在N 維空間中的費滋漢那古默系統之行波解

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國立臺灣大學理學院數學研究所博士論文

Department of Mathematics College of Science

National Taiwan University Doctoral Dissertation

在 N 維空間中的費滋漢那古默系統之行波解

Travelling wave solutions of the diffusive FitzHugh-Nagumo system in R^N

黃志強

Chih-Chiang Huang

指導教授：陳俊全博士 Advisor: Chiun-Chuan Chen Ph.D.

中華民國 101 年 6 月 June 2012

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Acknowledgements

I wish to express sincere gratitude to the following people.

· My advisor: Professor Chiun-Chuan Chen;

· Members of dissertation committee: Professor I-Liang Chern, Professor Jann-Long Chern, Professor Jong-Shenq Guo, Professor Chun-Hsiung Hsia, Professor Tai-Chia Lin, Professor Chih-Wen Shih;

· my classmates and friends: Chuen-Hsin Chang, Der-Chiun Chen, Li-Chang Hung, Chiun-Chang Lee;

· My family: my parents, my wife, my sister and my brother;

Without them, much of my dissertation could not have been written.

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Abstract

In this thesis, we study the existences of travelling waves of the diffusive FitzHugh- Nagumo system (DFHN) in R^N. This system has a skew-gradient structure as defined by Yanagida as well as a non-local gradient structure. In addition, by a suitable transfor- mation, it also has a monotone-system structure on some parameter ranges. For bounded domains, the variational approach is applied to construct steady states of (DFHN) with Dirichlet or/and Neumann condition. For unbounded cylindrical domains, we study the travelling wave solutions via all of the three structures mentioned above when the diffusion coefficients in the equations are equal. By using the nonlocal variational energy, we establish the existence of a travelling front solution for (DFHN). Our existence result also obtains a variational characterization for the wave speed. On the other hand, using the skew-gradient structure, we give a mini-max formulation of the travelling wave and its speed. For whole domains, we employ the method of super- and subsolutions to establish the existence of monostable-type traveling wave solutions in R^N. Moreover, we construct infinitely many standing periodic solutions in R¹ based on the reflection method.

keywords: FitzHugh-Nagumo system; travelling waves; skew-gradient structure; variational method; the method of super- and subsolutions

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Chapter 1 Introduction

In this thesis, we are concerned with the diffusive FitzHugh-Nagumo system (DFHN) in R^N.

u_t= u_ξξ+ ∆_yu + f (u) − v, (1.0.1) v_t= d(v_ξξ+ ∆_yv) + δ(u − γv), (1.0.2) where (ξ, y) ∈ Ω := R¹× Ω_y with Ω_y being R^{N −1} or being a bounded C^2,α⁰ domain in R^{N −1}, d ≥ 0, δ, γ > 0, and f (u) = u(1 − u)(u − β) for 0 < β < ¹₂. Moreover, if Ω_y is a bounded domain, we impose the Dirichlet or Neumann boundary condition on it.

(DFHN), a typical model for excitable media, arises as a simplification of the Hodgkin- Huxley for nerve-impulse propagation ([8], [12] and [32]). Here u is the membrane po- tential of the nerve cells and v represents the effects of sodium ions and potassium ions.

In the past decades, (DFHN) has become one of the frequently-used reaction-diffusion systems to describe different phenomena in many fields, such as physics, chemistry and biology ([13], [27], [34], [35] and [37]).

Here we interest in the existence of steady states, standing waves and travelling waves in R^N. By setting x = ξ − ct, (DFHN) is reduced to a elliptic system with a unknown variable c, called the ”wave speed”.

u_xx+ ∆_yu + cu_x+ f (u) − v = 0, (1.0.3) d(v_xx+ ∆_yv) + cv_x+ δ(u − γv) = 0. (1.0.4) Among variant interesting structures related to (DFHN), we list three ones we used in this thesis as follows.

1. The skew-gradient structure

For variational approach, the functions u and v need to be in the same weighted space, i.e., d = 1. Moreover, setting v =√

δ˜v and dropping the tilde we obtain system (1.0.3)-(1.0.4) enjoying the skew-gradient structure defined by Yanagida [45]; namely, the system

u_xx+ ∆_yu + cu_x+ f (u) −√

δv = 0, (1.0.5)

v_xx+ ∆_yv + cv_x+√

δu − δγv = 0 (1.0.6)

satisfying _∂v^∂ (f (u) −√

δv) = −_∂u^∂ (√

δu − δγv).

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The corresponding energy of the above system (1.0.5)-(1.0.6) is E₁[u, v] = 1

2 Z

Ω

e^cx(u²_x− v²_x) + 1 2

Z

Ω

e^cx(|∇_yu|²− |∇_yv|²) + Z

Ω

e^cxH(u, v), where H(u, v) = F (u) +√

δuv −¹₂γδv² and F (u) = −Ru

0 f (s)ds = ¹₄u⁴−^β+1₃ u³+ ^β₂u². 2. The nonlocal-gradient structure

Observing that (1.0.6) is a linear equation, we can formally solve v, expressed in term of u. Denote v by B_c[u]. Consequently, system (1.0.5)-(1.0.6) is reduced to a single equation u_xx+ ∆_yu + cu_x+ f (u) − B_c[u] = 0. (1.0.7) Moreover, equation (1.0.7) is the Euler-Lagrange equation of the nonlocal gradient energy E₂[u], defined by

E₂[u] = 1 2

Z

Ω

e^cx(u²_x+ |∇_yu|²) + Z

Ω

e^cxF (u) +

√ δ 2

Z

Ω

e^cxuB_c[u].

