Hung, L.-C.

Liao, X.

English

KITopen

NONLINEAR ESTIMATES FOR TRAVELING WAVE SOLUTIONSOF REACTION DIFFUSION EQUATIONS AND THEIRAPPLICATIONS TO MATHEMATICAL ECOLOGYLi-Chang Hung∗Department of Mathematics, National Taiwan UniversityTaipei, TaiwanXian LiaoInstitute for Analysis, Karlsruhe Institute for TechnologyKarlsruhe, Germany(Communicated by the associate editor name)Abstract. In this paper we will establish nonlinear a priori lower and upperbounds for the solutions to a large class of equations which arise from the studyof traveling wave solutions of reaction-diffusion equations, and we will applyour nonlinear bounds to the Lotka-Volterra system of two competing species asexamples. The idea used in a series of papers [2, 3, 4, 5, 6, 7] where the linearN-barrier maximum principle was established will also be used in the proof.1. IntroductionThe present paper is devoted to nonlinear a priori upper and lower bounds forthe solutions ui = ui(x) : R 7→ [0,∞), i = 1, · · · , n to the following boundary valueproblem of n equations(1)di (ui)xx + θ (ui)x + ulii fi(u1, u2, · · · , un) = 0, x ∈ R, i = 1, 2, · · · , n,(u1, u2, · · · , un)(−∞) = e−, (u1, u2, · · · , un)(∞) = e+.In the above, di, li > 0, θ ∈ R are parameters, fi ∈ C0([0,∞)n) are given functionsand the boundary values e−, e+ take value in the following constant equilibria set(2){(u1, · · · , un)∣∣∣ ulii fi(u1, · · · , un) = 0, ui ≥ 0, ∀i = 1, · · · , n}.Equations (1) arise from the study of traveling waves solutions of reaction-diffusion equations (see [16, 18]). A series of papers [2, 3, 4, 5, 6, 7] by Hunget al. have been contributed to the linear (N-barrier) maximum principle for the nequations (1), and in particular the lower and upper bounds for any linear combi-nation of the solutionsn∑i=1αi ui(x), ∀(α1, · · · , αn)have been established in terms of the parameters di, li, θ in (1).1991 Mathematics Subject Classification. Primary: 35B50; Secondary: 35C07, 35K57.Key words and phrases. Maximum principle, traveling wave solutions, reaction-diffusion equa-tions, Lotka-Volterra.∗ Corresponding author.12 HUNG AND LIAOHere we aim to derive nonlinear estimates for the polynomials of the solutions:n∏i=1(ui(x) + ki)αi , ∀(α1, · · · , αn)for some ki ≥ 0, which is related to the diversity indices of the species in ecology:Dq = (∑ni=1(ui)q)1/(1−q), q ∈ [1,∞). Observe that when either e+ = (0, · · · , 0)or e− = (0, · · · , 0), the trivial lower bound ofn∏i=1(ui(x))αi is 0. For ki > 0 thefollowing lower bound for the upper solutions of (1) holds.Proposition 1 (Lower bound). Suppose that (ui(x))ni=1 ∈ (C2(R))n with ui(x) ≥0, ∀i = 1, · · · , n is an upper solution of (1):(3)di(ui)xx + θ(ui)x + ulii fi(u1, u2, · · · , un) ≤ 0, x ∈ R, i = 1, 2, · · · , n,(u1, u2, · · · , un)(−∞) = e−, (u1, u2, · · · , un)(∞) = e+.and that there exist (¯ui)ni=1 ∈ (R+)n such that(4)fi(u1, · · · , un) ≥ 0, for all(u1, · · · , un) ∈ R̄ :={(ui)ni=1 ∈ ([0,∞))n |∑ni=1ui¯ui ≤ 1}.Then we have for any (αi)ni=1 ∈ (R+)n,(5)n∏i=1(ui(x) + ki)diαi ≥ eλ1 , x ∈ R,whereλ1 = min1≤j≤n(η dj +n∑i=1,i6=jαi(di − dj) ln ki),(6a)η = min1≤j≤n1dj(λ2 −n∑i=1,i6=jαi(di − dj) ln ki),(6b)λ2 = min1≤j≤n(αjdj ln(¯uj + kj) +n∑i=1,i6=jαidi ln ki).