Monostable-Type Travelling Wave Solutions of the Diffusive FitzHugh-Nagumo-Type System in 𝐑^𝑁

Abstract

This paper is concerned with monostable-type travelling wave solutions of the diffusive FitzHugh-Nagumo-type system (FHN) in 𝐑^𝑁 for the two components 𝑢 and 𝑣. By solving 𝑣 in terms of 𝑢, this system can be reduced to a nonlocal single equation for 𝑢. When the diffusion coefficients in the system are equal, we construct travelling wave solutions for the non-local equation by the method of super- and subsolutions developed by Morita and Ninomiya (2008) Moreover, we propose a condition for 𝛾, which is similar to the condition Reinecke and Sweers (1999) used to transform (FHN) into a quasimonotone system.

1. Introduction

In the present paper, we are concerned with the diffusive FitzHugh-Nagumo-type system (FHN) in 𝐑^𝑁 that is, 𝑢_𝑡=𝑢_𝜉𝜉+Δ_𝑦𝑣𝑢+𝑓𝑦,𝑢−𝑣,_𝑡=𝑑𝑣_𝜉𝜉+Δ_𝑦𝑣+𝛿𝑢−𝛾𝑣, (1.1) where (𝜉,𝑦)∈𝐑^𝑁=𝐑¹×𝐑^𝑁−1, 𝑁≥2, 𝛿,𝛾>0 and 𝑑≥0. A typical example of 𝑓(𝑦,𝑢) is 𝑓(𝑦,𝑢)=𝑢(1−𝑢)(𝑢−𝛽) for 0<𝛽<1/2. Throughout the paper we assume that 𝑓 is a 𝐶² function in 𝑢 and 𝑓, 𝑓_𝑢, and 𝑓_𝑢𝑢 are bounded in {(𝑦,𝑢)∣𝑦∈Ω_𝑦,|𝑢|≤𝐾} for some large constant 𝐾>0. In addition, 𝑓 satisfies (H1)−(H5).

FHN derived from the Hodgkin-Huxley model is a typical model for excitable media. In many fields, such as physics, chemistry, and biology, FHN has become one of the frequently used-reaction diffusion systems to describe interesting phenomena. The solutions of interest here are traveling wave solutions. Let 𝑥=𝜉−𝑐𝑡, then travelling wave solutions of (1.1) satisfy 𝑢_𝑥𝑥+𝑐𝑢_𝑥+Δ_𝑦𝑢+𝑓𝑦,𝑢−𝑣=0,𝑑𝑣_𝑥𝑥+𝑐𝑣_𝑥+Δ_𝑦𝑣+𝛿𝑢−𝛾𝑣=0. (1.2)(1.3)

Over the past decades, this system has been extensively studied. For instance, as 𝑁=1, under different assumptions, Systems (1.2) and (1.3) admit standing pulses in [1–3], infinitely many periodic solutions in [3], fronts, back waves in [4, 5] and travelling pulses in [5]. For the higher dimension case 𝑁≥2, symmetric standing waves were established by Reinecke and Sweers [6] and Wei and Winter [7].

As 𝛾→∞, if the solutions are assumed to be bounded, (1.2) and (1.3) tend to the single equation 𝑢_𝑥𝑥+𝑐𝑢_𝑥+Δ_𝑦𝑢+𝑓𝑦,𝑢=0. (1.4) Let 𝑓(𝑦,𝑢) be a 𝐶² function 𝑔(𝑢) which has the property that for some ∈(0,1)𝑔(0)=𝑔(𝜃)=𝑔(1)=0,𝑔_𝑢(0)<0,𝑔_𝑢(𝜃)>0,𝑔_𝑢(1)<0, 𝑔<0 on (0,𝜃) and 𝑔>0 on (𝜃,1). In addition to the planar waves, (1.4) admits other types of solutions, including travelling curved fronts (𝑁=2), conical shapes and pyramidal shapes (𝑁≥3) in [8–11]. Moreover, Hamel and Roquejoffre [12] established travelling wave solutions of (1.4) in 𝐑² which connect one unstable periodic solution at 𝑥→∞ (−∞) and one stable constant solution at 𝑥→−∞ (∞). On the other hand, travelling wave solutions of (1.4) in 𝐑^𝑁 connecting a unstable one-peak solution at 𝑥→∞ (−∞) and a stable constant solution 𝑥→−∞ (∞) were obtained by Morita and Ninomiya [13].

