A PARABOLIC EQUATION OF KPP TYPE IN HIGHER DIMENSIONS

Jean-François MALLORDY D.M.I., E.N.S 45, rue d’Ulm 75230 PARIS CEDEX 05 FRANCE

Jean-Michel ROQUEJOFFRE C.N.R.S., Ecole Polytechnique Centre de Mathématiques Appliquées 91128, PALAISEAU CEDEX FRANCE

April 1992

Abstract

The aim of this paper is to study the long-time behaviour of the solutions of a semilinear parabolic equation in an infinite cylinder, which generalises a well-known one dimensional model that was first investigated by Kolmogorov, Petrovskii and Piskunov.

We show that the asymptotic behaviour of the solutions strongly depend on the asymptotic behaviour of the initial datum at the ends of the cylinder. In particular, the equation admits a continuum of travelling wave solutions, each of them being stable with regards to adequate perturbations.

1 Introduction

This paper is concerned with the large time behaviour of the solutions of a class of semilinear parabolic equations in an infinite multidimensional cylinder. Namely, denoting by Σ the cylinder Σ = (x, y) ∈ IR × ω, ω being a bounded regular domain in IR^{N−1}, we consider the following problem

(P) : \{ \begin{matrix}∂_{t}u − Δu + α(y)∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(t = 0) = u_{0} in Σ \end{matrix}

Here ν denotes the outward unit normal on ∂Σ, α(y) is a given continuous function on ω̄, and g is a given source term satisfying g(0) = g(1) = 0 with g > 0 in (0, 1), and g′(0) > 0 > g′(1). The initial datum u_{0} is assumed to take its values in (0, 1).

Under the above assumptions, (P) has a unique global classical solution, denoted by S(t)u_{0}. These assumptions will always be understood to hold in the rest of the paper.

Our aim is to understand how the asymptotics of u, for large t, depend on the initial datum u_{0}. Indeed, (P) admits travelling wave solutions of speed c ≥ c_{∗}, for some c_{∗} depending on the source term g, and it is expected that each travelling wave is stable in some sense.

Such problems have a wide range of applications, such as population dynamics or combustion; quite a few authors have already contributed to the study of the one-dimensional problem

u_{t} − u_{xx} = g(u), u(−∞, t) = 0, u(+∞, t) = 1

The first paper dealing with the subject is the celebrated article by Kolmogorov, Petrovskii and Piskunov [13]. The source term which is taken into account in their paper is g(u) = u(1 − u), and S(t)H - H being the Heaviside step function - is shown to converge, in some weak sense, towards the travelling front with speed c_{∗}.

Subsequent work was then done by Aronson and Weinberger [2], where, in particular, the set of c’s such that u(x − ct, t) goes to 0 (resp. 1) as t → +∞ is characterised. Local exponential stability results were proved by Sattinger [19] for the waves with speed c > c_{∗} in weighted spaces; quite recently Kirchgässner [12] reached local algebraic stability for the waves with speed c_{∗}. In [10], Hagan investigated fully nonlinear equations of the form u_{t} = f(u, u_{x}, u_{xx}), with f satisfying suitable assumptions.

Finally, let us mention the work of Uchiyama [21], which covers all the above aspects, and which provides a classification of every possible asymptotic behaviour.

In the multidimensional framework, Aronson and Weinberger [3] investigated the unsteady problem in IR^{N}, with α ≡ 0. They obtained ”hair-trigger effect” type reults, i.e a nonnegative initial datum with even a small compact support evolves into a solution that tends to 1 on every compact subset.

Our purpose is to extend to (P) some of the above results. In particular, we will fully generalise [2], [19], and the global stability results of [21] for the waves with speed c > c_{∗}. To do this, we will heavily rely on the results of Berestycki and Nirenberg [8].

The paper is organised as follows. In Section 2, we recall the existence, uniqueness and qualitative properties results for the travelling wave solutions (P). In Sections 3 and 4, we investigate (P) with a source terms g such that g(s)/s is decreasing in s. Such a g will be called a KPP source-term, and we will see in Sections 3 and 4 that this additional assumption allows us to produce shorter and more elegant proofs. Section 5 is devoted to exponential local stability results, and no KPP assumption is made, unless otherwise stated. Finally, Section 6 is devoted to the extension of the results of Section 4 when g is no longer a KPP source term.

2 Travelling waves

All the material in this section comes from the work of Berestycki and Nirenberg [8]. Therefore no proof will be given, apart from the last exponential behaviour result.

When we look for travelling waves solutions of the form u(x, y, t) = ϕ(x+ ct, y), we have to deal with the elliptic problem

(E) : \{ \begin{matrix}−Δϕ + (α(y) + c)∂_{x}ϕ = g(ϕ) in Σ \\ ∂_{ν}ϕ = 0 on ∂Σ \end{matrix}

We set

β(y) = α(y) + c, ∀y ∈ ω̄

A_{c} = −Δ + β(y)∂_{x}

The resolution of Problem (E) requires the study of the linearised problem around 0, which is presented below.

The linearised problem. Here we are interested in finding positive solutions to the problem

(E_{L}) : \{ \begin{matrix}−Δz + β(y)∂_{x}z = g′(0)z in Σ \\ ∂_{ν}z = 0 on ∂Σ \end{matrix}

It is easy to see that z = e^{λx}φ(y), with φ > 0 on ω, is a solution of (E_{L}) if and only if

λ^{2} = μ_{1}(−Δ + λβ(y) − g′(0))

where μ_{1}(B) represents the first eigenvalue of the operator B defined on ω with Neumann condition on the boundary.

Lemma 1 Let m ∈ IR and μ_{1}(t) = μ_{1}(−Δ + tβ(y) + m), then the function t ↦ μ_{1}(t) satisfies:

1. μ_{1} is Lipschitz with coefficient ||β||_{∞}

2. μ_{1}(0) = m

3. μ_{1} is concave

4. μ_{1} is differentiable and μ′_{1}(0) = \frac{1}{|ω|} ∫_{ω} β(y)dy

5. m + t(inf_{ω} β) ≤ μ_{1}(t) ≤ m + t(sup_{ω} β)

We call λ ∈ IR a principal eigenvalue (p.e) if μ_{1}(λ) = λ^{2}. An immediate consequence of the lemma is the following theorem (remember m = −g′(0) < 0):

Theorem 1 There exists c_{0} ∈ IR such that

• c < c_{0} ⟹ there is no positive p.e

• c = c_{0} ⟹ there is exactly one positive p.e • c > c_{0} ⟹ there are exactly two positive p.e.

Notation: we call λ_{c} the smallest positive principal eigenvalue associated to c > c_{0} and λ_{0} the positive principal eigenvalue associated to c_{0}. We will also use the notation λ_{c}^{u} (u for ”usual”) instead of λ_{c}, and then λ_{c}^{a} (a for ”accidental”) will be the other p.e.

Solutions of Problem (E)

Theorem 2 There exists c^{∗} ≥ c_{0} such that the problem

(E) : \{ \begin{matrix}−Δϕ + (α(y) + c)∂_{x}ϕ = g(ϕ) in Σ \\ ∂_{ν}ϕ = 0 on ∂Σ \end{matrix}

with ϕ(−∞, y) = 0, ϕ(+∞, y) = 1 uniformly on ω̄, has a solution if and only if c ≥ c^{∗}. Furthermore the solution is unique (up to the x-translations), and increasing in x, more precisely: ∂_{x}ϕ > 0 on Σ̄.

Notation: we call ϕ_{c} the solution associated to c normalised by the condition: sup_{y∈ω̄} ϕ_{c}(0, y) = 1/2. If we now impose ϕ(−∞, y) = 1, ϕ(+∞, y) = 0

uniformly on ω̄, we will have solutions if and only if c ≤ c^{∗∗}, for a real c^{∗∗} < c^{∗}, this is obvious by changing x into −x. We call ϕ̃_{c} these decreasing solutions. We call λ_{00} the λ_{0} obtained when considering −α(y) instead of α(y) in the linearized problem.

