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

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 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 ^

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 eigen-function with kw_n,ik₂ = 1 for i = 0, 1. Then

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

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

For the Neumann condition,

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.

12

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

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 trav-elling 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

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.

15

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.

16

Let

L[u] = u_xx+ cu_x+ ∆_yu − (K₁+ µ +√

δ)u, (4.2.4)

where K₁ = − min_{{u−(y)≥u≥u+(y),y∈R}N −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.

Assume γ ≥ ^√2

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.

17

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.

Assume γ ≥ ^√2

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,

18

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.

19

20

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

weighted energy with a constraint. According to his ingenious setting, a minimizer of the problem corresponds to a travelling front solution while the Lagrange multiplier of the constraint corresponds to the wave speed. We explain this more precisely by considering (5.1.4)-(5.1.5) as follows. Using the new variable x = c(ξ − ct), a travelling wave solution of the single equation (5.1.4) satisfies

c²(u_xx+ u_x) + ∆_yu + f (u) = 0. (5.1.6) Heinze considered the weighted energy, acting on the Sobolev space H, which is the space W^1,2 with the weight e^xdx (See section 5.2.) φ ∈ H. According to this, Heinze viewed ˆu as a minimizer of I0[u] under the constraint A_k = {u ∈ H|J₀[u] = k} and _c¹2 as the corresponding Lagrange multiplier. Also it is easy to verify that the corresponding Euler-Lagrange equation is (5.1.6). On the other hand, multiplying (5.1.6) by uxe^x and taking integration one obtains c²I0[u] + J0[u] = 0.

This implies J₀[u] < 0 if u is a nontrivial solution of (5.1.6). By a suitable translation x₀, J₀[u(x − x₀, y)] = −1. Therefore Heinze chose k = −1 and solved the minimizing problem

Ainf−1

I0[u]. (5.1.7)

Moreover, he proved that A−1 6= ∅ is equivalent to inf

This condition also guarantees that the minimizer is nontrivial.

Later in a series of papers [24]-[31], Lucia, Muratov and Novaga further developed the variational approach and proved existence results via subtle ideas and techniques. In their approach (see [25]), the ξ variable is not scaled and the energy

Sˆ_c[u] := ˆI₀[u] + ˆJ₀[u] := 1

is considered on the Sobolev space with weight e^czdz, where z = ξ − ct. In [25], they assumed that there is a c^∗ > 0 such that ˆS_c^∗[u] ≤ 0 for a nontrivial u, which is equivalent to Heinze’s condition (5.1.8) (see proposition 6.2 in [25]). Then they minimized the energy ˆS_c∗[u] under the constraint ˆI₀[u] = 1. In their proof, to show the lower semi-continuity of ˆS_c[u] is one of the most crucial steps for the existence of inf{I₀[u]=1}Sˆ_c[u].

They proved a minimizer u_c∗ of ˆS_c^∗[u] under the constraint ˆI₀[u] = 1 can be achieved.

By the scaling property of the equations, the travelling wave solution can be obtained as u(x, y) = uc^∗(xp1 − Sc^∗[uc^∗], y) and the travelling wave speed equals c = c^∗p1 − Sc^∗[uc^∗].

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Lucia, Muratov and Novaga also gave a new criterion for the so called linear and nonlinear selection of mono-stable travelling waves as an application of their methods (see [24]).

For a gradient system, one disadvantage in applying these variational approaches comes from that one needs to assume the diffusion coefficients of all components are equal when the wave speed c 6= 0. On the other hand, the variational approaches have some advantages also. Besides they themselves provide very interesting and different viewpoints of the problem, these methods are more easily generalized to higher dimension cases, e. g. waves on cylindrical domains, and usually require only mild assumptions for the existence of a solution.

