ( u ( z ) , u ( z ) , · · · , u n ( z )) ∈ ¯ R by (2.48). Because of [ ¯H ] and F ( u , u , · · · , u n ) = n (cid:88) i =1 α i u l i i f i ( u , u , · · · , u n ), it is easy to see that (cid:110) ( u ( x ) , u ( x ) , · · · , u n ( x )) (cid:12)(cid:12)(cid:12) z ≤ x ≤ z (cid:111) ⊂ ¯ R ⊂ F − . (2.63)Therefore we have (cid:90) zz F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx < q (cid:48) ( z ) − q (cid:48) ( z ) + θ ( p ( z ) − p ( z )) + (cid:90) zz F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx < , (2.64)which contradicts (2.62). Therefore when θ ≤ q ( x ) ≤ λ for x ∈ R . For the case where θ ≥
0, integrating(2.47) with respect to x from z to z yields q (cid:48) ( z ) − q (cid:48) ( z ) + θ ( p ( z ) − p ( z )) + (cid:90) z z F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx ≥ . (2.65)In a similar manner, it can be shown that q (cid:48) ( z ) ≤ q (cid:48) ( z ) = 0, p ( z ) < η , p ( z ) > η , and (cid:90) z z F ( u ( x ) , u ( x ) , · · · , u n ( x )) dx < . (2.66)These together contradict (2.65). Consequently, (2.44) is proved and the proof is completed. (cid:3) Remark 2.2 ( N-barrier for upper bounds ) . We illustrate the construction of the N-barrier in Proposi-tion 2 for the case when m = n = 2 . For consistency, we use the setting in Remark 2.1. (i) Ellipse α d u + β d v = λ We first determine the ellipse α d u + β d v = λ by letting λ = max (cid:0) α d ¯ u , β d ¯ v (cid:1) . (2.67) The u -coordinate of the u -intercept and the v -coordinate of the v -intercept of the ellipse α d u + β d v = λ are (cid:114) λ α d and (cid:115) λ β d , respectively; the u -coordinate of the u -intercept and the v -coordinate of the line u ¯ u + v ¯ v = 1 are ¯ u and ¯ v , respectively. It turns out that – when max (cid:0) α d ¯ u , β d ¯ v (cid:1) = α d ¯ u , we have (cid:114) λ α d = ¯ u, (cid:115) λ β d = ¯ u (cid:115) α d β d ≥ ¯ u (cid:114) ¯ v ¯ u = ¯ v ; (2.68) when max (cid:0) α d ¯ u , β d ¯ v (cid:1) = β d ¯ v , we have (cid:115) λ β d = ¯ v, (cid:114) λ α d = ¯ v (cid:114) β d α d ≥ ¯ v (cid:114) ¯ u ¯ v = ¯ u. (2.69) This means that the ellipse α d u + β d v = λ lies entirely above the line u ¯ u + v ¯ v = 1 in the firstquadrant of the uv -plane. (ii) Line η = α u + β v Since the line η = α u + β v is tangent to the ellipse α d u + β d v = λ at ( u, v ) in the first quadrant of the uv -plane, we have the following equations: α d uβ d v = αβ , (2.70) α u + β v = η , (2.71) α d u + β d v = λ . (2.72) Employing Lemma 5.2 in Section 5, we obtain η = (cid:115) λ (cid:18) αd + βd (cid:19) . (2.73) We note that the line η = α u + β v lies entirely above the ellipse α d u + β d v = λ in the firstquadrant of the uv -plane. (iii) Ellipse α d u + β d v = λ We determine the ellipse α d u + β d v = λ by letting λ = η max (cid:18) d α , d β (cid:19) . (2.74) The u -coordinate of the u -intercept and the v -coordinate of the v -intercept of the ellipse α d u + β d v = λ are (cid:114) λ α d and (cid:115) λ β d , respectively; the u -coordinate of the u -intercept and the v -coordinate of the line η = α u + β v are η α and η β , respectively. It follows that – when max (cid:18) d α , d β (cid:19) = d α , we have (cid:114) λ α d = η α , (cid:115) λ β d = η (cid:115) d α β d ≥ η (cid:115) α d α β d = η β ; (2.75) – when max (cid:18) d α , d β (cid:19) = d β , we have (cid:115) λ β d = η β , (cid:114) λ α d = η (cid:115) d β α d ≥ η (cid:115) β d α β d = η α . (2.76) We see from the construction of the ellipse α d u + β d v = λ that the ellipse α d u + β d v = λ lies entirely above the line η = α u + β v in the first quadrant of the uv -plane.The two ellipses α d u + β d v = λ and α d u + β d v = λ , and the line η = α u + β v form theN-barrier. Finally, we find the tangent line of the ellipse α d u + β d v = λ in the first quadrant of the uv -plane by determining the line η = α u + β v as in step (ii): α d uβ d v = αβ , (2.77) α u + β v = η , (2.78) α d u + β d v = λ . (2.79) e obtain η = (cid:115) λ (cid:18) αd + βd (cid:19) (2.80) or η = (cid:18) αd + βd (cid:19)(cid:115) max (cid:18) d α , d β (cid:19) max (cid:16) α d ¯ u , β d ¯ v (cid:17) (2.81) by combining (2.67) , (2.73) , (2.74) and (2.80) .It is readily seen from that, depending on max (cid:0) α d ¯ u , β d ¯ v (cid:1) and max (cid:18) d α , d β (cid:19) , 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 n (cid:88) i =1 α i u i ( x ), respec-tively. Combining the results in Propositions 1 and 2, we immediately establish Theorem 1.2. (cid:3) Remark 2.3.