We end this paragraph by the asymptotic behaviour of ϕ_{c} as |x| → +∞.

Lemma 2

• if c > c^{∗} we have, near −∞, and uniformly in y ∈ ω̄,

ϕ_{c}(x, y) ∼ e^{λ_{−}x}φ_{−}(y), ∂_{x}ϕ_{c}(x, y) ∼ λ_{−}e^{λ_{−}x}φ_{−}(y)

furthermore, if c^{∗} > c_{0}, we have the same at c = c^{∗}

• if c ≥ c^{∗}, we have near +∞, and uniformly in y ∈ ω̄,

1 − ϕ_{c}(x, y) ∼ e^{−λ_{+}x}φ_{+}(y), ∂_{x}ϕ_{c}(x, y) ∼ λ_{+}e^{−λ_{+}x}φ_{+}(y) where e^{λ_{−}x}φ_{−}(y) is a positive solution of the linearised problem around 0 (E_{L}) with λ_{−} > 0, and e^{−λ_{+}x}φ_{+}(y) is a positive solution of the linearised problem around 1, with λ_{+} > 0.

In the case c > c^{∗}, we can be more precise.

Lemma 3 Let c > c^{∗}, then the exponential behaviour of ϕ_{c} near −∞ is given by λ_{c}, the smallest principal eigenvalue, ie:

ϕ_{c}(x, y) ∼ e^{λ_{c}x}φ_{c}(y) as x → −∞

where φ_{c} is a positive eigenfunction associated to λ_{c}.

PROOF. By the sliding method, as in [7]. Set ϕ_{c}(x, y) ∼ φ_{−}(y)e^{λ_{−}x} as x → −∞, and assume that λ_{−} > λ_{c}^{a}. Then we have

lim_{|x|→+∞} ϕ_{c^{∗}}/ϕ_{c} = +∞

(this follows from Theorem 4.1 in [8]). Moreover:

−Δ(ϕ_{c^{∗}} − ϕ_{c}) + (c + α(y))∂_{x}(ϕ_{c^{∗}} − ϕ_{c}) + \frac{g(ϕ_{c^{∗}}) − g(ϕ_{c})}{ϕ_{c^{∗}} − ϕ_{c}}(ϕ_{c^{∗}} − ϕ_{c})

= (c − c^{∗})∂_{x}ϕ_{c^{∗}} > 0

We choose a translate of ϕ_{c^{∗}} (still denoted by ϕ_{c^{∗}}), such that ϕ_{c^{∗}} ≥ ϕ_{c}, and such that the two functions coincide somewhere. The maximum principle and the Hopf boundary lemma yield ϕ_{c^{∗}} = ϕ_{c}, a contradiction. □

3 Long-time behaviour with rapidly decreasing initial data

Here we consider the problem

(P) : \{ \begin{matrix}∂_{t}u − Δu + α(y)∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(t = 0) = u_{0} in Σ \end{matrix}

where u_{0} decays faster than any travelling front at one or both ends of the cylinder. We recall that, in this section, g is assumed to be a KPP source term.

3.1 Decay at only one end

Theorem 3

• If u_{0}(x, y) = o(e^{λx}) as x → −∞ for every λ < λ^{∗}, then for each c > c^{∗} we have u(x−ct, y, t) → 0 when t → +∞ uniformly on each (−∞, a]×ω̄

• If lim inf_{x→+∞} u_{0} > 0, then for each c < c^{∗} we have u(x − ct, y, t) → 1 when t → +∞ uniformly on each [a, +∞) × ω̄.

If we now consider a u_{0} satisfying both conditions of Theorem 3, we can obviously define

φ(t) = sup{x ∈ IR; u(x, y, t) ≤ 1/2, ∀y ∈ ω}

for sufficiently large values of t. We then have the

Corollary 1 Under the conditions of Theorem 3, φ(t)/t → −c^{∗} as t → +∞.

We first prove the first part of the theorem, which is much easier. Set v(x, y, t) = u(x − ct, y, t) so that ∂_{t}v + A_{c}v = g(v); assume c > c^{∗}. Let c′ ∈ (c^{∗}, c) and e^{λx}φ(y) a positive solution of the linearised problem ∂_{t}v + A_{c′}v = g′(0)v, with the Neumann boundary condition (such a solution exists since c > c^{∗}). Using the condition at −∞, we know ∃x_{0} such that

x < x_{0} ⟹ u_{0}(x, y) < e^{λx}φ(y)

so, for some large M, u_{0}(x, y) < e^{λ(x+M)}φ(y) on Σ; now just apply the maximum principle to see that v(x, y, t) < e^{λ(x+M−(c′−c^{∗})t)}φ(y) on Σ, ∀t ≥ 0, wich yields the first assertion of theorem 3.

Before proving the second part of the theorem, we need a lemma:

Lemma 4 Let f be a C^{1} function such that f(0) = f(1) = 0, f′(0) > 0 > f′(1), and f > 0 on (0, 1); we suppose u satisfies ∂_{t}u + A_{c}u = f(u) in Σ with Neumann boundary conditions, and lim inf_{t→+∞} u(x, y, t) ≥ δ > 0 uniformly on each [a, +∞) × ω̄, then u(x, y, t) → 1 when t → +∞ uniformly on each [a, +∞) × ω̄.

PROOF. There exists a sequence t_{n} → +∞ such that ∀t ≥ t_{n}, u(t) ≥ h_{n}^{0} where

h_{n}^{0}(x, y) = \{ \begin{matrix}δ/2 if x > −n \\ 0 if x < −n, \end{matrix}

we denote by h_{n}(x, y, t) the solution of

∂_{t}h + A_{c}h = f(h) in Σ
∂_{ν}h = 0 on ∂Σ
h(t = 0) = h_{n}^{0} in Σ

By the maximum principle, we have u(t+t_{n}) ≥ h_{n}(t), ∀t ≥ 0. As n increases, h_{n}^{0} increases, and so h_{n}(t) increases too (another consequence of the maximum principle). Since h_{n}(t) is bounded by 1, we have: h_{n}(x, y, t) → h_{∞}(x, y, t) as n → +∞. Using classical parabolic estimates, we get: h_{n} → h_{∞}, ∂_{t}h_{n} → ∂_{t}h_{∞}, ∂_{x}h_{n} → ∂_{x}h_{∞}, and Δh_{n} → Δh_{∞}, uniformly on each [−a, +a] × ω̄ × [1/T, T]. This shows ∂_{t}h_{∞} + A_{c}h_{∞} = f(h_{∞}) with the Neumann boundary condition. Meanwhile h_{∞}(x, y, t = 0) ≥ δ/2 > 0. Obviously, h_{∞}(x, y, t) → 1 uniformly on Σ as t → +∞.

We conclude that h_{n}(x, y, t) → 1 uniformly on each [−a, +a] × ω̄ as n and t → +∞, but h_{n}(t) being increasing in x (maximum principle), the convergence is in fact uniform on each [−a, +∞) × ω̄. Remembering u(t + t_{n}) ≥ h_{n}(t), ∀t ≥ 0, we have our lemma. □

Now we prove the second assertion in Theorem 3; we first suppose that g is concave. Set g = 0 outside of [0, 1]. For ϵ > 0, consider the Problem ,with unknown (u, c),

(E_{ϵ}) : \{ \begin{matrix}−Δu + (α(y) + c)∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(−∞, y) = −ϵ and u(+∞, y) = 1 \end{matrix}

Setting w = (u + ϵ)/(1 + ϵ), we see that (E_{ϵ}) is equivalent to

(E′_{ϵ}) : \{ \begin{matrix}−Δw + (α(y) + c)∂_{x}w = g_{ϵ}(w) in Σ \\ ∂_{ν}w = 0 on ∂Σ \\ w(−∞, y) = 0 and w(+∞, y) = 1 \end{matrix}

where we have set g_{ϵ}(s) = g((1 + ϵ)s − ϵ)/(1 + ϵ). Using the concavity of g, we get that g_{ϵ} increases as ϵ decreases. Now, we just do as Berestycki and Nirenberg when they prove the existence of c^{∗} (see [8]); this yields: for a given ϵ > 0, there is exactly one solution (u_{ϵ}, c_{ϵ}) to Problem (E_{ϵ}) (defined modulo x-translations); this solution is increasing in x and c_{ϵ} increases to c^{∗} as ϵ decreases to 0.