3. The monotone-system structure

For technical restriction, we assume d = 1. By transforming w = −v + σu, σ > 0 determined later, system (1.0.3)-(1.0.4) is rewritten as

u_xx+ ∆_yu + cu_x+ f (u) − σu + w = 0, (1.0.8)

w_xx+ ∆_yw + cw_x+ σ(f (u) + (δγ − σ − δ

σ)u) + (σ − δγ)w = 0. (1.0.9) Usually, we expect the solution u is bounded, for example 0 ≤ u ≤ 1. To ensure system (1.0.8)-(1.0.9) is monotone, we impose A = f⁰(u) + (δγ − σ − _σ^δ) ≥ 0 on u ∈ [0, 1]. The condition that γ ≥ _σ¹ + ^σ_δ + ^1−β_δ is sufficient and necessary for A ≥ 0 on u ∈ [0, 1]. By choosing σ = √

δ, we obtain a optimal parameter range for γ such that system (1.0.8)- (1.0.9) is a monotone system on u ∈ [0, 1].

The above three structures will be discussed more in chapter 2-5. This thesis is organized as follows. In chapter 2, we survey the existence of waves from literature and focus on the statements of the main theorems and simple descriptions of proofs. Next, in Chapter 3, the steady states of (DFHN) in a bounded domain are established. We employ the direct method to obtain nontrivial minimizers and use the reflection method to construct periodic solutions in R¹. By applying the method of sub- and supersolutions, Chapter 4 is devoted to the existence of monostable-type travelling waves for (DFHN). By using the nonlocal structure of (DFHN), the existence of travelling frons is established in Chapter 5. Moreover, we obtain a variational characterization for the wave speed. From the skew-gradient structure, we set a mini-max formulation of the travelling wave and its speed.

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Chapter 2 Literature review

Among variant interesting problems related to (DFHN), the existence of wave solutions is one of the main issues. There has been tremendous work on this problem.

Assume N = 1. When d = 0, Rinzel and Terman [40] completely analyzed critical values of γ based on the results of Carpenter [2], Casten et al. [4] and Keener [18]. If δ is small, they showed that travelling pules, fronts and back waves exist for γ ∈ (0, γ₁), γ ∈ (γ₀, ∞) and γ ∈ (γ₀, γ₂) respectively (See Fig. 2.1 for the definition of γ_i, i = 0, 1, 2.).

I₂ I₁

I0

(0,0)

Figure 2.1: There are three critical values γ0, γ1 and γ2 of γ, which are defined as follows:

v = _γ¹

iu is the line passing through the origin and I_i ∈ {v = f (u)} for i = 0, 1, 2, where I₀ and I₁ are the local maximum and the inflection point of the curve v = f (u) respectively and I₂ satisfies the condition that the line I₁I₂ is parallel to the u axis.

By a direct calculation, γ₁ = _2β2−5β+2⁹ . As one will see, this γ₁ is also a crucial critical value in our main theorem.

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2.1 Steady states on bounded domains

In this section, assume Ω is a bounded smooth domain and the steady states of (DFHN) satisfy the Dirichlet boundary condition. Let = δ/d. The system we study is as follows:

∆u + f (u) − v = 0, (2.1.1)

∆v + (u − γv) = 0, (2.1.2)

u|_∂Ω= v|_∂Ω = 0. (2.1.3)

The corresponding energy of system (2.1.1)-(2.1.3) is Φ[u] = 1

2 Z

Ω

|∇u|²+ Z

Ω

F (u) + 1 2

Z

Ω

uB[u], (2.1.4)

where B[u] = (−∆ + γ)⁻¹[u].

2.1.1 Bistable cases

The existence results of the steady states for system (2.1.1)-(2.1.3) were constructed first by Klaasen and Mitidieri [20]. According to variational approaches, they obtained two pairs of solutions for (DFHN), where one is a minimizer of the energy and the other is a mountain pass solution. The main theorems in [20] are stated as follows.

T

HEOREM

2.1.1.

([20]) Assume γ > _2β2−5β+2⁹ =: γ₁. There exists R₀ > 0 such that if Ω contains a ball B_R₀ with the radius R₀ then system (2.1.1)-(2.1.3) has a nontrivial C²-solution pair (u₁, B[u₁]) which satisfies

inf

H₀¹(Ω)

Φ[u] = Φ[u₁] < 0. (2.1.5)

Moreover, there exists another nontrivial C²-solution pair (u2, B[u2]) which satisfies

σ∈Σinf max

0≤s≤1Φ[σ(s)] = Φ[u₂] > 0, (2.1.6)

where Σ = {σ ∈ C([0, 1]; H₀¹(Ω))|σ(0) = 0, σ(1) = u₁}.

On the other hand, nonexistence theorems were also established in [20].

T

HEOREM

2.1.2.

([20]) If Ω is a ball B_R(0) and one of the following assumption is supposed:

(i) , γ > 0 are fixed and R > 0 is sufficiently small;

(ii) γ² ≥ 1, γ < _(1−β)⁴ 2 and any R > 0;

(iii) γ² < 1, 2√

− γ > _(1−β)⁴ 2 and any R > 0.

Then system (2.1.1)-(2.1.3) has no nontrivial weak solutions.

Alternatively, Reinecke and Sweers [39] obtained the existence of steady states of system (2.1.1)-(2.1.3) by considering the following eigenvalue problem.

1

λ∆u + f (u) − v = 0, (2.1.7)

1

λ∆v + (u − γv) = 0, (2.1.8)

u|_∂Ω = v|_∂Ω = 0. (2.1.9)

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According to their existence results, the ”boundary layer solution” of system (2.1.7)- (2.1.9) was established by transforming (2.1.7)-(2.1.8) to a quasimonotone system. Con- sequently, the new system enjoys the maximum principle. By this structure, a pointwise estimate was obtained. Let

γ⁽⁾ =

( _1−β

+ ^√² if 0 < < γ₁²;

1−β

+ ^γ¹ + _γ¹

1 if ≥ γ₁². (2.1.10)

When γ is greater than γ⁽⁾, the existence theorem was established as follows.

T

HEOREM

2.1.3.