(6c)Remark 1 (Equal diffusion). When di = d for all i = 1, 2, · · · , n, thenλ1 = min1≤j≤n(αj ln(¯uj + kj) +n∑i=1,i6=jαi ln ki)d = λ2 = dη,and the lower bound (5) becomesn∏i=1(ui(x) + ki)αi ≥ min1≤j≤n((¯uj + kj)αj∏i 6=jkαii), x ∈ R.If furthermore αi = α, ∀i = 1, · · · , n, then the inequality of arithmetic and geomet-ric averages yieldsn∑i=1(ui + ki)α ≥ n( n∏i=1(ui + ki)α) 1n ≥ n min1≤j≤n((¯uj + kj)α∏i 6=jkαi) 1n.NONLINEAR ESTIMATES FOR REACTION DIFFUSION EQUATIONS 3On the other hand, we can find a upper bound ofn∏i=1(ui(x))αi for the lowersolutions of (1).Proposition 2 (Upper bound). Suppose that (ui(x))ni=1 ∈ (C2(R))n with ui(x) ≥0 ∀i = 1, · · · , n is a lower solution of (1):(7)di(ui)xx + θ(ui)x + ulii fi(u1, u2, · · · , un) ≥ 0, x ∈ R, i = 1, 2, · · · , n,(u1, u2, · · · , un)(−∞) = e−, (u1, u2, · · · , un)(∞) = e+,and there exist ūi > 0, i = 1, · · · , n, such that(8)fi(u1, · · · , un) ≤ 0, for all(u1, · · · , un) ∈ R̄ :={(ui)ni=1 ∈ ([0,∞))n |∑ni=1uiūi≥ 1}.Then we have for any mi ≥ 1 and αi > 0 (i = 1, 2, · · · , n)(9)n∑i=1αi(ui(x))mi ≤(max1≤i≤nαi(ūi)mi) max1≤i≤ndimin1≤i≤ndi, x ∈ R,and hence(10)n∏i=1(ui(x))mi/n ≤max1≤i≤nαi ūmiin(n∏i=1αi)1/n max1≤i≤n dimin1≤i≤ndi, x ∈ R.In particular, when αi = α for all i = 1, · · · , n, (10) becomes(11)n∏i=1(ui(x))mi/n ≤max1≤i≤nūmiinmax1≤i≤ndimin1≤i≤ndi, x ∈ R.The remainder of this paper is organized as follows. Section 2 is devoted to theproofs of Proposition 1 and Proposition 2. As an example to illustrate our mainresult, we use the Lotka-Volterra system of two competing species to conclude withSection 2.2. Proofs of Proposition 1 and Proposition 2Proof of Proposition 1. We first rewrite the inequality di(ui)′′ + θ(ui)′ + ulii fi ≤ 0in (3). If u(x) ≥ 0, then for any k > 0, a straightforward calculation gives(ln(u(x) + k))′ =u′(x)u(x) + k,(ln(u(x) + k))′′ =u′′(x)u(x) + k− (u′(x))2(u(x) + k)2.Hence we divide the inequality by ui + ki > 0 with ki > 0 to arrive atdi(ln(ui + ki))′′ + di((ui)′)2(ui + ki)2+ θ (ln(ui + ki))′ +uliiui + kifi ≤ 0.Thus (Ui)ni=1 := (ln(ui + ki))ni=1 satisfies the following inequalities:diU′′i + θ U′i +uliiui + kifi ≤ 0, i = 1, · · · , n.