Let δ > 0 such that lim inf_{x→+∞} > δ; set g_{δ}(s) = δg(s/δ) for s ∈ IR. Still using the concavity of g, we get g_{δ} ≤ g, ∀δ ∈ (0, 1]. Consider the problem

(E_{ϵ,δ}) : \{ \begin{matrix}−Δv + (α(y) + c)∂_{x}v = g_{δ}(v) in Σ \\ ∂_{ν}v = 0 on ∂Σ \\ v(−∞, y) = −ϵ and v(+∞, y) = δ \end{matrix}

Setting v = δu, we see that Problem (E_{ϵ,δ}) is equivalent to problem (E_{ϵ/δ}) (defined above); let us call (v_{ϵ,δ}, c_{ϵ,δ}) the solution of problem (E_{ϵ,δ}); we still have that c_{ϵ,δ} increases to c^{∗} as ϵ decreases to 0.

Let c < c^{∗}. Let ϵ > 0 such that c < c_{ϵ,δ} < c^{∗}; we now consider u the solution of

(Pc) : \{ \begin{matrix}∂_{t}u − Δu + (α(y) + c)∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(t = 0) = u_{0} in Σ \end{matrix}

and we set z(x, y, t) = v_{ϵ,δ}(x + (c_{ϵ,δ} − c)t, y). We have

−Δz + (α(y) + c)∂_{x}z = g_{δ}(z) ≤ g(z) in Σ
∂_{ν}z = 0 on ∂Σ
z(x, y, t = 0) ≤ u_{0}(x, y) in Σ up to some x-translation

which implies z(x, y, t) ≤ u(x, y, t), ∀t ≥ 0, ∀(x, y) ∈ Σ. This yields lim inf_{t→+∞} u(x, y, t) ≥ δ uniformly on each [−a, +∞) × ω̄. An application of Lemma 4 ends the proof, in the case where g is KPP and concave.

The result is still available when g is KPP but not concave: for η > 0, set c^{∗}(g′(0) − η) the critical speed obtained when replacing g′(0) by g′(0) − η. Choose some small η > 0 such that c < c^{∗}(g′(0) − η) < c^{∗}; choose g_{η} a KPP and concave function smaller than g satisfying g′_{η}(0) = g′(0) − η. Now just replace g by g_{η} in (E_{ϵ,δ}): this provides a subsolution with the same properties as z, and completes the proof of the second part of Theorem 3. □

3.2 Initial data with compact support

Theorem 4 We suppose that u_{0} satisfies u_{0}(x, y) = o(e^{λx}) as x → −∞, uniformly in y for each λ < λ^{∗} and u_{0}(x, y) = o(e^{−μx}) as x → +∞, uniformly in y for each μ < λ^{∗∗}, then u satisfies uniformly on every compact of Σ̄

1. u(x − ct, y, t) → 0 when t → +∞ if c ∉ [c^{∗∗}, c^{∗}],

2. u(x − ct, y, t) → 1 when t → +∞ if c ∈ (c^{∗∗}, c^{∗}).

The first assertion follows directly from Theorem 3. As a first step to the second assertion, we prove the following lemma, which was obtained independantly by Lachand-Robert [14] in a different context.

Lemma 5 There exists c′ < c^{∗} such that ∀c ∈ (c′, c^{∗}), ∃λ ∈ C^{|} \IR and φ ∈ C^{2}(ω̄, C^{|} ) with Re(φ) > 0 in ω̄ satisfying

\{ \begin{matrix}−Δφ + (λ(α(y) + c) − λ^{2} − g′(0))φ = 0 in ω \\ ∂_{ν}φ = 0 on ∂ω \end{matrix}

PROOF. Remember that for λ ∈ IR, we denote by μ_{1}(−Δ+λβ_{∗}) the first eigenvalue of the operator −Δ + λ(α(y) + c^{∗}); the eigenspace associated in L^{2}(ω) is one-dimensional. The family of operators −Δ+λ(α(y)+c^{∗}) depends analytically on λ, in the sense of Kato (see [16]). From Kato-Rellich theorem, ∃V a neighbourhood of λ^{∗} in C^{|} , such that there exists a simple eigenvalue μ̃_{1}(−Δ + λβ_{∗}) continuing μ_{1} on all V analytically; there also exists a family of eigenfunctions ψ_{λ} in L^{2}(ω), analytic in λ.

Let us study the zeros of F_{c}(λ) = μ̃_{1}(−Δ + λβ_{∗}) + λ(c − c^{∗}) − λ^{2} − g′(0). F_{c} is analytic and converges locally uniformly to F_{c^{∗}} as c → c^{∗}, λ^{∗} being a zero of F_{c}^{∗}, the Rouché theorem yields the existence, in a neighbourhood of c^{∗}, of a family λ_{c} such that F_{c}(λ_{c}) = 0 and λ_{c} → λ^{∗} as c → c^{∗}. We then have, for c near c^{∗}: μ̃_{1}(−Δ + λ_{c}β_{∗}) + λ_{c}(c − c^{∗}) − λ_{c}^{2} − g′(0) = 0;

setting ψ_{c} = ψ_{λ_{c}}, we get:

\{ \begin{matrix}−Δψ_{c} + (λ_{c}(α(y) + c) − λ_{c}^{2} − g′(0))ψ_{c} = 0 in ω \\ ∂_{ν}ψ_{c} = 0 on ∂ω \end{matrix}

Notice that ψ_{c} → ψ_{c^{∗}} as c → c^{∗}, in L^{2}(ω). Writing the equation satisfied by ψ_{c} − ψ_{c^{∗}}, and using classical elliptic estimates, we get:

||ψ_{c} − ψ_{c^{∗}}||_{L^{2}(ω)} → 0 ⟹ ||ψ_{c} − ψ_{c^{∗}}||_{∞} → 0

as c → c^{∗}. Without loss of generality, we can assume that ψ_{c^{∗}} is real and positive on ω̄ (since the eigenspace associated to λ^{∗} is generated by a real positive function). We have that Re(ψ_{c}) → ψ_{c^{∗}} in L^{∞}(ω), so Re(ψ_{c}) > 0 for c near c^{∗}. Now, we need only prove that for c ∈ IR, with c < c^{∗} near enough, it is impossible to have λ_{c} ∈ IR; if it was, we could write μ_{1}(−Δ + λ_{c}(α(y) + c) − g′(0)) = λ_{c}^{2}; but c < c^{∗} means that this never happens for λ ∈ IR (see the definition of c^{∗}). The lemma is proved with λ = λ_{c} and φ = ψ_{c}. □

Now we prove Theorem 4. Let ϵ_{0} > 0 such that c^{∗∗} < c^{∗} − ϵ_{0}. Let c_{δ} the c^{∗} obtained when replacing g′(0) by g′(0) − δ, for some δ > 0, in the principal eigenvalue problem. It is not difficult to see that c_{δ} increases to c^{∗} as δ decreases to 0. So, for δ small enough, c^{∗} − ϵ_{0} < c_{δ} < c^{∗}. We apply the lemma with g′(0) − δ instead of g′(0) and c_{δ} instead of c^{∗}. This provides, for c < c_{δ} near c_{δ}, (λ, φ) such that φ_{0} :=Re(e^{λx}φ(y)) is a real-valued solution of

\{ \begin{matrix}−Δφ_{0} + (α(y) + c)∂_{x}φ_{0} = (g′(0) − δ)φ_{0} in Σ \\ ∂_{ν}φ_{0} = 0 on ∂Σ \end{matrix}

Since Re(φ) > 0 in ω̄ and λ ∈ C^{|} \IR, there exist two functions in C^{1}(ω̄), x_{1}(y) and x_{2}(y), such that

∀y ∈ ω̄, , x_{1}(y) < x_{2}(y) and φ_{0}(x_{1}(y), y) = φ_{0}(x_{2}(y), y) = 0
∀y ∈ ω̄, ∀x ∈ (x_{1}(y), x_{2}(y)), φ_{0}(x, y) > 0.