In this paper, we use the method of super- and subsolutions developed in [13]. Due to technical restriction, we assume 𝑑=1. Since (1.3) is linear, 𝑣 can be solved formally in terms of 𝑢. With 𝑣 expressed in terms of 𝑢, Systems (1.2) and (1.3) are reduced to the non-local equation ℱ[𝑢]∶=𝑢_𝑥𝑥+𝑐𝑢_𝑥+Δ_𝑦𝑢+𝑓𝑦,𝑢−𝐵_𝑐[𝑢]=0, (1.5) where we denote 𝑣 by 𝐵_𝑐[𝑢]∶=𝛿(−𝜕²/𝜕𝑥²−𝑐(𝜕/𝜕𝑥)−Δ_𝑦+𝛿𝛾)⁻¹𝑢. It is readily seen that if 𝑢 is independent of 𝑥, then by the uniqueness theorem 𝐵_𝑐[𝑢]=𝛿(−Δ_𝑦+𝛿𝛾)⁻¹𝑢. As 𝑥→±∞, the asymptotic behaviors of travelling wave solutions of (1.5) formally satisfy Δ_𝑦𝑢+𝑓𝑦,𝑢−𝐵_𝑐[𝑢]=0, (1.6) where 𝐵_𝑐[𝑢]=𝛿(−Δ_𝑦+𝛿𝛾)⁻¹𝑢. Our main purpose is to look for monostable-type travelling wave solutions 𝑢(𝑥,𝑦) which connect a stable solution of (1.6) as 𝑥→−∞ (∞) and a unstable one as 𝑥→∞ (−∞). Without loss of generality, we may assume that 𝑢(+∞,𝑦) is an unstable solution. Throughout this paper, the following hypotheses are assumed.(H1) There are two solutions 𝑢_±(𝑦) of (1.6) satisfying 𝑢₋(𝑦)≥𝑢₊(𝑦). Moreover, there exist an eigenvalue 𝜇>0 and its corresponding eigenfunction 𝜙(𝑦)>0 with max_{{𝑦∈𝐑𝑁−1}}𝜙(𝑦)=1 and lim_|𝑦|→∞𝜙(𝑦)=0 such that Δ_𝑦𝜙+𝑓_𝑢𝑦,𝑢₊𝜙−𝐵_𝑐𝜙=𝜇𝜙. (1.7)(H2) There exists no other solution 𝑢(𝑦) of (1.6) with the property 𝑢₋(𝑦)≥𝑢(𝑦)≥𝑢₊(𝑦).(H3)𝑢₋(𝑦)≥𝑢₊(𝑦)+𝜖𝜙(𝑦) for some 𝜖>0.(H4) For all small 𝜂>0, there exists solutions 𝑢^𝜂₊(𝑦) satisfying lim_𝜂→0𝑢^𝜂₊(𝑦)=𝑢₊(𝑦), Δ_𝑦𝑢^𝜂₊+𝑓𝑦,𝑢^𝜂₊−𝐵_𝑐𝑢^𝜂₊𝑢+𝜂=0,_𝜂+𝑦≥𝑢₊𝑦+𝜂𝑀, (1.8) for some constant 𝑀>0. (H5)Δ_𝑦𝜓_𝑖−𝐾₁+√𝛿𝜓_𝑖≤0,𝑖=1,2,3, (1.9) where 𝐾₁=−min_{{𝑢−(𝑦)≥𝑢≥𝑢+(𝑦),𝑦∈𝐑𝑁−1}}𝑓_𝑢(𝑦,𝑢)>0,𝜓₁=𝜙,𝜓₂=𝑢₋−𝑢₊ and 𝜓₃=𝑢^𝜂₊−𝑢₊.

