# Monostable-Type Travelling Wave Solutions of the Diffusive FitzHugh-Nagumo-Type System inRN

Authors: Chih-Chiang Huang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is concerned with monostable-type travelling wave solutions of the diffusive FitzHugh-Nagumo-type system (FHN) inRN
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.


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## Body

## 1. Introduction

In the present paper, we are concerned with the diffusive FitzHugh-Nagumo-type system (FHN) inRN that is,
(1.1)ut=uξξ+Δyu+f(y,u)-v,vt=dvξξ+Δyv+δ(u-γv),
where (ξ,y)∈RN=R1×RN-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, fu, and fuu are bounded in {(y,u)∣y∈Ωy,|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. Letx=ξ-ct, then travelling wave solutions of (1.1) satisfy
(1.2)uxx+cux+Δyu+f(y,u)-v=0,(1.3)dvxx+cvx+Δyv+δ(u-γv)=0.Over the past decades, this system has been extensively studied. For instance, asN=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].Asγ→∞, if the solutions are assumed to be bounded, (1.2) and (1.3) tend to the single equation
(1.4)uxx+cux+Δyu+f(y,u)=0.
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,gu(0)<0,gu(θ)>0,gu(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 RN 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
(1.5)F[u]:=uxx+cux+Δyu+f(y,u)-Bc[u]=0,
where we denote v by Bc[u]:=δ(-∂2/∂x2-c(∂/∂x)-Δy+δγ)-1u. It is readily seen that if u is independent of x, then by the uniqueness theorem Bc[u]=δ(-Δy+δγ)-1u. As x→±∞, the asymptotic behaviors of travelling wave solutions of (1.5) formally satisfy
(1.6)Δyu+f(y,u)-Bc[u]=0,
where Bc[u]=δ(-Δy+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 solutionsu±(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
(1.7)Δyϕ+fu(y,u+)ϕ-Bc[ϕ]=μϕ.(H2)
There exists no other solutionu(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 smallη>0, there exists solutions u+η(y) satisfying limη→0u+η(y)=u+(y),
(1.8)Δyu+η+f(y,u+η)-Bc[u+η]+η=0,u+η(y)≥u+(y)+ηM,
for some constant M>0.(H5)
(1.9)Δyψi-(K1+δ)ψi≤0,i=1,2,3,
where K1=-min{u-(y)≥u≥u+(y),y∈RN-1}fu(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 termf(y,u) such that the minimum and maximum of fu(y,u) in {u(y)∈R,y∈RN-1} are the same as those in {u-(y)≥u≥u+(y),y∈RN-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}fu(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{2μ,2K2}>0 such that for all c≥c*, Systems (1.2) and (1.3) admit a pair of smooth solutions (u*,v*) which satisfie ux*≤0,vx*≤0 and the boundary conditions (u*,v*)(±∞,y)=(u±(y),v±(y)), where v±(y)=Bc[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 ux*≤0 and vx*≤0 in Theorem 1.1 need to be replaced by ux*≥0 and vx*≥0, respectively.Remark 1.3.
In fact,(H5) can be weakened to the following assumption:
(1.10)Δyψi-Miψi≤0,forsomeconstantsMi>0.
This condition holds if Δyψi does not decay faster than ψi 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 Theorem1.1, we use the super- and subsolutions constructed in [13]. By considering the following equation, we construct subsolutions of ℱ[u]. Let w(x) satisfy
(2.1)wxx+cwx+μw-w2=0,w(-∞)=μ,w(∞)=0.
For all c≥2μ, 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.
LetU_(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≥2μ.Proof.
LetV:=wBc[ϕ]-Bc[ϕw]≥0, then V≥0. Indeed, it is easy to see that Bc[ϕ]≥0 by the maximum principle and ϕ>0. A straightforward calculation gives
(2.2)Vxx+cVx+ΔyV-δγV=-w(μ-w)Bc[ϕ]≤0.
Using the maximum principle, we obtain V≥0. Therefore by (H1)
(2.3)F[U_]=σϕ(wxx+cwx)+(Δyu+-Bc[u+])+σwΔyϕ+f(y,u++σϕw)-σBc[ϕw]=σϕ(wxx+cwx+μw)+f(y,u++σϕw)-f(y,u+)-fu(y,u+)σϕw+σV≥σϕw2+G,
where G=f(y,u++σϕw)-f(y,u+)-fu(y,u+)σϕw.
LetM1=min{u-(y)≥u≥u+(y),y∈RN-1}fuu(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 ofℱ[u].Lemma 2.2.