(12)4 HUNG AND LIAOFor any (αi)ni=1 ∈ (R+)n, letp(x) =n∑i=1αiUi, q(x) =n∑i=1αi diUi,then the above inequality (12) reads as(13) q′′ + θp′ + F ≤ 0, F :=n∑i=1αi uliiui + kifi(u1, · · · , un).We are going to derive a lower bound forq =n∑i=1αidiUi =n∑i=1αi di ln(ui(x) + ki),and hence a lower bound for∏ni=1(ui + ki)diαi . The idea is similar as in the papers[2, 3, 4, 5, 6, 7], namely we are going to determine three parametersλ1, η, λ2to construct an N-barrier consisting of three hypersurfacesQ1 := {(ui)ni=1 | q = λ1}, P := {(ui)ni=1 | p = η}, Q2 := {(ui)ni=1 | q = λ2},such that the following inclusion relations hold:Q1 := {(ui)ni=1 ∈ ([0,∞))n | q ≤ λ1} ⊂ P := {(ui)ni=1 ∈ ([0,∞))n | p ≤ η}⊂ Q2 := {(ui)ni=1 ∈ ([0,∞))n | q ≤ λ2} ⊂ R̄ ={(ui)ni=1 ∈ ([0,∞))n |n∑i=1ui¯ui≤ 1}.It will turn out that if λ1, η, and λ2 are given respectively by (6a), (6b), and (6c),then λ1 determines a lower bound of q(x): q(x) ≥ λ1, which is exactly (5).More precisely, we follow the steps as in [2, 3, 4, 5, 6, 7] to determine λ2, η, λ1such that the above inclusion relations Q1 ⊂ P ⊂ Q2 ⊂ R̄ hold:(i) Determine λ2 The hypersurface Q2 intersects the uj-axis: {(ui)ni=1 |ui =0, ∀i 6= j} at the point{(ui)ni=1 |ui = 0, ∀i 6= j, u2,j = eλ2−∑ni=1,i 6=j αi di ln kiαj dj − kj}.If u2,j ≤¯uj , ∀j = 1, · · · , n, then by the monotonicity of the function ln(·+k),Q2 ⊂ R̄. That is, Q2 ⊂ R̄ if λ2 is chosen as in (6c):λ2 = min1≤j≤n(αjdj ln(¯uj + kj) +n∑i=1,i6=jαidi ln ki).(ii) Determine η As above, the hypersurface P intersects the uj-axis at{(ui)ni=1 |ui = 0, ∀i 6= j, u0,j = eη−∑ni=1,i 6=j αi ln kiαj − kj}.If u0,j ≤ u2,j , ∀j = 1, · · · , n, then P ⊂ Q2 and the hypersurface Q2 is abovethe hypersurface P . That is, P ⊂ Q2 if η is chosen as in (6b):η = min1≤j≤n1dj(λ2 −n∑i=1,i6=jαi(di − dj) ln ki).NONLINEAR ESTIMATES FOR REACTION DIFFUSION EQUATIONS 5(iii) Determine λ1 Replacing λ2 by λ1 in (i), the uj-intercept of the hypersurfaceQ1 is given by u1,j = eλ1−∑ni=1,i 6=j αi di ln kiαj dj −kj . Hence if we take λ1 as in (6a):λ1 = min1≤j≤n(η dj +n∑i=1,i6=jαi(di − dj) ln ki),then u1,j ≤ u0,j , ∀j = 1, · · · , n and hence Q1 ⊂ P.We now show q(x) ≥ λ1, x ∈ R by a contradiction argument. Suppose bycontradiction that there exists z ∈ R such that q(z) < λ1. Since ui(x) ∈ C2(R)(i = 1, · · · , n) and (u1, u2, · · · , un)(±∞) = e±, we may assume minx∈Rq(x) = q(z).We denote respectively by z2 and z1 the first points at which the solution trajectory{(ui(x))ni=1 |x ∈ R} intersects the hypersurface Q2 when x moves from z towards∞ and −∞. For the case where θ ≤ 0, we integrate (13) with respect to x from z1to z and obtain(14) q′(z)− q′(z1) + θ (p(z)− p(z1)) +∫ zz1F (u1(x), · · · , un(x)) dx ≤ 0.We also have the following facts from the construction of the hypersurfacesQ1, Q2, P :• q′(z) = 0 because of minx∈Rq(x) = q(z);• q(z1) = λ2 because of (ui(z1))ni=1 ∈ Q2.