([39]) Assume ∂Ω is C³. For all > 0 and γ > γ⁽⁾, there exist λ^∗ > 0 and a function Λ ∈ C¹([λ^∗, ∞), C²(Ω)×C²(Ω)) such that (u_λ, v_λ) := Λ(λ) is a pair of positive solution of system (2.1.7)-(2.1.9) for all λ ≥ λ^∗. Moreover, p₁ < max_Ωu_λ < p₂,

p1

γ < max_Ωv_λ < ^p_γ² and lim_λ→∞Λ(λ) = (p₂,^p_γ²) uniformly on all compact subsets of Ω, where 0 < p₁ < p₂ and p₁, p₂ solve u²− (β + 1)u + (β +_γ¹) = 0.

Remark. By scaling for space variables, that λ is large in Theorem 2.1.3 is equivalent to that Ω contains a large ball in Theorem 2.1.1.

As γ 1, Reinecke et al. [39] and Klaasen et al. [20] obtained the solutions of system (2.1.1)-(2.1.3) by different approaches. It is natural to ask what relations are between those solutions. Consequently, Matsuzawa [26] proved that the global minimizer in [20]

identifies the boundary layer solution in [39] under the following conditions.

(C1) γ > max{_β¹,^√² +^β}.

(C2) γ − 2√

> M := ^(1−β)₂ ² +^1+β₂ q

(1 − β)²+ ⁴_γ +³_γ. (C3) ^2β²^−5β+2₉ > ^γ−M₂ − ¹₂p(γ − M)²− 4.

T

HEOREM

2.1.4.

([26]) If the conditions (C1)-(C3) hold, then there exists λ^[ > 0 such that u_λ in Theorem 2.1.3 coincides with the global minimizer in Theorem 2.1.1 for all λ > λ^[.

2.1.2 Monostable cases

For monostable case (0 < γ < _(1−β)⁴ 2), by the nonexistence theorem (see Theorem 2.1.2), if Ω is sufficiently small or := ^δ_d ≥ _γ¹2 then system (2.1.1)-(2.1.3) only has trivial solution.

Klaasen [19] obtained a sufficient condition to insure the existence of steady states of (2.1.1)-(2.1.3).

T

HEOREM

2.1.5.

([19]) Let 0 < γ < _(1−β)⁴ 2 and δ > 0. If R and d are chosen such that

( R

R − 1)^N − 1

1 + δR⁴

d + δγR² + β³

6 (2 − β)

< 1 − 2β

6 . (2.1.11)

Then for all smooth domain containing B_R system (2.1.1)-(2.1.3) has two nontrivial classical solutions.

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2.2 Travelling waves in R

¹

In this section, we consider the following travelling wave equation.

u_xx+ cu_x+ f (u) − v = 0, (2.2.1) dv_xx+ cv_x+ δ(u − γv) = 0. (2.2.2)

2.2.1 Bistable cases

For the bistable case, there are also various types of solutions. By the shooting method, Klaasen and Troy [22] obtained the existences of standing pulses and infinitely many periodic solutions. The main theorem is stated as follows.

T

HEOREM

2.2.1.

([22]) Let γ > max{γ₁,^√²

δ+^1−β_δ }. Then system (2.2.1)-(2.2.2) with c = 0 has a nonconstant pair (u, v) satisfying (u, u_x, v, v_x)(±∞) = 0 and (u_x, v_x)(0) = 0.

Moreover, system (2.2.1)-(2.2.2) with c = 0 has an infinite number of periodic solution.

The global bifurcation structure of front and back waves were studied by Ikeda, Mimura and Nishiura [17] when γ > γ₀.

T

HEOREM

2.2.2.

([17]) Let d = _σ^τ, δ = τ σ and c = sτ , where τ, σ > 0 and s ∈ R are parameters. Assume τ = O(σ) or O(¹_σ), then there exists σ₀ > 0 such that for all 0 < σ ≤ σ₀ the ”following bifurcation phenomena”(see Fig. 2.2) holds.

(A) (B) s s

(C) (D)

s s

Figure 2.2: (A)γ ≥ γ₂ (B)γ₁ < γ < γ₂ (C)γ = γ₁ (D)γ₀ < γ < γ₁

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2.2.2 Monostable cases

For the monostable case, Ermentrout, Hastings and Troy [7] proved the existence of two standing pulses by using the shooting method. Later Dockery [6] obtained a similar result by a different approach: a geometric singular perturbation.

T

HEOREM

2.2.3.

([6] and [7]) Let 0 < γ < _(1−β)⁴ 2. If δ/d is sufficiently small, system (2.2.1)-(2.2.2) with c = 0 has least two nonconstant, bounded solutions satisfies the following:

(i) lim|x|→∞(u, u_x, v, v_x) = (0, 0, 0, 0);

(ii) u(x) and v(x) have exactly one relative maximum in R which occurs at x = 0.

Moreover, system (2.2.1)-(2.2.2) with c = 0 has a continuum of periodic solutions.

2.3 Standing waves in R

^N

For the higher dimension case Ω = R^N and γ is large, symmetric standing waves were obtained by Reinecke et al. [38] and Wei et al. [44]. The system in R^N is that

∆u + f (u) − v = 0, (2.3.1)

∆v + (u − γv) = 0, (2.3.2)

where = δ/d.

Reinecke and Sweers [38] constructed a entire solution of system (2.3.1)-(2.3.2) by using solutions to the system (2.3.1)-(2.3.2) with Dirichlet boundary condition on the ball B_R and letting R → ∞. The main theorem is stated as follows.

T

HEOREM

2.3.1.

([38])If γ > max{^√² + ^1−β ,_2β2−5β+2⁹ + ^2β²^−14β+11₉ }, there exists a pair of positive solution (u, v) ∈ C^∞(R^N)×C^∞(R^N) for system (2.3.1)-(2.3.2). Moreover, u and v are radially symmetric, decreasing and satisfy p₁ < max_x∈RNu(x) < p₂ and max_x∈R^Nv(x) < ^p_γ².