Set D = {(x, y) ∈ Σ; x_{1}(y) < x < x_{2}(y)}. Since g is C^{1}, there exists η > 0 such that ∀s ∈ (0, η), (g′(0) − δ)s < g(s); so, without loss of generality, we can assume that φ_{0} ≤ η on D, wich implies

−Δφ_{0} + (α(y) + c)∂_{x}φ_{0} = (g′(0) − δ)φ_{0} ≤ g(φ_{0}) in D.

Now we follow the idea of Aronson and Weinberger in [2]. Solve the following problem:

∂_{t}w − Δw + (α(y) + c)∂_{x}w = g(w) in Σ
∂_{ν}w = 0 on ∂Σ
w(t = 0) = φ_{0} in D, and 0 in Σ − D

Then w(x, y, t) ≥ φ_{0}(x, y) in D, ∀t ≥ 0 (apply the maximum principle in D, taking into account the Neumann condition on ∂D ∩ ∂Σ and the Dirichlet condition on ∂D ∩ Σ). Considering the function m(x, y, t) = w(x, y, t + h) − w(x, y, t) for h > 0, we see that m is positive ∀t ≥ 0 (maximum principle again), and so w is increasing in t. Then there exists a limit function w_{∞}(x, y) to which w(x, y, t) converges as t → +∞. Using classical parabolic estimates and Ascoli theorem, we get the uniform convergence on every compact subset of Σ̄, as well as for the derivatives ∂_{x}w, ∂_{t}w, and Δw. This implies

−Δw_{∞} + (α(y) + c)∂_{x}w_{∞} = g(w_{∞}) in Σ

with Neumann boundary conditions.

An argument similar to that of [5], Theorem 1 shows that w_{∞} has to go to a constant limit as |x| → +∞, which can only be 0 or 1. But from the theory of Berestycki and Nirenberg, it follows that w_{∞} is monotonic; since c ∈ (c^{∗∗}, c^{∗}), it must be 0 or 1 identically, so it is 1 identically. To sum up, we have just constructed a function with compact support, with arbitrarily small supremum norm, and such that the solution of the corresponding Cauchy problem goes to 1 uniformly on every compact subset of Σ̄. Applying the maximum principle, we generalize this result to any u_{0} having compact support.

Now proceed similarly with u(−x, y, t); we finally get that u(x−ct, y, t) → 1 for some real c ∈ (c^{∗∗}, c^{∗}) arbitrarily near c^{∗∗}. The second assertion in theorem 4 is then proved for reals c ∈ (c^{∗∗}, c^{∗∗} + ϵ) ∪ (c^{∗} − ϵ, c^{∗}) with any small ϵ > 0. The general result follows from an application of the maximum principle as in theorem 8, see section 4. □

NB: For the proof of the second assertion in theorem 4, we needn’t suppose that g(s)/s decreases.

4 Global stability of waves

In this section, we show that the travelling waves are stable in some sense, and give criteria for the creation of travelling fronts, as t grows to infinity.

In order to emphasize the main ideas, we first investigate particular cases, then generalize. Recall that g is still assumed to be a KPP term.

Lemma 6 Let c ∈ IR and β(y) = α(y) + c. Consider two solutions u and v of

\{ \begin{matrix}∂_{t}z − Δz + β(y)∂_{x}z = g(z) in Σ \\ ∂_{ν}z = 0 on ∂Σ \end{matrix}

Also suppose that u(t = 0) ≤ v(t = 0), with values in [0, 1] and that there exists x_{0} ∈ IR such that u(x, y, 0) = v(x, y, 0), ∀x ≤ x_{0}, ∀y ∈ ω; then

0 ≤ v(t) − u(t) ≤ Ce^{λ^{∗}[x−(c−c^{∗})t]} for some real C.

PROOF. The left handside follows from the maximum principle. For the right handside, we write for some w,

(∂_{t} + A_{c})(v − u) = g′(w)(v − u) ≤ g′(0)(v − u)

since v − u ≥ 0 and g is a KPP source term;

let then e^{λ^{∗}x}φ^{∗}(y) be a positive exponential solution of

−Δz + (α(y) + c^{∗})∂_{x}z = g′(0)z

with the Neumann boundary condition on ∂Σ; then h(x, y, t) = e^{λ^{∗}[x−(c−c^{∗})t]}φ^{∗}(y) satisfies (∂_{t} + A_{c} − g′(0))h = 0 with the same boundary condition; we then choose a constant Λ such that v(0) − u(0) ≤ Λh(0); the result then follows from the maximum principle. □

Theorem 5 Let c > c^{∗}, β(y) = α(y) + c, and consider u(t) the solution of the problem

(Pc) : \{ \begin{matrix}∂_{t}u − Δu + (α(y) + c)∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(t = 0) = u_{0} in Σ \end{matrix}

where u_{0}, with values in [0, 1], satisfies:

u_{0}(x, y) ∼ ϕ_{c}(x, y) uniformly on ω̄ when x → −∞,

then

u(x, y, t) → ϕ_{c}(x, y) when t → +∞, uniformly on (−∞, a] × ω̄, ∀a ∈ IR.

PROOF. Let a ∈ IR and ϵ > 0. Set ϕ := ϕ_{c} . Let δ > 0 such that

|ϕ(x, y) − ϕ(x − δ, y)| < ϵ/2 on Σ

since ϕ_{c}(x, y) ∼ e^{λ_{c}x}φ_{c}(y) as x → −∞, with λ_{c} > 0 and φ_{c} > 0 on ω̄, we have

ϕ^{δ}(x, y) := ϕ(x − δ, y) ∼ e^{−λ_{c}δ}ϕ(x, y) when x → −∞

so ū_{0} := inf(u_{0}, ϕ^{δ}) satisfies ū_{0} ≤ ϕ^{δ} and ū_{0}(x, y) = ϕ^{δ}(x, y), ∀x ≤ x_{0} (for some real x_{0}); we denote by ū the solution of the Cauchy problem with ū(0) = ū_{0}, and apply Lemma 6 which yields

0 ≤ ϕ^{δ} − ū(t) ≤ Ce^{λ^{∗}[(x−(c−c^{∗})t]}

similar considerations on ū_{0} := sup(u_{0}, ϕ^{−δ}) provide

0 ≤ ū(t) − ϕ^{−δ} ≤ Ce^{λ^{∗}[x−(c−c^{∗})t]}

now, we just have to apply the maximum principle:

ϕ^{δ} − Ce^{λ^{∗}[x−(c−c^{∗})t]} ≤ ū(t) ≤ u(t) ≤ ū(t) ≤ ϕ^{−δ} + Ce^{λ^{∗}[x−(c−c^{∗})t]} . □

Under an additional hypothesis, we now obtain uniform convergence in the whole cylinder.

Theorem 6 Let c > c^{∗}, β(y) = α(y) + c, and consider u(t) the solution of the problem

(Pc) : \{ \begin{matrix}∂_{t}u − Δu + (α(y) + c)∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(t = 0) = u_{0} in Σ \end{matrix}

with u_{0} (taking its values in [0, 1]) satisfying

1. u_{0}(x, y) ∼ ϕ_{c}(x, y) uniformly in ω as x → −∞

2. lim inf_{x→+∞} u_{0} > 0 uniformly in ω,

then:

sup_{(x,y)∈Σ̄} |u(x, y, t) − ϕ_{c}(x, y)| → 0 as t → +∞.

We will make use of the following lemma. This kind of sub and supersolutions result have been widely used in one space dimension; see [10], [21].