LetQ(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 thatQxx+cQx+K2Q=0 and 0<Bc[Q]<∞. Indeed, by the uniqueness theorem we have Bc[Q(x)]=δ(-∂2/∂x2-c(∂/∂x)+δγ)-1Q and
(2.4)Bc[Q]=δc2+4γδ∫-∞+∞‍e-(c2+4γδ/2)|x-ξ|+(c/2)(ξ-x)Q(ξ)dξ=δδγ+K2Q(x).
It follows from (H4) that
(2.5)F[U+]=(Qxx+cQx)+(Δyu+η-Bc[u+η])+f(y,u+η+Q)-Bc[Q]=-K2Q+f(y,u+η+Q)-f(y,u+η)-η-Bc[Q]={-K2+fu(y,u+η+θQ)-δδγ+K2}Q-η≤-η<0,
where 0≤θ≤1. The last second inequality is due to
(2.6)K2+δδγ+K2=max⁡{u-(y)≥u≥u+(y),y∈RN-1}fu(y,u).
We complete the proof of the lemma.Let(2.7)L[u]=uxx+cux+Δyu-(K1+μ+δ)u,
where K1=-min{u-(y)≥u≥u+(y),y∈RN-1}fu(y,u)>0.To show the existences of travelling wave solutions of (1.6), we use the following iteration process:
(2.8)un(x,y)=L-1(-f(un-1)+Bc[un-1]-(K1+μ+δ)un-1),n=1,2,…,u0(x,y)=U_.In the following lemma, we assert that the supersolutions ofℱ 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.
Assumeγ≥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
(2.9)U¯≥U_,L[U+-U_]≤0,L[u--U_]≤0.Proof.
For the caseU¯=u-(y) we take σ≤ϵ/μ, then
(2.10)U¯-U_=u-(y)-u+(y)-σϕ(y)w(x)≥u-(y)-u+(y)-ϵϕ(y)≥0.
The last inequality holds by (H3). On the other hand,
(2.11)L[u--U_]=Δy(u--u+)-(K1+μ+δ)(u--u+)+A,
where A=-σϕ(wxx+cwx)+(K1+μ+δ)σϕw-σwΔyϕ. According to (H5), |A|≤σCϕ for some positive constant C=C(μ,δ,K1). By choosing σ≤ϵμ/C, we obtain
(2.12)L[u--U_]≤Δy(u--u+)-(K1+δ)(u--u+)-μ(u--u+)+σCϕ≤-ϵμϕ+σCϕ≤0,
which holds due to assumptions (H3) and (H5).
For the caseU¯=u+η(y)+Q(x), given η>0 we choose σ≤η/μM and use assumption (H4), then
(2.13)U¯-U_=u+η(y)+Q(x)-u+(y)-σϕ(y)w(x)≥ηM-σμ≥0.
Moreover,
(2.14)L[U+-U_]=Δy(u+η-u+)-(K1+μ+δ)(u+η-u+)+A+Qxx+Qx-(K1+μ+δ)Q.
It is readily seen that Qxx+Qx-(K1+μ+δ)Q≤0. By (H4) and (H5),
(2.15)L[U+-U_]≤-ημM+σC≤ifσ≤ημMC.
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 Bc[u] by the local term u such that the iterative sequence un is comparable with un-1.Lemma 2.4.
Letu∈C2(RN) be nonnegative and solve uxx+cux+Δyu-au≤0 for some constant a. Assume γ≥a/δ+1/b for some b. Then bu-Bc[u]≥0.Proof.
Letv=Bc[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
(2.16)Uxx+cUx+ΔyU-ab+δbU≤-(δγ-a-δb)v≤0.
The last inequality follows from the hypothesis of γ and the nonnegativity of v. By the maximum principle, U≥0.Asγ becomes large, we claim that the iterative sequence un is increasing.Lemma 2.5.
Assumeγ≥2/δ+(K1+μ)/δ and c≥c*=max{2μ,2K2}, then for all η>0 and 0<σ≤min{σ1,σ2} one has un,x≤0 and
(2.17)u0≤u1≤⋯≤un≤⋯≤U¯.Proof.
We first claim thatun≤U¯ for all n. Indeed, by Lemmas 2.3 and 2.4 (take a=K1+μ+δ and b=δ) we obtain
(2.18)δ(U+-u0)-Bc[U+-u0]≥0.
Therefore Lemmas 2.2 and 2.3 yield
(2.19)L[U+-u1]≤-f(U+)+Bc[U+]+f(u0)-Bc[u0]-(K1+μ+δ)(U+-u0)≤{-fu(θU+(1-θ)u0)-K1}(U+-u0)≤0,
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 un≤U¯ for all n by induction.
Next obvert thatℒ[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
(2.20)δ(u1-u0)-Bc[u1-u0]≥0.
Therefore
(2.21)L[u2-u1]=-(f(u1)-f(u0))+Bc[u1-u0]-(K1+μ+δ)(u1-u0)≤{-fu(θu1+(1-θ)u0-K1}(u1-u0)-δ(u1-u0)+Bc[u1-u0]≤0,
where 0≤θ≤1. Thus u2≥u1. By induction, the sequence of functions {un} is nondecreasing. On the other hand, obvert that u0,x=σϕwx<0. Therefore by (H5), we obtain
(2.22)L[-u0,x]=σϕ(μwx-2wwx)-σwxΔyϕ+(K1+μ+δ)σϕwx=-σwx{Δyϕ-(K1+δ)ϕ+(-2μ+2w)ϕ}≤0.
Using Lemma 2.4 again, we have
(2.23)δ(-u0,x)-Bc[-u0,x]≥0,L[u1,x]=-fu(u0)u0,x+Bc[u0,x]-(K1+μ+δ)u0,x≥0.
Then u1,x≤0 by the maximum principle. Inducting in n, we obtain un,x≤0.Proof of Theorem1.1.
By Lemma2.5, we define u*(x,y)=limn→∞un(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), ux*≤0 and u*(±∞,y)=u±(y). Let v*=Bc[u*], then vx*=Bc[ux*]≤0 by the maximum principle. We complete the proof of the theorem.



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