To simplify the proof of the main theorem in this paper, we modify the nonlinear term 𝑓(𝑦,𝑢) such that the minimum and maximum of 𝑓_𝑢(𝑦,𝑢) in {𝑢(𝑦)∈𝐑,𝑦∈𝐑^𝑁−1} are the same as those in {𝑢₋(𝑦)≥𝑢≥𝑢₊(𝑦),𝑦∈𝐑^𝑁−1}. For convenience, we still denote 𝑓(𝑦,𝑢) for the new modification of 𝑓. Let 𝐾^∗∶=max_{{𝑢−(𝑦)≥𝑢≥𝑢+(𝑦),𝑦∈𝐑𝑁−1}}𝑓_𝑢(𝑦,𝑢)>0 and let 𝐾₂>0 satisfy 𝐾₂+𝛿/(𝛿𝛾+𝐾₂)=𝐾^∗. We state the main theorem as follows.

Theorem 1.1. Assume𝛾≥2/√𝛿+(𝐾₁+𝜇)/𝛿 and (H1)–(H5) hold. Then there exists 𝑐^∗=max{2√𝜇,2𝐾₂}>0 such that for all 𝑐≥𝑐^∗, Systems (1.2) and (1.3) admit a pair of smooth solutions (𝑢^∗,𝑣^∗) which satisfie 𝑢^∗_𝑥≤0,𝑣^∗_𝑥≤0 and the boundary conditions (𝑢^∗,𝑣^∗)(±∞,𝑦)=(𝑢_±(𝑦),𝑣_±(𝑦)), where 𝑣_±(𝑦)=𝐵_𝑐[𝑢_±(𝑦)].

Remark 1.2. In (H1), when the inequality 𝑢₋(𝑦)≥𝑢₊(𝑦) is reversed, that is, 𝑢₋(𝑦)≤𝑢₊(𝑦), a result similar to Theorem 1.1 can be proved except that the inequalities 𝑢^∗_𝑥≤0 and 𝑣^∗_𝑥≤0 in Theorem 1.1 need to be replaced by 𝑢^∗_𝑥≥0 and 𝑣^∗_𝑥≥0, respectively.

Remark 1.3. In fact, (H5) can be weakened to the following assumption: Δ_𝑦𝜓_𝑖−𝑀_𝑖𝜓_𝑖≤0,forsomeconstants𝑀_𝑖>0. (1.10) This condition holds if Δ_𝑦𝜓_𝑖 does not decay faster than 𝜓_𝑖 as |𝑦|→∞. In this case, if we choose 𝛾≥1/√𝛿+(𝐾₃+𝜇)/𝛿, where 𝐾₃=max{𝑀₁,𝑀₂,𝑀₃,𝐾₁+√𝛿}, then a similar result can be proved.

It is not easy to find an example which satisfies assumptions (H1)-(H5) even for the case 𝑓(𝑦,𝑢)=𝑢(1−𝑢)(𝑢−𝛽) since the stability of the radially symmetric solutions obtained in [6, 7] has not yet been studied. However, we believe that for 𝛾≫1 the structure of System (1.2) and (1.3) are similar to that of (1.5). Accordingly, we extend the result of Theorem 2.1 in [13] to the one in Theorem 1.1.

2. Proof of the Main Theorem

To prove the Theorem 1.1, we use the super- and subsolutions constructed in [13]. By considering the following equation, we construct subsolutions of ℱ[𝑢]. Let 𝑤(𝑥) satisfy 𝑤_𝑥𝑥+𝑐𝑤_𝑥+𝜇𝑤−𝑤²𝑤()(∞)=0,−∞=𝜇,𝑤=0. (2.1) For all 𝑐≥2√𝜇, the above boundary value problem admits a unique solution 𝑤(𝑥) (up to a translation) which is strictly increasing in 𝑥. Subsolutions of ℱ[𝑢] are established as follows.