Lemma 7 Let c > c^{∗} and ϵ_{0} ∈ (0, 1), then ∃s, K > 0 such that ψ(x, y, t) := h(t)ϕ_{c}(x − m(t), y) satisfies ∂_{t}ψ + A_{c}ψ − g(ψ) ≤ 0 sur Σ, ∀t ≥ 0, with h(t) := 1 − ϵ_{0}e^{−st} and m(t) := K(1 − e^{−st}).

PROOF. Let u_{1} ∈ (0, 1) such that ∀ϵ_{0} ∈ (0, 1), ∃T > 0 satisfying

(1 − ϵ)g(u) − g((1 − ϵ)u) ≤ −T ϵu, ∀u ∈ [u_{1}, 1], ∀ϵ ∈ [0, ϵ_{0}]

(such an u_{1} exists as soon as g′(0) > 0 > g′(1)).

Set N[ψ] = ∂_{t}ψ+A_{c}ψ−g(ψ), and φ(x, y, t) = ϕ_{c}(x−m(t), y); we compute

N[ψ] = sϵ_{0}e^{−st}φ − hsK∂_{x}φ + hg(φ) − g(hφ)

There exists X_{0} such that ∀x ≥ X_{0}, ∀y ∈ ω, ϕ_{c}(x, y) ≥ u_{1}, so, if x − m(t) ≥ X_{0}, we have hg(φ) − g(hφ) ≤ −T ϵ_{0}e^{−st}φ, and then N[ψ] ≤ e^{−st}φ(s − T)ϵ_{0}; choosing s = T, we have N[ψ] ≤ 0; if x − m(t) ≤ X_{0}, notice there exists a constant C > 0 such that ∂_{x}φ/φ ≥ C, because of the asymptotic behaviour of ϕ_{c} as x → −∞. Write

N[ψ] ≤ sϵ_{0}e^{−st}φ − hsCKe^{−st}φ + g(φ) − g(hφ)
N[ψ] ≤ e^{−st}φ(sϵ_{0} − hsCK + C′ϵ_{0})

Now choose K = (s + C′)ϵ_{0}/(1 − ϵ_{0})sC, which yields N[ψ] ≤ 0 too. □

Here is the proof of Theorem 6. In the case where u_{0}(x, y) > 0 in Σ̄, the hypothesis guarantees there exists some h_{0} ∈ (0, 1) and some x_{0} ∈ IR satisfying h_{0}ϕ_{c}(x − x_{0}, y) ≤ u_{0}(x, y) in Σ. Apply Lemma 7 with ϵ_{0} = 1 − h_{0}. Then ψ(x, y, t) ≤ u(x, y, t), ∀t ≥ 0; this, together with Theorem 5, proves that the convergence is uniform on Σ̄.

If we do not assume u_{0}(x, y) > 0 in Σ̄, we still have, by the maximum principle, that u(x, y, t_{0}) > 0 as soon as t_{0} > 0. Notice that the condition 1 is preserved at all positive times (use the same arguments as in the proof of Theorem 5); the second condition is in fact also preserved: just compare u_{0} to some C^{∞} and increasing function h_{0}(x, y) which is 0 if x < M and ϵ if x > M + 1; solve the corresponding Cauchy problem, which gives some h(x, y, t) increasing in x, so that obviously lim inf_{x→+∞} h(x, y, t) > 0, for each positive t.

Both conditions being preserved,we use the semigroup property and prove Theorem 6 in the general case. □

Now we give a more powerful version of this result.

Theorem 7 Let c > c^{∗}, β(y) = α(y) + c, and consider u(t) the solution of the problem

(Pc) : \{ \begin{matrix}∂_{t}u − Δu + (α(y) + c)∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(t = 0) = u_{0} in Σ \end{matrix}

with u_{0} (taking its values in [0, 1]) satisfying

1. u_{0}(x, y) ∼ a(y)e^{λ_{c}x} uniformly on ω as x → −∞, where a(y) is continuous and not identically zero;

2. lim inf_{x→+∞} u_{0} > 0 uniformly on ω,

then:

∃x_{0} ∈ IR such that sup_{(x,y)∈Σ̄} |u(x, y, t) − ϕ_{c}(x + x_{0}, y)| → 0 as t → +∞.

First, let us state a lemma.

Lemma 8 Let u be a solution of (P_{c}), with c > c^{∗}; suppose u(x, y, 0) satisfies the first condition in theorem 7; Let ψ(y, t) be the solution of

\{ \begin{matrix}∂_{t}ψ − Δψ + (λ_{c}β(y) − λ_{c}^{2})ψ = g′(0)ψ in ω \\ ∂_{ν}ψ = 0 on ∂ω \\ ψ(y, 0) = a(y) \end{matrix}

then:

u(x, y, t) ∼ ψ(y, t)e^{λ_{c}x} as x → −∞, ∀t ≥ 0, uniformly in ω.

PROOF. Set u(x, y, t) = e^{λ_{c}x}v(x, y, t). Now notice that v(x, y, t) is bounded on every [0, T]. To prove it, notice that the hypothesis on u_{0} implies the existence of a wave ϕ_{c} and a real x_{0} such that (u_{0} − ϕ_{c})(x, y) < 0 for all x < x_{0}; apply Lemma 6:

u(x, y, t) < ϕ_{c}(x, y) + Ce^{λ^{∗}[x−(c−c^{∗})t]}, ∀t > 0

so

v(t) < e^{−λ_{c}x}ϕ_{c}(x, y) + Ce^{(λ^{∗}−λ)x−λ^{∗}(c−c^{∗})t}

since λ_{c} < λ^{∗}, this yields an upper bound for v(t) on IR^{−} × ω, ∀t ∈ [0, T], whereas v(t) is clearly bounded on IR^{+} × ω (remember u ≤ 1).

For θ ∈ IR, set w(x, y, t) = v(x, y, t)−θψ(y, t) and write the PDE satisfied by w:

Lw := ∂_{t}w − Δw + β(y)∂_{x}w + (λ_{c}β(y) − λ_{c}^{2} − g′(0))w = e^{−λ_{c}x}g(e^{λ_{c}x}v) − g′(0)v

g being C^{2}, we can write g(s) = g′(0)s+(s^{2}/2)g′′(ηs), so |Lw| ≤ Ke^{−λ_{c}x}e^{2λ_{c}x}v^{2}, but we know that v is bounded on each [0, T], therefore:

|Lw| ≤ K(T)e^{λ_{c}x}, ∀t ∈ [0, T].

Let M a real, and h(x, t) = e^{λ_{c}x+Mt}; choose T > 0. We have Lh = (M + 2(λ_{c}β(y) − λ_{c}^{2}) − g′(0))h, so for some large enough M > 0, we have:

Lh ≥ K(T)h = K(T)e^{λ_{c}x+Mt} ≥ K(T)e^{λ_{c}x}.

If θ > 1, we have w(x, y, 0) < 0 for x < x(θ) (a real depending on θ), and since w(x, y, 0) is bounded in Σ, we get w(x, y, 0) < Nh(x, 0) for N large enough. The maximum principle then implies:

w(x, y, t) < Nh(x, t), ∀t ∈ [0, T]
v(x, y, t) < θψ(y, t) + Ne^{λ_{c}x+Mt}

meanwhile, a(y) being nonnegative and not identically zero, we know that ψ(y, t) ≥ ϵ(T) > 0 for all t ∈ [1/T, T], so

v(x, y, t) < (θ + N(T)e^{λ_{c}x})ψ(y, t), ∀t ∈ [1/T, T],

then lim sup_{x→−∞}(v(x, y, t)/ψ(y, t)) ≤ θ, uniformly in ω × [1/T, T]; the same

treatment for θ < 1 provides lim inf_{x→−∞}(v(x, y, t)/ψ(y, t)) ≥ θ, uniformly in ω × [1/T, T], which ends the proof of Lemma 8. □

We keep the notations of Lemma 8 for the proof of Theorem 7: consider the operator T = −Δ + (λ_{c}β(y) − λ_{c}^{2} − g′(0)), with Neumann boundary condition; T is self-adjoint with 0 as first eigenvalue; then it is classical to see that, in L^{2}(ω), ψ(y, t) → φ_{c}(y), as t → +∞, where φ_{c}(y) is the only positive eigenfunction satisfying ∫_{ω} aφ_{c} = ∫_{ω} φ_{c}^{2}.