The tanh method [20, 8, 31, 32] allows us to find exact solutions to (BVP) with certain classof the nonlinearity. For instance, when m = n = 2 , (BVP) with Zeldovich-type reaction terms ( [42, ? , 41] )becomes d ( u ) xx + θ u x + u ( σ − c u − c v ) = 0 , x ∈ R ,d ( v ) xx + θ v x + v ( σ − c u − c v ) = 0 , x ∈ R , ( u, v )( −∞ ) = (cid:16) σ c , (cid:17) , ( u, v )(+ ∞ ) = (cid:16) , σ c (cid:17) . (2.82) Applying Theorem 5.3 (see Appendix 5.2), we see that when c = 1 , c = 2 , d = 3 , and d = 4 , (5.38) gives θ = 0 , k = 60 , k = 8 , σ = 240 , σ = 32 , c = 27 , and c = 25 , and hence (2.82) admits thesolution (see Figure 3) u ( x ) = 60 (cid:0) − tanh x (cid:1) , x ∈ R ,v ( x ) = 8 (cid:0) x (cid:1) , x ∈ R . (2.83) Letting α = 12 and β = 13 , it follows immediately that α u ( x ) + β v ( x ) = 30 tanh x − x + 983 ismonotonically decreasing in x . As a result,
163 = α u ( ∞ ) + β v ( ∞ ) ≤ α u ( x ) + β v ( x ) ≤ α u ( −∞ ) + β v ( −∞ ) = 120 , x ∈ R . (2.84) On the other hand, upper and lower bounds given by Corollary 1.3 turn out to be . ≈ √ λ ≤ α u ( x ) + β v ( x ) ≤ ¯ λ = 180 √ ≈ . , x ∈ R . (2.85) Thus, we verify Corollary 1.3 in this case. ¯ u = 80 , ¯ v = 809 , ¯ u = 240 , ¯ v = 16 , Application to the nonexistence of three species traveling waves: proof of Theorem 1.4
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 and w xx ( x ) ≤
0. Since w ( x ) satisfies d ( w ) xx + θ w x + w ( σ − c u − c v − c w ) = 0 (3.1)and ( w ) xx = 2 ( w x + w w xx ), we obtain σ − c u ( x ) − c v ( x ) − c w ( x ) ≥ . (3.2) z z u v (a) d α > d β , α d ¯ u < β d ¯ v : d = 4, β = 2, λ = 8, λ = 20, η = 2 (cid:114)
53 , η = 5 (cid:114)
23 . z z z u v (b) d α < d β , α d ¯ u < β d ¯ v : d = 4, β = 1, λ = 4, λ = 283 , η = (cid:114)
73 , η = 73 . zz z u v (c) d α < d β , α d ¯ u > β d ¯ v : d = 4, β = 12 , λ = 3, λ = 11, η = 12 (cid:114)
112 , η = 112 √ zz z u v (d) d α > d β , α d ¯ u > β d ¯ v : d = 2, β = 34 , λ = 3, λ = 518 , η = 12 (cid:114)
172 , η = 178 . Figure 2.