Although (DFHN) is not a gradient system, by replacing v by√

δv, this system has the skew-gradient structure defined by Yanagida [45]. More precisely, under this replacement, (DFHN) becomes

u_t= ∆u + f (u) −√

δv, (5.1.9)

v_t = d∆v +√

δv − γδv, (5.1.10)

of which the steady states correspond to the critical points of the energy S[u, v] = 1

2 Z

Ω

|∇u|² − d|∇v|²+ H(u, v)
, (5.1.11)

where H(u, v) = F (u) +√

δuv −¹₂γδv². Restricted to the u direction, (5.1.9) is the gradi-ent flow of the energy (5.1.11) while restricted in the v direction, (5.1.10) is the gradigradi-ent flow of the minus of (5.1.11). Along the orbits of (5.1.9) and (5.1.10), the ”u-part” of the energy (5.1.11) decreases and the ”v-part” of the energy increases. Yanagida [45] called a system of reaction-diffusion equations with an energy like this a skew-gradient system. He developed a theory for a skew-gradient system and found that the correct notion in such a system corresponding to a minimizer in a gradient system should be a mini-maximizer.

With this structure in mind, the authors feel curious about the following problem:

Question 1 Can variational methods be applied to find wave front or pulse solutions of a skew-gradient system? For example, applied to (DFHN)?

The setting of Heinze [11] and the setting of Lucia, Muratov and Novaga [24]-[31] in applying variational methods mentioned above are slightly different. The former uses the change of variables x = c(ξ − ct) to scale out the factor c in the weight of the energy while the latter does not make any scaling in ξ. Although these two settings are almost parallel to each other, they posses different advantages and weakness in some subtle situations. We will comment on this later. Following their approaches, we assume that d = 1 and consider the following problem. Let us first assume c > 0 and make the change of variables x = c(ξ − ct) as in [10]. Then a travelling wave solution of (5.1.9) and (5.1.10) satisfies

c²(u_xx+ u_x) + ∆_yu + f (u) −√

δv = 0, (5.1.12)

c²(v_xx+ v_x) + ∆_yv +√

δu − γδv = 0. (5.1.13)

u|_∂Ω = v|_∂Ω = 0 (5.1.14)

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To answer Question 1, we consider the weighted energy Then as in the case of a gradient system, the first problem is to determine the speed c. In general, this is a difficult problem. Assuming that we know the value of c, it is easy to check that a suitable critical point of this energy is a solution of (5.1.12)-(5.1.14).

Since (DFHN) is a skew-gradient system, it is expected that a mini-max approach should be proper to solve the problem. Unfortunately compared to a minimization problem, there are much less methods to solve a mini-maximization problem. Recently Chen-Hu [5] succeeded in applying a mini-max approach to solve (5.1.12)-(5.1.13) with c = 0 on a bounded domain in R^N. In their study, the Sobolev space on which the energy is defined is decomposed into a ”positive” space and a ”negative” space, which both have infinite dimensions. It is a very interesting problem whether one can generalize their method to the case c 6= 0.

To further explore the existence problem, we can also consider another variational setting for the steady state of (DFHN), which has been used in many literatures. That is, to solve the equation for v first and substitute it in the u’s equation. Then we obtain an equation with only one unknown function u and a non-local term. For our problem c 6= 0, we can solve (5.1.13) under the boundary condition (5.1.14) first and denote the solution

Substituting this into (5.1.12), we obtain the non-local equation for u c²(u_xx+ u_x) + ∆_yu + f (u) −√

δB_c[u] = 0. (5.1.17) For this equation, we consider the weighted energy

Φ^∗_c[u] = 1 Fortunately the bilinear form of this non-local term in this energy is symmetric even c 6= 0. Therefore one can readily check that once the speed c is known, a suitable critical point of (5.1.18) corresponds to a solution of (5.1.12)-(5.1.14). This consideration leads to the question.

Question 2 Can variational methods be applied to find wave front or pulse solutions of a system with a non-local term? For example, applied to the non-local formulation (5.1.17).

In this chapter, the authors are concerned with finding travelling front solutions of (DFHN). The major part will focus on Question 2 for (DFHN) and rely on the non-local energy (5.1.18) to solve (5.1.12)-(5.1.14). As for Question 1, it seems more complicated to apply the energy Ψc[u, v] to obtain a travelling wave solution. We have only partial

24

answer for it. In section 5.6, after converting the Lagrange multiplier formulation of Heinze for the wave speed into a quotient form of energy, we use the energy Ψ_c[u, v] to describe a mini-max formulation for the wave speed and show that a mini-maximizer, as in Yanagida’s theory, of the speed functional is a solution of (5.1.12)-(5.1.14). However we do not know how to find a mini-maximizer in general when c 6= 0.