Once again, we use the classical parabolic estimates to prove that the convergence happens in L^{∞}(ω).

Let ϵ > 0, so there exists t_{0} such that ∀t ≥ t_{0}, (1 − ϵ)φ(y) < ψ(y, t) < (1 + ϵ)φ(y). Let ϕ_{c} and ϕ′_{c} the travelling waves, with speed c, satisfying

ϕ_{c} ∼ (1−ϵ)φ(y)e^{λ_{c}x} and ϕ′_{c} ∼ (1+ϵ)φ(y)e^{λ_{c}x} as x → −∞. We apply theorem 6 with initial data inf{ϕ_{c}(x, y); u(x, y, t_{0})} and sup{ϕ′_{c}(x, y); u(x, y, t_{0})}; u(x, y, t) stays between the solutions of the corresponding Cauchy problems (maximum principle), but these converge uniformly on Σ̄ to ϕ_{c} and ϕ′_{c} respectively. Just notice now that ||ϕ_{c} − ϕ′_{c}||_{L^{∞}(Σ)} is arbitrarily small when ϵ is. This proves Theorem 7. □

Keeping the notations of Theorem 7, we denote by ϕ_{c} the only travelling front such that ϕ_{c}(x, y) ∼ e^{λ_{c}x}φ_{c}(y) as x → −∞. We have an explicit expression for x_{0}:

x_{0} = \frac{1}{λ_{c}} (Log ∫_{ω} aφ_{c} − Log ∫_{ω} φ_{c}^{2}).

Our last result concerns the creation of fronts at both ends of the cylinder.

Theorem 8 Let u(x, y, t) be a solution of (P); suppose there exist λ ∈ (0, λ^{∗}) and λ̃ ∈ (0, λ^{∗∗}) satisfying

u_{0}(x, y) ∼ a(y)e^{λx} near − ∞ uniformly in ω
u_{0}(x, y) ∼ b(y)e^{−λ̃x} near + ∞ uniformly in ω

for some continuous, non identically zero a(y) and b(y) ; then if we denote by c and c′ the wave speeds associated to λ and λ̃ respectively, with c′ < c^{∗∗} < c^{∗} < c ,we have

sup_{(x,y)∈Σ} |u(x, y, t) − inf{ϕ_{c}(x + ct, y), ϕ̃_{c}(x + c′t, y)}| → 0 as t → +∞.

PROOF. Using Lemma 8, and proceeding as in the proof of Theorem 7, we have that u(x − ct, y, t) → ϕ_{c}(x, y), uniformly on each (−∞, a] × ω and u(x − c′t, y, t) → ϕ̃_{c′}(x, y), uniformly on each [a, +∞) × ω.

Let ϵ > 0; there exist X_{1} and X_{2} such that ϕ_{c}(x, y) ≥ (1 − ϵ), ∀x ≥ X_{1}, ∀y ∈ ω, and ϕ̃_{c′}(x, y) ≥ (1−ϵ), ∀x ≤ X_{2}, ∀y ∈ ω. Set D = {(x, y, t)/X_{1}−ct < x < X_{2} − c′t}; choose some large t_{0} satisfying t ≥ t_{0} ⟹ u(X_{1} − ct, y, t) ≥ ϕ_{c}(X_{1}, y) − ϵ ≥ 1 − 2ϵ and u(X_{2} − c′t, y, t) ≥ ϕ̃_{c′}(X_{2}, y) − ϵ ≥ 1 − 2ϵ. Set g_{ϵ}(s) = ms(1 − 2ϵ − s) with m small enough to have g_{ϵ} ≤ g; we are going to apply the maximum principle on D ∩ {t ≥ t_{0}}: let h(t) satisfy ḣ = g_{ϵ}(h) and h(0) = inf{1 − 2ϵ; inf_{D∩{t=t_{0}}} u(x, y, t_{0})}; now write

(∂_{t} + A)(u − h) = g(u) − g_{ϵ}(h) ≥ g(u) − g(h),

considering the boundary conditions, this shows that u(t) ≥ h(t + t_{0}) in D ∩ {t = t_{0}}.

Therefore, there exists t_{1} such that ∀t ≥ t_{1}, u(x, y, t) ≥ 1 − 3ϵ in D. This ends the proof of Theorem 8. □

5 Local stability in weighted spaces

In this section, we adopt another point of view, based upon the spectral analysis of operator A_{c} in a suitable weighted space. This kind of analysis appeared for the first time in the work of Sattinger [19], with a KPP source term. This section generalises [19]; furthermore, we show how to improve very easily the uniform convergence results of the last section in the case of a KPP source term, when the initial datum has a suitable behaviour as x → +∞. The general case is more intricate and is presented in the next section.

5.1 Framework

We linearise Problem (P_{c}) near ϕ_{c}, by setting u_{0} = ϕ_{c}+v_{0} and u(t) = ϕ_{c}+v(t), ∀t > 0, c > c^{∗} being fixed once and for all. We obtain

∂_{t}u + A_{c}u = g(u) ⇔ ∂_{t}v + A_{c}v = g(ϕ_{c} + v) − g(ϕ_{c})

Set L = A_{c} − g′(ϕ_{c}) and f(v) = g(ϕ_{c} + v) − g(ϕ_{c}) − vg′(ϕ_{c}), which leads to the formulation

\{ \begin{matrix}∂_{t}v + Lv = f(v) in Σ \\ ∂_{ν}v = 0 on ∂Σ \\ v(0) = v_{0} \end{matrix}

Choice of a space. Set β_{η} = β + η − β̄, (β̄ denotes the average of β); for 0 < η < β̄ sufficiently close to β̄, the following problem

(E_{η}) : \{ \begin{matrix}−Δ_{y}φ − g′(0)φ + λβ_{η}φ = λ^{2}φ in ω \\ ∂_{ν}φ = 0 on ∂ω \end{matrix}

admits exactly two positive principal eigenvalues; we denote by r = r(η) the smallest one, and consider an associated eigenfunction ψ, which is positive on ω.

Remark 1 r > λ_{c} and r(η) ↘ λ_{c} as η ↗ β̄.

Let Γ ∈ C^{∞}(IR) such that 0 ≤ Γ ≤ 1, Γ = 0 on (−∞, 0] and Γ = 1 on [1, +∞);

let

w_{1}(x, y) = (1 − Γ(x))e^{rx}ψ(y) + Γ(x + 1) and w(x, y) = 1/w_{1}(x, y)

set

X = {u ∈ C_{0}(Σ̄); wu ∈ C_{0}(Σ̄)}

where C_{0}(Σ̄) is the set of all bounded uniformly continuous functions on Σ̄, which tend to 0 as |x| → ∞, uniformly on ω̄.

We equip X with the norm ||u||_{X} = ||wu||_{∞}, so that

M : X ⟶ C^{0}(Σ̄)
u ↦ wu

is an isometric isomorphism between (X, ||.||_{X}) and (C^{0}(Σ̄), ||.||_{∞}).

Remark 2 ∂_{x}ϕ_{c} ∉ X

This is clear from the behaviour of ∂_{x}ϕ_{c} as x → −∞.