Red line: 1 − u − a v = 0; blue line: 1 − a u − v = 0; green curve: F ( u, v ) = 0; brownline: uu + vv = 1, where u and v are given by (2.26) and (2.27); magenta ellipses : α d u + β d v = λ , λ , where λ (below) is given by (2.67) and λ (above) by (2.74); yellow lines: α u + β v = η , η ,where η (below) is given by (2.73) and η (above) by (2.80); dashed orange curve: the solution( u ( x ) , v ( x )); dotted lines: √ α d u + (cid:112) β d v = √ λ (below), √ λ (above); u = v = 1; d = 3, a = 2, a = 3, α = 1. (cid:45) x Figure 3.
Red: u ( x ) = 60 (cid:0) − tanh x (cid:1) ; green: v ( x ) = 8 (cid:0) x (cid:1) . This lead to an upper bound of w ( x ), i.e. w ( x ) ≤ w ( x ) ≤ c (cid:0) σ − c u ( x ) − c v ( x ) (cid:1) < σ c , x ∈ R . (3.3)By virtue of the inequality w ( x ) < σ c , the last two equations in (1.23) become d ( u ) xx + θ u x + u ( σ − c σ c − − c u − c v ) ≤ , x ∈ R ,d ( v ) xx + θ v x + v ( σ − c σ c − − c u − c v ) ≤ , x ∈ R . (3.4)By means of [ H0 ] and [ H2 ], we can employ Corollary 1.3 with u = u , u = v and α = c , α = c toobtain a lower bound of c u ( x ) + c v ( x ), i.e. c u ( x ) + c v ( x ) ≥ d d ¯ u ∗ ¯ v ∗ min (cid:16) c d , c d (cid:17) (cid:114) c c ( c d ¯ u ∗ + c d ¯ v ∗ ) ( c d + c d ) , x ∈ R . (3.5)However, [ H3 ] yields c u ( x ) + c v ( x ) ≥ λ ∗ ≥ σ , x ∈ R , (3.6)which contradicts (3.2). This completes the proof of (i) . To prove (ii) , an easy observation leads to d ( u ) xx + θ u x + u ( σ − c u − c v ) > , x ∈ R ,d ( v ) xx + θ v x + v ( σ − c u − c v ) > , x ∈ R , (3.7)since w ( x ) > x ∈ R . Letting u = u , u = v and α = c , α = c , an upper bound of c u ( x ) + c v ( x )given by Corollary 1.3 is c u ( x ) + c v ( x ) ≤ (cid:18) c d + c d (cid:19)(cid:115) max (cid:18) d c , d c (cid:19) max (cid:16) c d ¯ u ∗ , c d ¯ v ∗ (cid:17) := λ ∗ , x ∈ R , (3.8)where ¯ u ∗ and ¯ v ∗ are defined in [ H5 ]. It follows from the last inequality that0 = d ( w ) xx + θ w x + w ( σ − c u − c v − c w ) (3.9) ≥ d ( w ) xx + θ w x + w ( σ − λ ∗ − c w ) . On the other hand, [ H4 ] leads to the fact that w x ( x ) = 0 and w xx ( x ) ≥
0, and hence σ − λ ∗ − c w ( x ) ≤ . (3.10)or w ( x ) ≥ w ( x ) ≥ c (cid:0) σ − λ ∗ (cid:1) , x ∈ R . (3.11)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 nonex-istence of three species waves in (1.23) under certain conditions. In particular, the upper and lower boundsgiven 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 concerningNBMP. We point out some of them for further study: • NBMP for periodic solutions : As we can see from [13], (NDC) admits periodic stationary solutionsunder 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 certainconditions on the parameters. The question is how to correct the N-barrier method adapted forperiodic solutions? • NBMP for multi-dimensional equations : The N-barrier method has not yet been applied to multi-dimensional equations since there is still a lack of systematic formulation of the method in the multi-dimensional case. The difficulty is to construct appropriate N-barriers corresponding to operatorlike ∆ u , ∇ u , ∆( u ) etc.. • NBMP for strongly-coupled equations : The N-barrier method developed to study (1.1) can also beapplied 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. 5.
Appendix
Algebraic solutions.Lemma 5.1.
For Θ , Λ > , if α i d i ¯ u i u m − i = α j d j ¯ u j u m − j , i, j = 1 , , · · · , n ; (5.1) n (cid:88) i =1 u i ¯ u i = Θ; (5.2) n (cid:88) i =1 α i d i u mi = Λ , (5.3) we have Λ = Θ m (cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − m . (5.4) Proof.