To study this problem, we refer to some papers written by Lucia, Muratov and Novaga [24]-[31]. They obtained many good viewpoint for gradient systems, with a local energy.

The first two term of the energy Φ^∗_c[u] is the same as their energy. Therefore the process of this paper is a little similar to [25]. However, we discuss our energy containing a nonlocal term, which causes that the boundedness of solution is not easy. Moreover, combining the advantages of [11] and [25], we obtain the travelling wave speed according to the minimal energy in subsection 5.3.2 In [25], travelling wave solutions are obtained by scaling a minimizer with negative energy. However, we show the existence of travelling wave by choosing a ”good” minimizing sequence to approximate it. If some lemmas are similar to [25], we skip the proof.

For the existence of travelling waves to (DFHN), we need to assume that Ωy is large enough. Our main theorem is as follows.

T

HEOREM

5.1.1.

Assume γ > _2β2−5β+2⁹ . Then there exists R₀ > 0 such that (5.1.12)-(5.1.14) has a solution (u₀, v₀) with c = c₀ > 0 for some c₀ if Ω contains a ball with radius R₀. Moreover, (u₀, v₀) decays exponentially to 0 uniformly in y as x → +∞.

To understand the behavior of u₀, v₀ as x → −∞, we need to investigate the equations (5.1.12)-(5.1.14) without x-coordinate, i.e., Assume γ > _2β2−5β+2⁹ , Klaasen and Mitidieri [20] showed that E[u] has at least two critical points if the domain Ω_y contains a ”large” ball. One is a minimizer with negative energy and the other one derived from the Mountain Pass theorem has positive energy.

Due to the nonlocal term of (5.1.17), the asymptotic behavior of the solution (u₀, v₀) obtained in Theorem 5.1.1 as x → −∞ is much more complicated than the behavior of a gradient system. It seems (u₀, v₀) may not tend to a steady state satisfying (5.1.19)-(5.1.21) as x → −∞ in general. However, when γ²δ > 1, we can obtain the L²-estimate for u_0,x (see Lemma 5.5.3). Using this estimate and following the ideas of Proposition 6.6 and Corollary 6.8 in [25], we obtain the following two theorems.

T

HEOREM

5.1.2.

Assume γ²δ > 1 and the assumptions in Theorem 5.1.1 hold. Let (u0, v0) be the solution in Theorem 5.1.1. Then there exists a sequence xn → −∞ such that lim_n→+∞(u₀, v₀)(x_n, y) exists and solves (5.1.19)-(5.1.21). Moreover, if all critical points of E[u] with negative energy are discrete in H₀¹(Ω_y), then the above limit is a full limit, that is, limx→−∞(u0, v0)(x, y) exists and E[u0(−∞, y)] < 0.

25

T

HEOREM

5.1.3.

Let N = 1 and γ²δ > 1. Then γ > _2β2−5β+2⁹ if and only if there exist c₀ > 0 and a pair of classical solutions (u₀, v₀) for (5.1.12)-(5.1.13) which satisfies u0 ∈ H, u0,x∈ L²(Ω) (u0, v0)(+∞) = (0, 0) and (u0, v0)(−∞) = (p2, q2) (see figure 2).

This chapter is organized as follows. In Section 5.2, we recall some Poincar´e-type inequalities of weighted Sobolev spaces and investigate the properties of the non-local operator B_c[u]. Also the maximum principle on an unbounded cylinder is proved for equation (5.1.13). In Section 5.3 we define an energy functional associated with equations (5.1.12)-(5.1.13) and show the boundedness and low semicontinuity of the energy. Next the continuity of the minimal energy and the estimate of the travelling speed are obtained.

This chapter is organized as follows. In Section 5.2, we recall some Poincar´e-type inequalities of weighted Sobolev spaces and investigate the properties of the non-local operator B_c[u]. Also the maximum principle on an unbounded cylinder is proved for equation (5.1.13). In Section 5.3 we define an energy functional associated with equations (5.1.12)-(5.1.13) and show the boundedness and low semicontinuity of the energy. Next the continuity of the minimal energy and the estimate of the travelling speed are obtained.

在文檔中 在N 維空間中的費滋漢那古默系統之行波解 (頁 18-34)