• q′(z1) < 0 because z1 is the first point for q(x) taking the value λ2 when xmoves from z to −∞, such that q(z1 + δ) < λ2 for z − z1 > δ > 0;• p(z) < η since (ui(z))ni=1 is below the hypersurface P ;• p(z1) > η since (ui(z1))ni=1 is above the hypersurface P ;• F (u1(x), · · · , un(x)) =n∑i=1αi uliiui + kifi(u1, · · · , un) ≥ 0, ∀x ∈ [z1, z]. Indeed,since (ui(z1))ni=1 ∈ Q2 ⊂ Q2 ⊂ R̄ and (ui(z))ni=1 ∈ Q1 ⊂ R̄, we derive thatF (u1(x), · · · , un(x))|x∈[z1,z] ≥ 0 by the hypothesis (4).We hence have the following inequality from the above facts when θ ≤ 0q′(z)− q′(z1) + θ (p(z)− p(z1)) +∫ zz1F (u1(x), · · · , un(x)) dx > 0,which contradicts (14). Therefore when θ ≤ 0, q(x) ≥ λ1 for x ∈ R. For the casewhere θ ≥ 0, we simply integrate (13) with respect to x from z to z2 to arrive atq′(z2)− q′(z) + θ (p(z2)− p(z)) +∫ z2zF (u1(x), · · · , un(x)) dx ≤ 0.Then we apply the facts that q′(z2) > 0, q′(z) = 0, p(z2) > η, p(z) < η andF (u1(x), · · · , un(x))|x∈[z,z2] ≥ 0, as well as a similar contradiction argument asabove, to derive q ≥ λ1.Proof of Proposition 2. We prove Proposition 2 in a similar manner to the proof ofProposition 1. We first rewrite the inequality di(ui)′′ + θ(ui)′ + ulii fi ≥ 0 in (7). Astraightforward calculation shows(um)′ = mum−1u′,(um)′′ = m((m− 1)um−2(u′)2 + um−1u′′(x)).6 HUNG AND LIAOHence we multiply the inequality by mi umi−1(x) to arrive atdi(uimi)′′ − dimi(mi − 1)uimi−2(ui′)2 + θ (uimi)′ +mi uimi−1ulii fi ≥ 0.For notational simplicity, we will adopt the same notations as in the proof of Propo-sition 1. Since ui ≥ 0, ∀i = 1, · · · , n, for any (mi)ni=1 ∈ ([1,∞))n, the vector field(Ui)ni=1 := (umii )ni=1 satisfies the following inequalitiesdiU′′i + θ U′i +mi uimi−1ulii fi ≥ 0, ∀i = 1, · · · , n.(15)For any (αi)ni=1 ∈ (R+)n, p(x) =∑ni=1 αi Ui and q(x) =∑ni=1 αi diUi satisfy(16) q′′ + θ p′ + F ≥ 0, F :=n∑i=1αimi uimi−1ulii fi(u1, u2, · · · , un).We are going to show the upper bound q ≤ λ1 by employing the N-barriermethod as in the proof of Proposition 1. That is, we are going to construct thethree hyperellipsoidsQ1 := {(ui)ni=1 | q = λ1}, P := {(ui)ni=1 | p = η}, Q2 := {(ui)ni=1 | q = λ2},such that the following inclusion relations hold:Q1 := {(ui)ni=1 ∈ ([0,∞))n | q ≥ λ1} ⊃ P := {(ui)ni=1 ∈ ([0,∞))n | p ≥ η}⊃ Q2 := {(ui)ni=1 ∈ ([0,∞))n | q ≥ λ2} ⊃ R̄ ={(ui)ni=1 ∈ ([0,∞))n |n∑i=1uiūi≥ 1},and the upper bound q ≤ λ1 follows by a contradiction argument. More precisely,we take(17) λ2 = max1≤i≤nαi di(ūi)mi ,such that the uj-intercept of the hyperellipsoid Q2u2,j =(λ2αj dj)1/mi≥ ūj , j = 1, 2, · · · , n.Then we take(18) η =λ2min1≤i≤ndi,such that the uj-intercept of the hyperellipsoid Pu0,j =(ηαj)1/mi≥ u2,j , j = 1, 2, · · · , n.Finally we take(19) λ1 = η max1≤i≤ndisuch that the uj-intercept of the hyperellipsoid Q1u1,j =(λ1αj dj)1/mi≥ u0,j , j = 1, 2, · · · , n.Combining (17), (18), and (19), we have(20) λ1 =(max1≤i≤nαi di(ūi)mi) max1≤i≤ndimin1≤i≤ndi.NONLINEAR ESTIMATES FOR REACTION DIFFUSION EQUATIONS 7We follow exactly the same contradiction argument to prove q(x) ≤ λ1 for x ∈ Ras in the proof of Proposition 1, which is omitted here. Since (αi)ni=1 ∈ (R+)n isarbitrary, q(x) =n∑i=1αi di(ui(x))mi ≤ λ1 implies the upper bound (9). Now we usethe inequality of arithmetic and geometric means to obtainn∑i=1αi(ui(x))mi ≥ n( n∏i=1αi(ui(x))mi) 