Lemma 2.1. Let 𝑈(𝑥,𝑦)=𝑢₊(𝑦)+𝜎𝜙(𝑦)𝑤(𝑥). Then there exists 𝜎₁>0 such that ℱ[𝑈]≥0 for all 0<𝜎≤𝜎₁ and 𝑐≥2√𝜇.

Proof. Let 𝑉∶=𝑤𝐵_𝑐[𝜙]−𝐵_𝑐[𝜙𝑤]≥0, then 𝑉≥0. Indeed, it is easy to see that 𝐵_𝑐[𝜙]≥0 by the maximum principle and 𝜙>0. A straightforward calculation gives 𝑉_𝑥𝑥+𝑐𝑉_𝑥+Δ_𝑦𝑉−𝛿𝛾𝑉=−𝑤𝜇−𝑤𝐵_𝑐𝜙≤0. (2.2) Using the maximum principle, we obtain 𝑉≥0. Therefore by (H1) ℱ𝑈=𝜎𝜙𝑤_𝑥𝑥+𝑐𝑤_𝑥+Δ_𝑦𝑢₊−𝐵_𝑐𝑢₊+𝜎𝑤Δ_𝑦𝜙+𝑓𝑦,𝑢₊+𝜎𝜙𝑤−𝜎𝐵_𝑐𝑤𝜙𝑤=𝜎𝜙_𝑥𝑥+𝑐𝑤_𝑥+𝜇𝑤+𝑓𝑦,𝑢₊+𝜎𝜙𝑤−𝑓𝑦,𝑢₊−𝑓_𝑢𝑦,𝑢₊𝜎𝜙𝑤+𝜎𝑉≥𝜎𝜙𝑤²+𝐺, (2.3) where 𝐺=𝑓(𝑦,𝑢₊+𝜎𝜙𝑤)−𝑓(𝑦,𝑢₊)−𝑓_𝑢(𝑦,𝑢₊)𝜎𝜙𝑤.
Let 𝑀₁=min_{{𝑢−(𝑦)≥𝑢≥𝑢+(𝑦),𝑦∈𝐑𝑁−1}}𝑓_𝑢𝑢(𝑦,𝑢). By choosing 𝜎≤𝜖/𝜇 and using (H3), we obtain 𝑢₊≤𝑢₊+𝜎𝜙𝑤≤𝑢₊+𝜖𝜙≤𝑢₋. According to the mean value theorem, we have 𝐺≥0 if 𝑀₁≥0 and 𝐺≥𝑀₁𝜎²𝜙²𝑤² if 𝑀₁<0. Therefore ℱ[𝑈]≥0 if 𝜎≤𝜎₁, where 𝜎₁=𝜖/𝜇 as 𝑀₁≥0 and 𝜎₁=min{𝜖/𝜇,−1/𝑀₁} as 𝑀₁<0. The proof is completed.

In what follows we construct supersolutions of ℱ[𝑢].

Lemma 2.2. Let 𝑄(𝑥)=𝑒^{−((𝑐−(𝑐2−4𝐾2)/2))𝑥} and 𝑈⁺(𝑥,𝑦)=𝑢^𝜂₊(𝑦)+𝑄(𝑥), where 𝐾₂>0 satisfies 𝐾₂+𝛿/(𝛿𝛾+𝐾₂)=𝐾^∗ and 𝑐≥2𝐾₂. Then ℱ[𝑈⁺]<0.