By a perturbation for δ, Wei and Winter [44] established the following existence.

T

HEOREM

2.3.2.

([44]) For all α ∈ (0, β), there exists a ₀ = ₀(α, β) such that for all 0 < < ₀ and γ = ^α system (2.3.1)-(2.3.2) has a unique standing wave (u, v) in R^N. Moreover, u and v are radially symmetric.

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Chapter 3 Steady states on bounded domains and periodic solutions in R ^N

3.1 Introduction

In this chapter, we are interested in using a variational approach to study the steady states of (DFHN) on a bounded domain Ω in R^N. Let = δ/d. The system we study is as follows:

∆u + f (u) − v = 0, (3.1.1)

∆v + (u − γv) = 0, (3.1.2)

u|_∂Ω= v|_∂Ω= 0 or ∂u

∂ν = ∂v

∂ν = 0, (3.1.3)

where ν is the outer normal of Ω.

In [20], the nonexistence theorem (see Theorem 2.1.2) suggests us that Ω is sufficiently large and is small if we would like to look for a nontrivial solution as γ ≤ γ₁. Some arguments in proving the existences of a minimizer and a mountain pass solution will be omitted if the proofs have showed in [20] and [19]. Our main theorem is stated as follows.

T

HEOREM

3.1.1.

Assume γ ≤ γ₁. There exists R₀ = R₀(β, N ) > 0 such that if B_R₀ ⊂ Ω, then we have the following existence result.

(i) There exists ₀ = ₀(β, N, γ, |Ω|) so that for all 0 < ≤ ₀ system (3.1.1)-(3.1.2) with Dirichlet condition has two pair of classical solutions.

(ii) There exist k₁ = k₁(β, N, γ) and ₁ = ₁(β, N, γ, Ω) such that for all Ω satisfying

|Ω| ≥ k₁ and for all 0 < ≤ ₁, system (3.1.1)-(3.1.2) with Neumann condition has two pair of classical solutions.

In addition, we can construct the following existence theorems of periodic solutions in R¹ by applying the above theorem to the domains Ω = [0, L₁].

C

OROLLARY

3.1.2.

Assume γ ≤ γ₁ and Ω = R¹. Then there exists ₂ = ₂(β, γ) > 0 such that for all 0 < < ₂ system (3.1.1)-(3.1.2) has infinitely many periodic solutions.

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3.2 Proof of the main theorem

We first observe that from (3.1.2), v can be solved formally expressed in term of u. With v expressed in terms of u, system (3.1.1)-(3.1.2) is reduced to the single equation

∆u + f (u) − B[u] = 0, (3.2.1)

where v = B[u] is a solution of (3.1.2). In this section, we consider (3.2.1) under either the Dirichlet or Neumann condition. When the Dirichlet (resp., Neumann) problem of (3.2.1) is taken into account, we denote v = B0[u] (resp., v = B1[u]) by the solution of (3.1.2). With no cause for ambiguity, we continue to denote B[u] by replacing B₀[u] or B₁[u].

For Dirichlet (resp., Neumman) problem, we define the energy functional Φ0[u] : H₀¹(Ω) → R((resp., Φ₁[u] : H¹(Ω) → R) by

Φ_i[u] = 1 2

Z

Ω

|∇u|²+ Z

Ω

F (u) + 1 2

Z

Ω

uB_i[u], i = 0, 1 (3.2.2) where F (u) = −Ru

0 f (s)ds = ¹₄u⁴−^β+1₃ u³+^β₂u². With no cause for ambiguity, we continue to denote Φ0[u] or Φ1[u] by Φ[u].

From the a priori estimate for a classical solution u, Klaasen and Mitidieri [20] modified F (u) such that the growth of F (u) is not greater than the quadratic function ku²for some k > 0 as u → ±∞, i.e. for large |u|, we have

|F (u)| ≤ ku². (3.2.3)

We denote the modification of F (u) in [20] by ˜F (u). Let ˜Φ[u] be the energy, replacing the term F (u) by ˜F (u) in Φ[u]. Then ˜F (u) and ˜Φ[u] enjoy the properties that ˜Φ[u] is weakly lower semicontinuous and R

ΩF (u) =˜ ^β₂ R

Ωu² + o(kuk²) at u = 0. The second statement benefits the geometry of ˜Φ[u] in mountain pass theorem.

The following lemma asserts that B[u] is a bounded operator in L² space and the operator norm of B[u] tends to 0 as → 0, which is curial in showing that the term R

ΩuB[u] in ˜Φ[u] is small as is small.

L

EMMA

3.2.1.

Let λ_1,0 (resp., λ_1,1) be the first eigenvalue of −∆ with Dirichlet (resp., Neumman) condition on the ∂Ω. Then kB₀[u]k₂ ≤ _λ

1,0(Ω)+γkuk₂ and kB₁[u] −

R

Ωu γ|Ω|k₂ ≤

λ1,1(Ω)+γkuk₂.

Proof. Let λ_n,0 (resp., λ_n,1) be the positive eigenvalue sequence of −∆ with Dirichlet (resp., Neumman) boundary condition and w_n,0 (resp., w_n,1) be the corresponding eigenfunction with kw_n,ik₂ = 1 for i = 0, 1. Then

B0[u] = Σ^∞_n=1 R

Ωuw_n,0

λ_n,0+ γwn,0 and B1[u] = R

Ωu

γ|Ω|+ Σ^∞_n=1 R

Ωuw_n,1 λ_n,1+ γwn,1. Therefore, for i = 1, 2

kB₀[u]k²₂ or kB₁[u] − R

Ωu γ|Ω|k²₂

= Σ^∞_n=1| R

Ωuw_n,i

λn,i+ γ|² ≤ ²

(λ1,i+ γ)²Σ^∞_n=1| Z

Ω

uw_n,i|² ≤ ²

(λ1,i+ γ)²kuk²₂.