1n ≥ n( n∏i=1αi) 1nn∏i=1(ui(x))min ,(21)which together with (9) yields (10).The construction of the N-barrier for the case n = 2 is illustrated in the followingexample, which provides an intuitive idea of the construction of the N-barrier inmulti-species cases.To illustrate Proposition 2 for the case n = 2, we use the Lotka-Volterra systemof two competing species coupled with Dirichlet boundary conditions:(22)d1uxx + θ ux + u (1− u− a1 v) = 0, x ∈ R,d2vxx + θ vx + λ v (1− a2 u− v) = 0, x ∈ R,(u, v)(−∞) = ei, (u, v)(+∞) = ej ,where a1, a2, λ > 0 are constants. In (22), the constant equilibria are e1 = (0, 0),e2 = (1, 0), e3 = (0, 1) and e4 = (u∗, v∗), where (u∗, v∗) =(1− a11− a1 a2,1− a21− a1 a2)is the intersection of the two straight lines 1 − u − a1 v = 0 and 1 − a2 u − v = 0whenever it exists. We call the solution (u(x), v(x)) of (22) an (ei, ej)-wave.Tang and Fife ([17]), and Ahmad and Lazer ([1]) established the existence ofthe (e1, e4)-waves. Kan-on ([10, 11]), Fei and Carr ([8]), Leung, Hou and Li ([15]),and Leung and Feng ([14]) proved the existence of (e2, e3)-waves using differentapproaches. (e2, e4)-waves were studied for instance, by Kanel and Zhou ([13]),Kanel ([12]), and Hou and Leung ([9]).For the above-mentioned (e1, e4)-waves, (e2, e3)-waves, and (e2, e4)-waves, weshow by Proposition 2 that an upper bound of u(x)v(x) exists for all of these waves(see (24) below). To this end, lettingū = max(1,1a2),v̄ = max(1,1a1),we see that (8) in Proposition 2 is satisfied. According to (10), letting α1 = α2 =m1 = m2 = 1 leads to(23)√u(x)v(x) ≤ 12max (ū, v̄)max (d1, d2)min (d1, d2), x ∈ Ror(24) u(x)v(x) ≤ 14(max (ū, v̄))2(max (d1, d2)min (d1, d2))2, x ∈ R.8 HUNG AND LIAOFor the equal diffusion case d1 = d2 = 1 with the bistable condition a1, a2 > 1 Forthe bistable condition a1, a2 > 1 and the equal diffusion case d1 = d2 = 1, (24) issimplified to(25) u(x)v(x) ≤ 14, x ∈ R.If we further consider the boundary conditions in the (e2, e4)-waves (also (e3, e4)-waves) or the (e4, e4)-waves, the upper bound given by (25) is optimal for the casea := a1 = a2 > 1 since as a→ 1+, we have(26) (u∗, v∗) =(1− a11− a1 a2,1− a21− a1 a2)=(11 + a,11 + a)→ (1/2, 1/2) .Acknowledgements. The authors are grateful to the anonymous referees for manyhelpful comments and valuable suggestions on this paper. L.-C. Hung thanks forthe hospitality he received from KIT while visiting KIT.References[1] S. Ahmad and A. C. Lazer, An elementary approach to traveling front solutions to a systemof N competition-diffusion equations, Nonlinear Anal., 16 (1991), pp. 893–901.[2] C.-C. Chen, T.-Y. Hsiao, and L.-C. Hung, Discrete n-barrier maximum principle for alattice dynamical system arising in competition models, to appear in Discrete Contin. Dyn.Syst. A.[3] C.-C. Chen and L.