Proof. Note that 𝑄_𝑥𝑥+𝑐𝑄_𝑥+𝐾₂𝑄=0 and 0<𝐵_𝑐[𝑄]<∞. Indeed, by the uniqueness theorem we have 𝐵_𝑐[𝑄(𝑥)]=𝛿(−𝜕²/𝜕𝑥²−𝑐(𝜕/𝜕𝑥)+𝛿𝛾)⁻¹𝑄 and 𝐵_𝑐[𝑄]=𝛿𝑐²+4𝛾𝛿^+∞−∞𝑒^{−(𝑐2+4𝛾𝛿/2)|𝑥−𝜉|+(𝑐/2)(𝜉−𝑥)}𝑄(𝜉)𝛿𝑑𝜉=𝛿𝛾+𝐾₂𝑄(𝑥). (2.4) It follows from (H4) that ℱ𝑈⁺=𝑄_𝑥𝑥+𝑐𝑄_𝑥+Δ_𝑦𝑢^𝜂₊−𝐵_𝑐𝑢^𝜂₊+𝑓𝑦,𝑢^𝜂₊+𝑄−𝐵_𝑐[𝑄]=−𝐾₂𝑄+𝑓𝑦,𝑢^𝜂₊+𝑄−𝑓𝑦,𝑢^𝜂₊−𝜂−𝐵_𝑐[𝑄]=−𝐾₂+𝑓_𝑢𝑦,𝑢^𝜂₊−𝛿+𝜃𝑄𝛿𝛾+𝐾₂𝑄−𝜂≤−𝜂<0, (2.5) where 0≤𝜃≤1. The last second inequality is due to 𝐾₂+𝛿𝛿𝛾+𝐾₂=max_{𝑢−𝑦≥𝑢≥𝑢+𝑦,𝑦∈𝐑𝑁−1}𝑓_𝑢𝑦,𝑢. (2.6) We complete the proof of the lemma.

Let ℒ[𝑢]=𝑢_𝑥𝑥+𝑐𝑢_𝑥+Δ_𝑦𝑢−𝐾₁+𝜇+√𝛿𝑢, (2.7) where 𝐾₁=−min_{{𝑢−(𝑦)≥𝑢≥𝑢+(𝑦),𝑦∈𝐑𝑁−1}}𝑓_𝑢(𝑦,𝑢)>0.

To show the existences of travelling wave solutions of (1.6), we use the following iteration process: 𝑢_𝑛𝑥,𝑦=ℒ⁻¹−𝑓𝑢_𝑛−1+𝐵_𝑐𝑢_𝑛−1−𝐾₁+𝜇+√𝛿𝑢_𝑛−1𝑢,𝑛=1,2,…,₀𝑥,𝑦=𝑈. (2.8)

In the following lemma, we assert that the supersolutions of ℱ are greater than or equal to the subsolutions of ℱ. Moreover, we show that both 𝑈⁺−𝑈 and 𝑢₋−𝑈 are supersolutions of ℒ, which is useful in the proof of iteration process.

Lemma 2.3. Assume 𝛾≥2/√𝛿+(𝐾₁+𝜇)/𝛿 and let 𝑈∶=min{𝑈⁺(𝑥,𝑦),𝑢₋(𝑦)}. Then for all 𝜂>0 there exists 𝜎₂>0 depending on 𝜂 such that for all 0<𝜎≤𝜎₂ one has 𝑈≥𝑈,ℒ𝑈⁺−𝑈≤0,ℒ𝑢₋−𝑈≤0. (2.9)