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Proof of theorem 3.1.1. By standard variational arguments, ˜Φ[u] is weakly lower semicontinuous, coercive and bounded below. Therefore, ˜Φ[u] attains a global minimum on H₀¹(Ω) or H¹(Ω). Next, we prove the minimizer is nontrivial if there exists a nontrivial test function u_R₀(x) such that ˜Φ[u_R₀] ≤ 0, where R₀is determined later. For R ≥ 1, define u_R= u_R(x) by

u_R(x) =







β₀, if x ∈ B_R−1 β₀(R − |x|), if x ∈ C_R 0, if x ∈ Ω − B_R

(3.2.4)

where β₀ = ^2(1+β)₃ , B_R is a ball with radius R and C_R = B_R − B_R−1. Then, by Lemma 3.2.1,

Φ˜₀[u_R] = 1 2

Z

Ω

|∇u_R|²+ Z

Ω

F (u_R) + 1 2

Z

Ω

u_RB₀[u_R]

≤ β₀²

2 |C_R| + F (β₀)|B_R−1| + F (β)|C_R| + β₀²

2(λ_1,0(Ω) + γ)|B_R|

= k(β)|C_R| + [F (β₀) + β₀²

2(λ_1,0(Ω) + γ)]|B_R|,

where k(β) = ^β₂⁰² − F (β₀) + F (β). To choose R₀ which depends only on β and N , we assume _2(λ ^β⁰²

1,0(Ω)+γ) ≤ −^{F (β}₂⁰⁾ or ≤ ^λ_2γ^1,0^(Ω)

1−γ . Here, we already use the assumption γ ≤ γ1 =

9

2β²−5β+2 = _{−2F (β}^β⁰²

0). By the Rayleigh-Faber-Krahn inequality, that is, λ_1,0(Ω) ≥ λ_1,0(B_ρ), it follows that λ1,0(Ω) ≥ ^λ^1,0_ρ^(B2 ¹⁾, where |Bρ| = |Ω|. Therefore, we choose

₀ = λ_1,0(B₁)|B₁|^2/N

(2γ₁− γ)|Ω|^2/N. (3.2.5)

It folloews from the fact |C_R| ≤ |B₁|2^{N −1}R^{N −1}, for all 0 < ≤ ₀ that Φ˜₀[u_R] ≤ |B₁|R^{N −1}{k(β)2^{N −1}+F (β₀)

2 R}

Taking

R₀ = k(β)2^N

−F (β₀),

The proof of the existence of a minimizer of ˜Φ₀ is completed.

For the Neumann condition, 1

2 Z

Ω

u_RB₁[u_R] = 1 2

Z

Ω

u_R(B₁[u_R] − R

Ωu_R γ|Ω| ) + (R

Ωu_R)² 2γ|Ω|

≤ β₀²

2(λ1,1(Ω) + γ)|B_R| + β₀²|B_R|² 2γ|Ω|

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Similarly, ˜Φ₁[u_R₀] ≤ 0 if we assume that β₀²

2(λ_1,1(Ω) + γ) +β₀²|BR0|

2γ|Ω| ≤ −F (β0) 2 . A sufficient condition of the above inequality is

β₀²

2(λ_1,1(Ω) + γ) ≤ −F (β₀)

4 and β₀²|B_R₀|

2γ|Ω| ≤ −F (β₀)

4 (3.2.6)

Therefore, we can choose

₁ ≤ λ_1,1(Ω)

4γ₁− γ and k₁ = |B_R₀| 4γ₁γ.

We also need to exclude the probability of constant solutions for Neumann problem.

Let (u, v) = (p, q) solve u − γv = 0 and f (u) − v = 0. Then ˜Φ(p) = R

Ω[F (p) + _2γ^p²] = R

Ω p²

4 [(p−β₀)²+_γ²−_γ²

0] ≥ 0, which shows that a minimizer of ˜Φ₁is nonconstant. Employing the mountain pass theorem, the other solution can be obtained. Moreover, those solutions are C²-functions. The detail of the existence of the minimizer and the mountain pass solution can be found in [20].

Proof of corollary 3.1.2. Let the domain Ω₁ = [0, L₁]. Then λ_1,1(Ω) = ^π_L²2

1. To chose

independent of Ω₁, we estimate L₁ by using (3.2.6). Then _4γ^2R⁰

1γ ≤ L₁ ≤ √ ^π

(4γ1−γ). On the other hand, if Ω₁ contains a ball B_R₀, then L₁ > 2R₀. To ensure the existence of L₁, we let √ ^π

(4γ1−γ) > max{2R₀,_4γ^2R⁰

1γ} := M or < _M2(4γ^π²1−γ) := ₂. Then for all 0 < < ₂, choose L₁ such that M < L₁ ≤ √ ^π

(4γ1−γ) Then the Neumann problem is solvable.

By even reflections with respect to the boundary of Ω, the domain of this solution can be extended to a larger one. Continuing in this manner, we obtain a solution in the whole domain R¹. Since L1 can be arbitrarily chosen in a interval, the system (3.1.1)-(3.1.2) has infinitely many solutions.

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Chapter 4 Monostable-type solutions in R ^N

This chapter is concerned with monostable-type travelling wave solutions of (DFHN) in R^N for the two components u and v. By solving v in terms of u, this system can be reduced to a non-local 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 [29]. Moreover, we propose a condition for γ, which is similar to the condition Reinecke and Sweers [38]

used to transform (DFHN) into a quasimonotone system.

4.1 Introduction

In the present work, we are concerned with (DFHN) in R^N i.e.,

u_t = u_ξξ+ ∆_yu + f (y, u) − v, (4.1.1) v_t = dv_ξξ+ ∆_yv + δ(u − γv), (4.1.2) 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 < ¹₂. Throughout the chapter 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).