Due to (5.1), we may assume u i = m − (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) n (cid:89) j =1 α j d j ¯ u j α i d i ¯ u i K, i = 1 , , · · · , n (5.5)for some K >
0. It follows immediately from (5.2) that K is determined by K = Θ n (cid:88) i =1 (cid:32) u i m − (cid:115) (cid:81) nj =1 α j d j ¯ u j α i d i ¯ u i (cid:33) , (5.6) nd hence u i = m − (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) n (cid:89) j =1 α j d j ¯ u j α i d i ¯ u i Θ n (cid:88) i =1 (cid:32) u i m − (cid:115) (cid:81) nj =1 α j d j ¯ u j α i d i ¯ u i (cid:33) (5.7)= 1 m − √ α i d i ¯ u i Θ n (cid:88) i =1 (cid:18) u i m − √ α i d i ¯ u i (cid:19) = Θ n (cid:88) i =1 (cid:18) m − (cid:112) α i d i ¯ u mi (cid:19) m − √ α i d i ¯ u i . Therefore, Λ is given byΛ = n (cid:88) i =1 α i d i u mi = n (cid:88) i =1 α i d i Θ n (cid:88) i =1 (cid:18) m − (cid:112) α i d i ¯ u mi (cid:19) m − √ α i d i ¯ u i m = Θ m (cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − m (cid:32) n (cid:88) i =1 α i d i m − (cid:112) ( α i d i ¯ u i ) m (cid:33) = Θ m (cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − m (cid:32) n (cid:88) i =1 m − (cid:112) ( α i d i ) m − m − (cid:112) ( α i d i ) m (¯ u i ) m (cid:33) = Θ m (cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − m (cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) = Θ m (cid:32) n (cid:88) i =1 m − (cid:112) α i d i ¯ u mi (cid:33) − m . (cid:3) Lemma 5.2.
For Θ , Λ > , if d i u m − i = d j u m − j , i, j = 1 , , · · · , n ; (5.8) n (cid:88) i =1 α i u i = Θ; (5.9) n (cid:88) i =1 α i d i u mi = Λ , (5.10) we have Λ = Θ m (cid:32) n (cid:88) i =1 α i m − √ d i (cid:33) − m . (5.11) Proof.
Lemma 5.2 follows from letting ¯ u i = 1 α i in Lemma 5.1. (cid:3) .2. Exact solutions using Tanh method.
Enlightened by the works of [20, 8, 31, 32], our idea is to lookfor a monotone solution with a hyperbolic tangent profile. We make the following ans¨atz for solving (2.82): u ( x ) = k (cid:0) − tanh x (cid:1) , x ∈ R ,v ( x ) = k (cid:0) x (cid:1) , x ∈ R , (5.12)where k and k are positive constants to be determined. Since the derivative of tanh x is expressible interms of itself, i.e. ddx tanh x = 1 − tanh x , we see that the n th derivative of a polynomial in tanh x withany order is also a a polynomial in tanh x . Inserting ans¨atz (5.12) into (2.82), this fact enables us to get d ( u ) xx + θ u x + u ( σ − c u − c v ) = u (cid:0) ζ + ζ T ( x ) + ζ T ( x ) + ζ T ( x ) (cid:1) ,d ( v ) xx + θ v x + v ( σ − c u − c v ) = v (cid:0) ξ + ξ T ( x ) + ξ T ( x ) + ξ T ( x ) (cid:1) , where T ( x ) := tanh x , ζ = − c k − c k k + 12 d k − θ + σ k , (5.13) ζ = 4 c k + c k k + 8 d k − θ − σ k , (5.14) ζ = − c k + c k k − d k + σ k , (5.15) ζ = 4 c k − c k k − d k , (5.16) ζ = 20 d k − c k , (5.17)and ξ = − c k − c k k + 2 d k + θ + σ k , (5.18) ξ = − c k + c k k − d k − θ + σ k , (5.19) ξ = − c k + c k k − d k , (5.20) ξ = 6 d k − c k k . (5.21)Equating the coefficients of powers of T ( x ) to zero yields a system of 9 equations: ζ i = 0 ( i = 0 , , , , , ξ i = 0 ( i = 0 , , , . (5.22)It turns out that, with d , d , c , and c being free parameters, (5.22) can be solved to give k = 20 d c , σ = 80 d , c = 18 c d d , θ = 0 , (5.23) k = 4 d c , σ = 8 d , c = 3 c d d . 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.
Exact solutions of (SKT-tw).