Transformation of operator L. To the operator L defined on X, we associate L̃ = MLM^{−1} defined in C^{0}(Σ̄), in other words ∀v ∈ C^{0}(Σ̄) smooth, L̃v = wL(v/w); recall that L = −Δ + β(y)∂_{x} − g′(ϕ_{c}), therefore we get L̃

L̃v = −Δv + B(x, y).∇v + C(x, y)v

where

B(x, y).∇v = β(y)∂_{x}v + 2\frac{∇w}{w}.∇v

and

C(x, y) = \frac{Δw}{w} − 2\frac{|∇w|^{2}}{w^{2}} − β(y)\frac{∂_{x}w}{w} − g′(ϕ_{c})

thanks to our choice for w, we get that B(x, y) is bounded, and

C(x, y) = \{ \begin{matrix}r(β̄ − η) + g′(0) − g′(ϕ_{c}) for x < −1 \\ −g′(ϕ_{c}) for x > 1 \end{matrix}

set γ(x) = r(β̄ − η)(1 − Γ(x)) − g′(1)Γ(x + 1); notice that {inf \atop IR} γ > 0 and

that |C(x, y) − γ(x)| → 0 as |x| → ∞, uniformly on ω̄; set

T̃v = −Δv + B(x, y).∇v + γ(x)v
S̃v = [C(x, y) − γ(x)]v

write L̃ = S̃+T̃ and L = S +T with S = M^{−1}S̃M and T = M^{−1}T̃M; finally define the domains:

D(L̃) = D(T̃) = {u ∈ C_{0}(Σ̄)∩( ⋂_{p≥1} W_{loc}^{2,p}(Σ)); Δu ∈ C^{0}(Σ̄) and ∂_{ν}u = 0 on ∂Σ}

D(L) = M^{−1}(D(L̃)) = {u ∈ X ∩( ⋂_{p≥1} W_{loc}^{2,p}(Σ)); Δu ∈ X and ∂_{ν}u = 0 on ∂Σ}

Lemma 9 D(L) is dense in X.

5.2 Spectral properties of L

Lemma 10 L is sectorial in X, more precisely ∃a < 0, γ_{0} > 0 and α ∈ [0, \frac{π}{2}) such that

1. σ(L) (the spectrum of L in X) is contained in the cône

C_{a,α} = {z ∈ C^{|} ; |Arg(z − a)| < α}

2. if λ ∈ σ(L) and Re(λ) < γ_{0}, then λ is an eigenvalue for L.

The proof is similar to the one given in [17], Theorem 4.1.

Theorem 9 If λ is an eigenvalue for L in X, then either λ = 0 or Re(λ) > 0.

PROOF. It is nearly the proof of Theorem 1 in [6], and in fact much simpler since there is nothing to prove as x → −∞.

Let u ∈ X such that Lu = λu; differentiating A_{c}ϕ_{c} = g(ϕ_{c}) yields L(∂_{x}ϕ_{c}) = 0; u being in X, we immediatly have

\frac{u}{∂_{x}ϕ_{c}} → 0 when x → −∞

the only point is to prove the same as x → +∞. Indeed, if we denote by λ^{+} the exponent which rules the behaviour of ϕ_{c} at +∞, then if Re(λ) ≤ 0 and λ ≠ 0, there exists ϵ > 0 such that |u(x, y)| ≤ e^{−(λ^{+}+ϵ)x} in Σ. (see Lemmas 5.1, 5.2, 5.3 and 5.6 in [6]).

Finally, the maximum principle applied to the function Re(e^{−λt}u(x, y))/∂_{x}ϕ_{c} shows that we have u = 0. □

Proposition 1 L : D(L) ⟶ X is one-to-one.

PROOF. Let u ∈ D(L) such that Lu = 0; from [7] and Lemma 3, there exist α_{−}, α_{+}, such that

\frac{u}{ϕ_{x}} → α_{−} as x → −∞

\frac{u}{ϕ_{x}} → α_{+} as x → +∞.

Now proceed as in [17], Proposition 3.4: this yields \frac{u}{ϕ_{x}} =constant. Now, since u is supposed to decay faster than ϕ_{x}, u = 0. □

Remark 3 This proves that Lu = 0, u ∈ C_{0}(Σ̄) implies that u is proportional to ∂_{x}ϕ_{c}.

5.3 Conclusion: local stability and exponential convergence

Proposition 2 L is a sectorial operator in X with spectrum σ(L) contained in a cone C_{a,α} with a > 0 and α ∈ [0, \frac{π}{2}).

In particular, Re(σ(L)) > a > 0.

This is an obvious consequence of the above paragraph.

Theorem 10 ∃ω > 0, ρ > 0 and M ≥ 0 such that, if ||v_{0}||_{X} ≤ ρ/2M, then the problem

\{ \begin{matrix}dv/dt + Lv = f(v), ∀t > 0 \\ v(0) = v_{0} \end{matrix}

admits exactly one solution v defined on (0, +∞) and satisfying ||v(t)||_{X} ≤ 2Me^{−ωt}||v_{0}||_{X}.

PROOF. Just use theorem 5.1.1 in [11]. □

Now we may use the global stability theorems of Section 4 to get an exponential global stability result. Assume for simplicity that g is concave.

Proposition 3 Let u_{0} be uniformly continuous in Σ̄ such that u_{0}(x, y) = ϕ_{c}(x, y)(1 + O(e^{rx})) as x → −∞, for some positive real r, and lim inf_{x→+∞} u_{0} > 0. Then

||u(t) − ϕ_{c}||_{∞} ≤ Ke^{−κt}

where K and κ are two positive real numbers.

PROOF. Let w(x, y) be a positive exponential solution of

\{ \begin{matrix}−Δw + β_{η}∂_{x}w = g′(0)w in Σ \\ ∂_{ν}w = 0 on ∂Σ \end{matrix}

with η close enough to β̄ so that

w(x, y)/e^{(λ_{c}+r)x} → +∞ as x → −∞.

We have

(−Δ + β∂_{x})(ϕ_{c} + Cw) = g(ϕ) + ηC∂_{x}w + g′(0)Cw
≥ g(ϕ) + g′(0)Cw ≥ g(ϕ + Cw),

for each constant C (here we have used the concavity of g). Symmetrically, we have

(−Δ + β∂_{x})(ϕ_{c} − Cw) = g(ϕ) − ηC∂_{x}w − g′(0)Cw
≤ g(ϕ) − g′(0)Cw ≤ g(ϕ − Cw).

We infer from the maximum principle that, for C large enough, the following inequalities hold

ϕ_{c} − Cw ≤ u(t) ≤ ϕ_{c} + Cw.

Now, we use the global stability theorem, then we are in a position to apply the local exponential stability theorem. □

6 Extensions to non-KPP source terms

We drop here the KPP assumption. The global stability results for the fast waves - i.e. c > c_{∗} - are the same as in the preceeding sections, but the proofs are more technical and appeal to some ideas presented in [18]. New stability results are derived when an additional assumption is made on g.

6.1 Global stability for rapid waves

We want to prove a result similar to Theorem 7; let us recall it.

Theorem 11 Let c > c^{∗}, β(y) = α(y) + c, and consider u(t) the solution of problem

(Pc) : \{ \begin{matrix}∂_{t}u − Δu + (α(y) + c)∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(t = 0) = u_{0} in Σ \end{matrix}

with u_{0} (taking its values in [0, 1]) satisfying

1. u_{0}(x, y) ∼ a(y)e^{λ_{c}x} uniformly on ω as x → −∞, where a(y) is continuous and not identically zero;

2. lim inf_{x→+∞} u_{0} > 0 uniformly on ω,

then:

∃x_{0} ∈ IR such that sup_{(x,y)∈Σ̄} |u(x, y, t) − ϕ_{c}(x + x_{0}, y)| → 0 as t → +∞.

We need only prove this when a(y) = φ_{c}(y) and u_{0}(x, y) = φ_{c}(y)e^{λ_{c}x}(1 + O(e^{rx})). To see this, one may proceed as in [18], Lemma 2.3; this is the way we will follow.

Before we start the proofs, notice that, just as in the proof of Theorem 7, we can obtain an explicit expression for the asymptotic shift x_{0}.

Let Γ ∈ C^{∞}(IR) such that 0 ≤ Γ ≤ 1, Γ = 0 on (−∞, 0] and Γ = 1 on [1, +∞).