Proof. For the case 𝑈=𝑢₋(𝑦) we take 𝜎≤𝜖/𝜇, then 𝑈−𝑈=𝑢₋𝑦−𝑢₊𝑦−𝜎𝜙𝑦𝑤(𝑥)≥𝑢₋𝑦−𝑢₊𝑦−𝜖𝜙𝑦≥0. (2.10) The last inequality holds by (H3). On the other hand, ℒ𝑢₋−𝑈=Δ_𝑦𝑢₋−𝑢₊−𝐾₁+𝜇+√𝛿𝑢₋−𝑢₊+𝐴, (2.11) where 𝐴=−𝜎𝜙(𝑤_𝑥𝑥+𝑐𝑤_𝑥)+(𝐾₁+𝜇+√𝛿)𝜎𝜙𝑤−𝜎𝑤Δ_𝑦𝜙. According to (H5), |𝐴|≤𝜎𝐶𝜙 for some positive constant 𝐶=𝐶(𝜇,𝛿,𝐾₁). By choosing 𝜎≤𝜖𝜇/𝐶, we obtain ℒ𝑢₋−𝑈≤Δ_𝑦𝑢₋−𝑢₊−𝐾₁+√𝛿𝑢₋−𝑢₊−𝜇𝑢₋−𝑢₊+𝜎𝐶𝜙≤−𝜖𝜇𝜙+𝜎𝐶𝜙≤0, (2.12) which holds due to assumptions (H3) and (H5).
For the case 𝑈=𝑢^𝜂₊(𝑦)+𝑄(𝑥), given 𝜂>0 we choose 𝜎≤𝜂/𝜇𝑀 and use assumption (H4), then 𝑈−𝑈=𝑢^𝜂₊𝑦+𝑄(𝑥)−𝑢₊𝑦𝑦𝑤(𝑥)≥𝜂−𝜎𝜙𝑀−𝜎𝜇≥0. (2.13) Moreover, ℒ𝑈⁺−𝑈=Δ_𝑦𝑢^𝜂₊−𝑢₊−𝐾₁+𝜇+√𝛿𝑢^𝜂₊−𝑢₊+𝐴+𝑄_𝑥𝑥+𝑄_𝑥−𝐾₁+𝜇+√𝛿𝑄. (2.14) It is readily seen that 𝑄_𝑥𝑥+𝑄_𝑥−(𝐾₁+𝜇+√𝛿)𝑄≤0. By (H4) and (H5), ℒ𝑈⁺−𝑈≤−𝜂𝜇𝑀+𝜎𝐶≤if𝜎≤𝜂𝜇.𝑀𝐶 (2.15) Setting 𝜎₂=min{𝜖/𝜇,𝜖𝜇/𝐶,𝜂/𝜇𝑀,𝜂𝜇/𝑀𝐶}, the lemma holds.

To generalize the result of Theorem 2.1 in [13], the nonlocal term of (1.5) needs to be better estimated. More precisely, we point wisely control 𝐵_𝑐[𝑢] by the local term 𝑢 such that the iterative sequence 𝑢_𝑛 is comparable with 𝑢_𝑛−1.

Lemma 2.4. Let 𝑢∈𝐶²(𝐑^𝑁) be nonnegative and solve 𝑢_𝑥𝑥+𝑐𝑢_𝑥+Δ_𝑦𝑢−𝑎𝑢≤0 for some constant 𝑎. Assume 𝛾≥𝑎/𝛿+1/𝑏 for some 𝑏. Then 𝑏𝑢−𝐵_𝑐[𝑢]≥0.

Proof. Let 𝑣=𝐵_𝑐[𝑢] and 𝑈=𝑏𝑢−𝑣. Then 𝑣≥0 because of 𝑢≥0 and the maximum principle. Our main purpose is to claim 𝑈≥0. By the assumption of 𝑢 and the definition of 𝑣, we have 𝑈_𝑥𝑥+𝑐𝑈_𝑥+Δ_𝑦𝑈−𝑎𝑏+𝛿𝑏𝛿𝑈≤−𝛿𝛾−𝑎−𝑏𝑣≤0. (2.16) The last inequality follows from the hypothesis of 𝛾 and the nonnegativity of 𝑣. By the maximum principle, 𝑈≥0.

As 𝛾 becomes large, we claim that the iterative sequence 𝑢_𝑛 is increasing.

Lemma 2.5. Assume 𝛾≥2/√𝛿+(𝐾₁+𝜇)/𝛿 and 𝑐≥𝑐^∗=max{2√𝜇,2𝐾₂}, then for all 𝜂>0 and 0<𝜎≤min{𝜎₁,𝜎₂} one has 𝑢_𝑛,𝑥≤0 and 𝑢₀≤𝑢₁≤⋯≤𝑢_𝑛≤⋯≤𝑈. (2.17)