-C. Hung, A maximum principle for diffusive lotka-volterra systems oftwo competing species, J. Differential Equations, 261 (2016), pp. 4573–4592.[4] , Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solu-tions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl.Anal., 15 (2016), pp. 1451–1469.[5] , An n-barrier maximum principle for elliptic systems arising from the study of travel-ing waves in reaction-diffusion systems, Discrete Contin. Dyn. Syst. B, 22 (2017), pp. 1–19.[6] C.-C. Chen, L.-C. Hung, and C.-C. Lai, An n-barrier maximum principle for autonomoussystems of n species and its application to problems arising from population dynamics, Com-mun. Pure Appl. Anal., 18 (2019), pp. 33–50.[7] C.-C. Chen, L.-C. Hung, and H.-F. Liu, N-barrier maximum principle for degenerate ellipticsystems and its application, Discrete Contin. Dyn. Syst. A, 38 (2018), pp. 791–821.[8] N. Fei and J. Carr, Existence of travelling waves with their minimal speed for a diffusingLotka-Volterra system, Nonlinear Anal. Real World Appl., 4 (2003), pp. 503–524.[9] X. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusionsystem and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), pp. 2196–2213.[10] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), pp. 340–363.[11] , Fisher wave fronts for the Lotka-Volterra competition model with diffusion, NonlinearAnal., 28 (1997), pp. 145–164.[12] J. I. Kanel, On the wave front solution of a competition-diffusion system in populationdynamics, Nonlinear Anal., 65 (2006), pp. 301–320.[13] J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed fora competition-diffusion system, Nonlinear Anal., 27 (1996), pp. 579–587.[14] A. W. Leung, X. Hou, and W. Feng, Traveling wave solutions for lotka-volterra systemre-visited, Discrete & Continuous Dynamical Systems-B, 15 (2011), pp. 171–196.[15] A. W. Leung, X. Hou, and Y. Li, Exclusive traveling waves for competitive reaction-diffusionsystems and their stabilities, J. Math. Anal. Appl., 338 (2008), pp. 902–924.[16] J. D. Murray, Mathematical biology, vol. 19 of Biomathematics, Springer-Verlag, Berlin,second ed., 1993.[17] M. Tang and P. Fife, Propagating fronts for competing species equations with diffusion,Archive for Rational Mechanics and Analysis, 73 (1980), pp. 69–77.NONLINEAR ESTIMATES FOR REACTION DIFFUSION EQUATIONS 9[18] A. I. Volpert, V. A. Volpert, and V. A. Volpert, Traveling wave solutions of para-bolic systems, vol. 140 of Translations of Mathematical Monographs, American MathematicalSociety, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda.Email address: lichang.hung@gmail.comEmail address: xian.liao@kit.edu

Nonlinear estimates for traveling wave solutions of reaction diffusion equations

https://publikationen.bibliothek.kit.edu/1000119591/120430690

Nonlinear estimates for traveling wave solutions of reaction diffusion equations

Abstract

Similar works

Full text

Available Versions

KITopen