The solutions of interest here are traveling wave solutions. Let x = ξ − ct, then travelling wave solutions of (4.1.1)-(4.1.2) satisfy

u_xx+ cu_x+ ∆_yu + f (y, u) − v = 0, (4.1.3) dv_xx+ cv_x+ ∆_yv + δ(u − γv) = 0. (4.1.4) Over the past decades, this system has been extensively studied. For instance, as N = 1, under different assumptions, system (4.1.3)-(4.1.4) admits standing pulses in [6], [7] and [22], infinitely many periodic solutions in [22], fronts, back waves in [17] and [21]

and travelling pulses in [21]. For the higher dimension case N ≥ 2, symmetric standing waves were established by Reinecke and Sweers [38] and Wei and Winter [44].

As γ → ∞, if the solutions are assumed to be bounded, the equations (4.1.3)-(4.1.4) tend to the single equation

u_xx+ cu_x+ ∆_yu + f (y, u) = 0. (4.1.5) 13

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Let f (y, u) be a C² function g(u) which has the property that for some θ ∈ (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, (4.1.5) admits other types of solutions, including travelling curved fronts (N = 2), conical shapes and pyramidal shapes (N ≥ 3) in [14], [23], [33]

and [41]. Moreover, Hamel and Roquejoffre [15] established travelling wave solutions of (4.1.5) 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 (4.1.5) in R^N connecting a unstable one-peak solution at x → ∞ (−∞) and a stable constant solution x → −∞ (∞) were obtained by Morita and Ninomiya [29].

In this paper, we use the method of super- and subsolutions developed in [29]. Due to technical restriction, we assume d = 1. Since equation (4.1.4) is linear, v can be solved formally in terms of u. With v expressed in terms of u, system (4.1.3)-(4.1.4) is reduced to the non-local equation

F [u] := u_xx+ cu_x+ ∆_yu + f (y, u) − B_c[u] = 0, (4.1.6) where we denote v by B_c[u] := δ(−_∂x^∂²2 − c_∂x^∂ − ∆_y+ δγ)⁻¹u. It is readily seen that if u is independent of x, then by the uniqueness theorem B_c[u] = δ(−∆_y+δγ)⁻¹u. As x → ±∞, the asymptotic behaviors of travelling wave solutions of (4.1.6) formally satisfy

∆_yu + f (y, u) − B_c[u] = 0, (4.1.7) where B_c[u] = 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 (4.1.7) as x → −∞ (∞) and a unstable one as x → ∞ (−∞). Without loss of generality, we may assume that u(+∞, y) is an unsatble solution. Throughout this paper, the following hypotheses are assumed.

(H1) There are two solutions u_±(y) of (4.1.7) satisfying u₋(y) ≥ u₊(y). Moreover, there exist an eigenvalue µ > 0 and its corresponding eigenfunction φ(y) > 0 with max_{y∈R^{N −1}_}φ(y) = 1 and lim|y|→∞φ(y) = 0 such that

∆yφ + fu(y, u+)φ − Bc[φ] = µφ. (4.1.8) (H2) u−(y) ≥ u₊(y) + φ(y) for some > 0.

(H3) There exists no other solution u(y) of (4.1.7) with the property u−(y) ≥ u(y) ≥ u+(y).

(H4) For all small η > 0, there exist solutions u^η₊(y) satisfying lim_η→0u^η₊(y) = u₊(y),

∆_yu^η₊+ f (y, u^η₊) − B_c[u^η₊] + η = 0 (4.1.9) and

u^η₊(y) ≥ u₊(y) + η

M (4.1.10)

for some constant M > 0.

(H5)

∆_yψ_i− (K₁+√

δ)ψ_i ≤ 0, i = 1, 2, 3, (4.1.11) 14

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where

K₁ = − min

{u−(y)≥u≥u+(y),y∈R^{N −1}}f_u(y, u) > 0, (4.1.12) 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 . Set

K^∗ := max

{u−(y)≥u≥u+(y),y∈R^{N −1}}f_u(y, u) > 0 (4.1.13) and let K₂ > 0 satisfy K₂+_δγ+K^δ

2 = K^∗. We state the main theorem as follows.

T

HEOREM

4.1.1.

Assume γ ≥ ^√²

δ + ^K¹_δ^+µ and (H1)-(H5) hold. Then there exists c^∗ = max{2√

µ, 2√

K₂} > 0 such that for all c ≥ c^∗, system (4.1.3)-(4.1.4) admits a pair of smooth solutions (u^∗, v^∗) which satisfies 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. In (H1), when the inequality u−(y) ≥ u+(y) is reversed,i.e., u−(y) ≤ u+(y), a result similar to Theorem 4.1.1 can be proved except that the inequalities u^∗_x ≤ 0 and v_x^∗ ≤ 0 in Theorem 4.1.1 need to be replaced by u^∗_x ≥ 0 and v^∗_x≥ 0 respectively.

Remark 2. In fact, (H5) can be weakened to the following assumption.

∆_yψ_i− M_iψ_i ≤ 0, for some constants M_i > 0. (4.1.14) This condition holds if ∆_yψ_i does not decay faster than ψ_i as |y| → ∞. In this case, if we choose γ ≥ ^√¹

δ+^K³_δ^+µ, 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 [38] and [44] has not yet been studied. However, we believe that for γ 1 the structure of system (4.1.3)-(4.1.4) is similar to that of equation (4.1.6). Accordingly, we extend the result of theorem 2.1 in [29] to the one in Theorem 4.1.1.

4.2 Proof of the main theorem

To prove the Theorem 4.1.1, we use the super- and subsolutions constructed in [29]. By considering the following equation, we construct subsolutions of F [u]. Let w(x) satisfy

w_xx+ cw_x+ µw − w² = 0, (4.2.1)

w(−∞) = µ, w(∞) = 0. (4.2.2)

For all c ≥ 2√

µ, the above boundary value problem admits an unique solution w(x) (up to a translation) which is strictly increasing in x. Subsolutions of F [u] are established as follows.