Inspired by the exact periodic solutions proposed in [13], we makethe ans¨atz for solving (1.23) as follows: u ( x ) = k + m cos ( µ x ) , x ∈ R ,v ( x ) = k + m cos ( µ x ) , x ∈ R ,w ( x ) = k + m cos ( µ x ) , x ∈ R , (5.24)where µ (cid:54) = 0, k , k , k > m (cid:54) = 0, m (cid:54) = 0, m (cid:54) = 0 with | m | ≤ k , | m | ≤ k , and | m | ≤ k areconstants to be determined. Inserting ans¨atz (5.24) into (1.23), we obtain d ( u ) xx + θ u x + u ( σ − c u − c v − c w ) = ζ + ζ C ( x ) + ζ C ( x ) + ζ S ( x ) ,d ( v ) xx + θ v x + v ( σ − c u − c v − c w ) = ξ + ζ C ( x ) + ξ C ( x ) + ξ S ( x ) ,d ( w ) xx + θ w x + w ( σ − c u − c v − c w ) = ς + ς C ( x ) + ς C ( x ) + ς S ( x ) , here C ( x ) := cos ( µ x ), S ( x ) := sin ( µ x ) and ζ = − c k − c k k − c k k + 2 d µ m + k σ , (5.25) ζ = − c k m − c k m − c k m − c k m (5.26) − c k m − d k µ m + m σ ,ζ = − c m − c m m − c m m − d µ m , (5.27) ζ = θ µ m , (5.28) ξ = − c k − c k k − c k k + 2 d µ m + k σ , (5.29) ξ = − c k m − c k m − c k m − c k m (5.30) − c k m − d k µ m + m σ ,ξ = − c m − c m m − c m m − d µ m , (5.31) ξ = θ µ m , (5.32) ς = − c k − c k k − c k k + 2 d µ m + k σ , (5.33) ς = − c k m − c k m − c k m − c k m (5.34) − c k m − d k µ m + m σ ,ς = − c m − c m m − c m m − d µ m , (5.35) ς = θ µ m . (5.36)Equating the coefficients of powers of C ( x ) and S ( x ) to zero yields a system of 12 equations: ζ i = 0 ( i = 0 , , , , ξ i = 0 ( i = 0 , , , , ς i = 0 ( i = 0 , , , . (5.37)It turns out that, with m i , d i , c ij ( i, j = 1 , , , i (cid:54) = j ), and µ being free parameters, (5.37) can be solved togive k = − m , σ = 2 (cid:0) c m + c m + 3 d µ m (cid:1) , c = − m − ( c m + c m + 4 d µ m ) , (5.38) k = m , σ = − (cid:0) c m + 3 d µ m (cid:1) , c = − m − ( c m + c m + 4 d µ m ) ,k = m , σ = − (cid:0) c m + 3 d µ m (cid:1) , c = − m − ( c m + c m + 4 d µ m ) ,θ = 0 . We note that ζ = ξ = ς = 0 immediately leads to θ = 0. The result obtained is summarized in thefollowing Theorem 5.4.
System (1.23) has a solution of the form (5.24) provided that (5.38) holds.
In view of Theorem 5.4, (1.23) has the solution u ( x ) = 110 (cid:0) − cos (2 x ) (cid:1) , x ∈ R ,v ( x ) = 111 (cid:0) x ) (cid:1) , x ∈ R ,w ( x ) = 112 (cid:0) x ) (cid:1) , x ∈ R , (5.39)when d i = σ i = c ii = 1 ( i = 1 , , c = 106760 , c = 1, c = 17511 , c = 611 , c = 15, c = 1112 , and θ = 0. The resulting profiles of (5.39) are shown in Figure 4. (cid:45) x Figure 4.
Red: u ( x ) = 110 (cid:0) − cos (2 x ) (cid:1) ; green: v ( x ) = 111 (cid:0) x ) (cid:1) ; blue: w ( x ) = 112 (cid:0) x ) (cid:1) . References [1]
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Y. B. Zeldovich , Theory of flame propagation (1951). i-Chang HungDepartment of Mathematics, National Taiwan University, Taipei, Republic of Taiwan E-mail address : [email protected] Hsiao-Feng LiuDepartment of Chemical Engineering, National Taiwan University, Taipei, Taiwan
E-mail address : [email protected] Chiun-Chuan ChenDepartment of Mathematics, National Taiwan University, Taipei, Taiwan
E-mail address : [email protected]@ntu.edu.tw