Proof. We first claim that 𝑢_𝑛≤𝑈 for all 𝑛. Indeed, by Lemmas 2.3 and 2.4 (take 𝑎=𝐾₁+𝜇+√𝛿 and 𝑏=√𝛿) we obtain √𝛿𝑈⁺−𝑢₀−𝐵_𝑐𝑈⁺−𝑢₀≥0. (2.18) Therefore Lemmas 2.2 and 2.3 yield ℒ𝑈⁺−𝑢₁≤−𝑓𝑈⁺+𝐵_𝑐𝑈⁺+𝑓𝑢₀−𝐵_𝑐𝑢₀−𝐾₁+𝜇+√𝛿𝑈⁺−𝑢₀≤−𝑓_𝑢𝜃𝑈⁺(1−𝜃)𝑢₀−𝐾₁𝑈⁺−𝑢₀≤0, (2.19) where 0≤𝜃≤1. According to the maximum principle, 𝑈⁺−𝑢₁≥0. It follows form the proof of 𝑈⁺−𝑢₁≥0 that 𝑢₋−𝑢₁≥0. Therefore 𝑢₁≤𝑈. Continuing this process, we have 𝑢_𝑛≤𝑈 for all 𝑛 by induction.
Next obvert that ℒ[𝑢₁−𝑢₀]=−ℱ[𝑈]≤0 due to Lemma 2.1. By the maximum principle, 𝑢₁−𝑢₀≥0. Applying Lemma 2.4 to 𝑢₁−𝑢₀, we have √𝛿𝑢₁−𝑢₀−𝐵_𝑐𝑢₁−𝑢₀≥0. (2.20) Therefore ℒ𝑢₂−𝑢₁=−𝑓𝑢₁−𝑓𝑢₀+𝐵_𝑐𝑢₁−𝑢₀−𝐾₁+𝜇+√𝛿𝑢₁−𝑢₀≤−𝑓_𝑢𝜃𝑢₁+(1−𝜃)𝑢₀−𝐾₁𝑢₁−𝑢₀−√𝛿𝑢₁−𝑢₀+𝐵_𝑐𝑢₁₋𝑢₀≤0, (2.21) where 0≤𝜃≤1. Thus 𝑢₂≥𝑢₁. By induction, the sequence of functions {𝑢_𝑛} is nondecreasing. On the other hand, obvert that 𝑢_0,𝑥=𝜎𝜙𝑤_𝑥<0. Therefore by (H5), we obtain ℒ−𝑢_0,𝑥=𝜎𝜙𝜇𝑤_𝑥−2𝑤𝑤_𝑥−𝜎𝑤_𝑥Δ_𝑦𝜙+𝐾₁+𝜇+√𝛿𝜎𝜙𝑤_𝑥=−𝜎𝑤_𝑥Δ_𝑦𝜙−𝐾₁+√𝛿𝜙+−2𝜇+2𝑤𝜙≤0. (2.22) Using Lemma 2.4 again, we have √𝛿−𝑢_0,𝑥−𝐵_𝑐−𝑢_0,𝑥ℒ𝑢≥0,_1,𝑥=−𝑓_𝑢𝑢₀𝑢_0,𝑥+𝐵_𝑐𝑢_0,𝑥−𝐾₁+𝜇+√𝛿𝑢_0,𝑥≥0. (2.23) Then 𝑢_1,𝑥≤0 by the maximum principle. Inducting in 𝑛, we obtain 𝑢_𝑛,𝑥≤0.

Proof of Theorem 1.1. By Lemma 2.5, we define 𝑢^∗(𝑥,𝑦)=lim_𝑛→∞𝑢_𝑛(𝑥,𝑦). Following the proof of Theorem 2.1 in [13], (H2) and (H3), for all 𝑐≥𝑐^∗ we obtain that 𝑢^∗(𝑥,𝑦) is a smooth solution of (1.5), 𝑢^∗_𝑥≤0 and 𝑢^∗(±∞,𝑦)=𝑢_±(𝑦). Let 𝑣^∗=𝐵_𝑐[𝑢^∗], then 𝑣^∗_𝑥=𝐵_𝑐[𝑢^∗_𝑥]≤0 by the maximum principle. We complete the proof of the theorem.

Acknowledgments

C.-C. Huang wishes to express his sincere gratitude to Dr. Li-Chang Hung for his careful reading of the paper and valuable suggestions for improvement, and to his supervisor, Professor Chiun-Chuan Chen for his warm encouragement and invaluable advice throughout his Ph.D. program.