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L

EMMA

4.2.1.

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+ ∆_yV − δγV = −w(µ − w)B_c[φ] ≤ 0. (4.2.3) Using the maximum principle, we obtain V ≥ 0. Therefore by (H1)

F [U ]

= σφ(w_xx+ cw_x) + (∆_yu₊− B_c[u₊]) + σw∆_yφ + f (y, u₊+ σφw) − σB_c[φw]

= σφ(w_xx+ cw_x+ µw) + f (y, u₊+ σφw) − f (y, u₊) − f_u(y, u₊)σφw + σV

≥ σφw²+ G,

where G = f (y, u₊+ σφw) − f (y, u₊) − f_u(y, u₊)σφw.

Let M1 = min{u−(y)≥u≥u+(y),y∈R^{N −1}}fuu(y, u). By choosing σ ≤ _µ and using (H2), we obtain u₊≤ u₊+ σφw ≤ u₊+ φ ≤ 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 = _µ as M1 ≥ 0 and σ1 = min{_µ,_M⁻¹

1} as M1 < 0. The proof is completed.

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

L

EMMA

4.2.2.

Let Q(x) = e⁻

c−

√

c2−4K2

2 x and U⁺(x, y) = u^η₊(y) + Q(x), where K₂ > 0 satisfies K₂+_δγ+K^δ

2 = K^∗ and c ≥ 2√

K₂. 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(−_∂x^∂²2 − c_∂x^∂ + δγ)⁻¹Q and

B_c[Q] = δ pc²+ 4γδ

Z +∞

−∞

e⁻

√

c2+4γδ

2 |x−ξ|+^c₂(ξ−x)Q(ξ)dξ = δ

δγ + K₂Q(x).

It follows from (H4) that

F [U⁺] = (Q_xx+ cQ_x) + (∆_yu^η₊− B_c[u^η₊]) + f (y, u^η₊+ Q) − B_c[Q]

= −K₂Q + f (y, u^η₊+ Q) − f (y, u^η₊) − η − B_c[Q]

= {−K₂+ f_u(y, u^η₊+ θQ) − δ δγ + K2

}Q − η ≤ −η < 0, where 0 ≤ θ ≤ 1. The last second inequality is due to

K₂+ δ

δγ + K₂ = max

{u−(y)≥u≥u+(y),y∈R^{N −1}}f_u(y, u).

We complete the proof of the lemma.

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Let

L[u] = u_xx+ cu_x+ ∆_yu − (K₁+ µ +√

δ)u, (4.2.4)

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 (4.1.7), we use the following iteration process:

u_n(x, y) = L⁻¹(−f (u_n−1) + B_c[u_n−1] − (K₁+ µ +√

δ)u_n−1), n = 1, 2, · · · ,

u₀(x, y) = U . (4.2.5)

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.

L

EMMA

4.2.3.

δ+^K¹_δ^+µ 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 we have

U ≥ U , L[U⁺− U ] ≤ 0 and L[u−− U ] ≤ 0. (4.2.6) Proof. For the case U = u−(y) we take σ ≤ _µ, then

U − U = u−(y) − u+(y) − σφ(y)w(x) ≥ u−(y) − u+(y) − φ(y) ≥ 0. (4.2.7) The last inequality holds by (H2). On the other hand,

L[u₋− U ] = ∆_y(u₋− u₊) − (K₁+ µ +√

δ)(u₋− u₊) + A, (4.2.8) where A = −σφ(w_xx+cw_x)+(K₁+µ+√

δ)σφw −σw∆_yφ. According to (H5), |A| ≤ σCφ for some positive constant C = C(µ, δ, K1). By choosing σ ≤ ^µ_C, we obtain

L[u−− U ] ≤ ∆y(u−− u+) − (K1+

√

δ)(u−− u+) − µ(u−− u+) + σCφ (4.2.9)

≤ −µφ + σCφ ≤ 0, (4.2.10)

which holds due to assumptions (H2) and (H5).

For the case U = u^η₊(y) + Q(x), given η > 0 we choose σ ≤ _µM^η and use assumption (H4), then

U − U = u^η₊(y) + Q(x) − u₊(y) − σφ(y)w(x) ≥ η

M − σµ ≥ 0. (4.2.11) Moreover,

L[U⁺− U ] = ∆_y(u^η₊− u₊) − (K₁+ µ +√

δ)(u^η₊− u₊) + A + Q_xx+ Q_x

− (K₁+ µ +√ δ)Q.

It is readily seen that Q_xx+ Q_x− (K₁+ µ +√

δ)Q ≤ 0. By (H4) and (H5), L[U⁺− U ] ≤ −ηµ

M + σC ≤ 0 if σ ≤ ηµ M C. Setting σ2 = min{_µ,^µ_C,_µM^η ,_{M C}^ηµ }, the lemma holds.

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To generalize the result of Theorem 2.1 in [29], the nonlocal term of (4.1.6) needs to be better estimated. More precisely, we pointwisely control B_c[u] by the local term u such that the iterative sequence u_n is comparable with u_n−1.

L

EMMA

4.2.4.

Let u ∈ C²(R^N) be nonnegative and solve u_xx+ cu_x+ ∆_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+ ∆_yU − ab + δ

b U ≤ −(δγ − a −δ

b)v ≤ 0. (4.2.12) 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 un is increasing.

L

EMMA

4.2.5.

δ+ ^K¹_δ^+µ and c ≥ c^∗ = max{2√ µ, 2√

K₂}, then for all η > 0 and 0 < σ ≤ min{σ₁, σ₂} we have u_n,x ≤ 0 and

u₀ ≤ u₁ ≤ . . . ≤ u_n≤ . . . ≤ U . (4.2.13) Proof. We first claim that u_n ≤ U for all n. Indeed, by Lemma 4.2.3 and Lemma 4.2.4 (take a = K1+ µ +√

δ and b =√

δ) we obtain

√

δ(U⁺− u₀) − B_c[U⁺− u₀] ≥ 0. (4.2.14) Therefore Lemma 4.2.2 and Lemma 4.2.3 yield

L[U⁺− u₁] ≤ −f (U⁺) + B_c[U⁺] + f (u₀) − B_c[u₀] − (K₁+ µ +√

δ)(U⁺− u₀)

≤ {−f_u(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 that L[u₁ − u₀] = −F [U ] ≤ 0 due to Lemma 4.2.1. By the maximum principle, u1− u0 ≥ 0. Applying Lemma 4.2.4 to u1− u0, we have

√

δ(u₁− u₀) − B_c[u₁− u₀] ≥ 0. (4.2.15) Therefore

L[u₂− u₁] = −(f (u₁) − f (u₀)) + B_c[u₁− u₀] − (K₁+ µ +√

δ)(u₁− u₀)

≤ {−f_u(θu₁+ (1 − θ)u₀− K₁}(u₁− u₀) −√

δ(u₁− u₀) + B_c[u₁− u₀]

≤ 0,

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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] = σφ(µw_x− 2ww_x) − σw_x∆_yφ + (K₁+ µ +√

δ)σφw_x (4.2.16)

= −σw_x{∆_yφ − (K₁+√

δ)φ + (−2µ + 2w)φ} ≤ 0. (4.2.17) Using Lemma 4.2.4 again, we have

√

δ(−u_0,x) − B_c[−u_0,x] ≥ 0 (4.2.18) and

L[u_1,x] = −f_u(u₀)u_0,x+ B_c[u_0,x] − (K₁+ µ +√

δ)u_0,x ≥ 0. (4.2.19) Then u1,x ≤ 0 by the maximum principle. Inducting in n, we obtain un,x ≤ 0.

Proof of Theorem 4.1.1. By Lemma 4.2.5, we define u^∗(x, y) = limn→∞un(x, y).

Following the proof of theorem 2.1 in [29] , (H2) and (H3), for all c ≥ c^∗ we obtain that u^∗(x, y) is a smooth solution of (4.1.6), u^∗_x ≤ 0 and u^∗(±∞, y) = u±(y). Let v^∗ = B_c[u^∗], then v_x^∗ = Bc[u^∗_x] ≤ 0 by the maximum principle. We complete the proof of the theorem.

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20

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Chapter 5 Travelling waves in a cylinder for bistable cases

5.1 Introduction

In this chapter, we are concerned with (DFHN) with Dirichlet boundary condition in a cylinder Ω

u_t= u_ξξ+ ∆_yu + f (u) − v, (5.1.1) v_t= d(v_ξξ+ ∆_yv) + δ(u − γv), (5.1.2)

u|∂Ω= v|∂Ω= 0, (5.1.3)

where (ξ, y) ∈ Ω := R¹ × Ω_y with Ω_y being a bounded C^2,α⁰ domain in R^{N −1}, d ≥ 0, δ, γ > 0, and f (u) = u(1 − u)(u − β) for 0 < β < ¹₂. We also consider this system with Neumann boundary condition in Section 5.7.

As γ → ∞, if the solutions are assumed to be bounded, the equations (5.1.1)-(5.1.3) tend to the single equation

u_t = u_ξξ+ ∆_yu + f (u), (5.1.4)

u|_∂Ω= 0, (5.1.5)

which is a gradient system. For N = 2, the existence of travelling waves of (5.1.4) with boundary condition (5.1.5) were obtained by Gardner [9] when Ω_y = [0, L] and L is large.

His result indicates that large Ω_y seems to be necessary for the existence of a travelling wave with the Dirichlet boundary condition. For higher dimension cases, existence results of travelling waves of (5.1.4)-(5.1.5) were obtained by Volpert, A. and Volpert, V [43], Heinze [11], and Lucia, Muratov and Novaga [25].

In this chapter, we are interested in using a variational approach to study the travelling front solution of (DFHN) and also interested in the higher dimension case N > 1. Let’s first consider the case of a gradient system. For a gradient system, when the wave speed is zero, it is natural to consider the solution as a critical point of the corresponding energy of the system. However when the speed is not zero, how to use the variational method becomes a very subtle problem. Let c denote the wave speed. Assume c > 0. Heinze [11]

made the change of variable x = c(ξ − ct) and considered a minimization problem of a 21

在N 維空間中的費滋漢那古默系統之行波解

Department of Mathematics College of Science

National Taiwan University Doctoral Dissertation

Acknowledgements

Abstract

Contents

Chapter 1 Introduction

Chapter 2

Literature review

2.1 Steady states on bounded domains

2.1.1 Bistable cases

T

2.1.1.

T

2.1.2.

T

2.1.3.

T

2.1.4.

2.1.2 Monostable cases

T

2.1.5.

2.2 Travelling waves in R

2.2.1 Bistable cases

T

2.2.1.

T

2.2.2.

2.2.2 Monostable cases

T

2.2.3.

2.3 Standing waves in R

T

2.3.1.

T

2.3.2.

Chapter 3

Steady states on bounded domains and periodic solutions in R N

3.1 Introduction

T

3.1.1.

C

3.1.2.

3.2 Proof of the main theorem

L

3.2.1.

Chapter 4

Monostable-type solutions in R N

4.1 Introduction

T

4.1.1.

4.2 Proof of the main theorem

L

4.2.1.

L

4.2.2.

L

4.2.3.

L

4.2.4.

L

4.2.5.

Chapter 5

Travelling waves in a cylinder for bistable cases

5.1 Introduction

Steady states on bounded domains and periodic solutions in R ^N

Monostable-type solutions in R ^N