Lemma 11 There exist three positive reals M, q_{0},and s, such that

u(x, y, t) ≤ ϕ_{c}(x + M, y) + q_{0}e^{−st}Γ(x + M).

PROOF. Just as in [18], Lemma 5.1. □ Next, we need an even sharper control on the decay of the solution

u(x, y, t) as x → −∞.

Lemma 12 ∀r′ > r, ∃C > 0 such that, ∀t ≥ 0,

|u(x, y, t) − ϕ_{c}(x, y)| ≤ Ce^{(λ_{c}+r′)x}.

PROOF. Let δ and x_{0} such that

• g′(0) < δ;

• the equation μ_{1}(−Δ + λβ − λ^{2} − δ) = 0 admits solutions; let w(x, y) be a corresponding exponential solution;

• ∀(x, y, t) ∈ (−∞, x_{0}] × ω × IR^{+}, we have

\frac{g(u(x, y, t)) − g(ϕ_{c}(x, y))}{u(x, y, t) − ϕ_{c}(x, y)} ≤ δ.

Set v = u − ϕ_{c}. The function w is an upper solution for the equation on v. So, for C large enough, we have Cw ≥ u − ϕ_{c}. The same method works if we look for a subsolution; just consider ϕ_{c} − u. □

Proposition 4 There exist three positive real numbers M, q_{0},and s, such that

u(x, y, t) ≥ ϕ_{c}(x − M, y) − q_{0}e^{−st}.

PROOF. Let ϵ > 0 such that the ball with radius ϵ and center ϕ_{c} (in the norm 1 + e^{−(λ_{c}+r/2)x}), be in the attraction domain of ϕ_{c}. Choose t_{0} > 0 such that ∀t ≥ t_{0}, lim_{x→+∞} u(x, y, t) ≥ 1 − ϵ/2. We can construct ū_{0}(x, y) satisfying the following properties:

• ū_{0}(x, y) = ϕ_{c}(x, y)(1 + e^{2rx/3}(1 + o(1))) for x ≤ −M;

• lim inf_{x→+∞} ū_{0}(x, y) ≥ 1 − ϵ/2;

• sup_{(x,y)∈Σ} |ū_{0}(x, y) − ϕ_{c}(x, y)|.(1 + e^{−(λ_{c}+r/2)x}) ≤ ϵ.

The solution of the Cauchy problem, with initial datum ū_{0}(x − M, y), will converge exponentially to ϕ_{c}(x − M, y). Moreover, for large enough M, ū_{0}(x, y) ≤ u_{0}(x, y). □

Now we prove Theorem 11. Let

X_{δ,r} = {u_{0} ∈ UC(Σ̄); u_{0} ↗, lim_{x→+∞} u_{0}(x, y) ≥ δ,

and |u_{0}(x, y) − ϕ_{c}(x, y)| ≤ Ce^{(λ_{c}+2r/3)x}},

where UC is the set of bounded uniformly continuous functions. Also set

C_{δ,r} = {u_{0} ∈ X; S(t)u_{0} → ϕ_{c} in the weighted norm 1 + e^{−(λ_{c}+r/2)x}}.

It is clear from the local stability theorems that C_{δ,r} is open in X_{δ,r}; in order to prove that it is also closed, we state the following proposition.

Proposition 5 Let u_{10} ∈ X_{δ,r}. There exists ε_{0} such that, for every ε < ε_{0},

\begin{matrix}∀u_{20} ∈ X_{δ,r}, ||u_{10} − u_{20}||_{X_{δ,r}} ≤ ϵ \\ ⟹ sup_{(x,y)∈Σ}(1 + e^{−λ_{c}x/2})|S(t)u_{10} − S(t)u_{20}| ≤ Cϵ. \end{matrix}

PROOF. Just do as in [18], Proposition 5.3. The proof involves the construction of upper and lower solutions, and the use of parabolic Harnack inequlities ”up to the boundary” □

Corollary 2 If u_{0} ∈ X_{δ,r}, then S(t)u_{0} converges towards ϕ_{c}.

To end the proof of the theorem, just notice that it is always possible to find ū_{0} ∈ X_{δ,r} and ū_{0} ∈ X_{δ′,r′}, such that ū_{0} ≤ u_{0} ≤ ū_{0}. □

6.2 Stability of the wave with speed c^{∗}

Here we suppose that c^{∗} > c_{0}. Since the results that are stated can be proved by the almost same methods as in the above sections, we will only give the main lines of the proofs, leaving it to the interested reader to check the details.

We first show that the case c^{∗} > c_{0} is indeed a possible one, and proceed as in as Berestycki and Nirenberg in [8]. We multiply by ∂_{x}ϕ the equality

−Δϕ + (c^{∗} + α(y))∂_{x}ϕ = g(ϕ)

and we integrate on Σ, which yields

∫_{Σ} (c^{∗} + α(y))[∂_{x}ϕ]^{2}dxdy = G(1)

where G(s) = ∫_{0}^{s} g(σ)dσ; the multiplication by 1 − ϕ yields

(c^{∗}+ < α >) ≥ \frac{2}{|ω|} ∫_{Σ} |∇ϕ|^{2}dxdy

(c^{∗} + sup α)^{2} ≥ \frac{2}{|ω|} ∫_{Σ} |∇ϕ|^{2}(c^{∗} + sup α)dxdy

so we get (c^{∗} + sup α)^{2} ≥ \frac{2}{|ω|}G(1), that is to say

c^{∗} ≥ \sqrt{\frac{2}{|ω|}G(1)} − sup α.

Now remember that c_{0}(g′(0)) → 0 as g′(0) → 0; then we just have to choose g′(0) small and G(1) large enough. □

As for the asymptotic behaviour of ϕ_{c^{∗}} as x → −∞, we have to take into account the case when ϕ_{c^{∗}} decays at the usual rate, and the case when ϕ_{c^{∗}} decays at the accidental rate.

Case 1. If ϕ_{c^{∗}} decreases at the usual rate, all the results for rapid waves are still available for ϕ_{c^{∗}}. We leave it to the interested reader to check the analogue of Theorem 11 for ϕ_{c^{∗}}.

Case 2. If ϕ_{c_{∗}} decreases at the accidental rate, we get the same results as in the ”ignition temperature” case, see [18]: local stability in weighted spaces with small exponents, and convergence to travelling waves for initial data that are increasing in x. Namely, the weight functions that are suitable in this case are not sufficient to kill the translation invariance. Here is the precise result.

Theorem 12 Let u(t) be the solution of

(P_{∗}) : \{ \begin{matrix}∂_{t}u − Δu + (α(y) + c_{∗})∂_{x}u = g(u) in Σ \\ ∂_{ν}u = 0 on ∂Σ \\ u(t = 0) = u_{0} in Σ \end{matrix}

Assume moreover that the initial datum u_{0} satisfies

1. u_{0}(x, y) = O(e^{rx}) as x → −∞, with r > 0;

2. lim inf_{x→+∞} u_{0} > 0 uniformly on ω.

Then the following is true.

1. If u_{0} has the form u_{0}(x, y) = ϕ_{c_{∗}}(x, y) + εv_{0}(x, y) (with v_{0} decaying as e^{rx} as x → −∞), then for ε_{0} > 0 small enough there exists γ(ε) ∈ C^{1}(−ε_{0}, ε_{0}) and

sup_{(x,y)∈Σ̄} |u(x, y, t) − ϕ_{c}(x + εγ(ε), y)| = O(e^{−ωt}) as t → +∞.

2. If u_{0} takes its values in (0, 1) is increasing in the x direction then ∃x_{0} ∈ IR and ω > 0 such that

sup_{(x,y)∈Σ̄} |u(x, y, t) − ϕ_{c}(x + x_{0}, y)| = O(e^{−ωt}) as t → +∞.

PROOF. Point 1 may be proved by the same method as in Section 5 as far as the linear stability is concerned. Nonlinear stability results are then proved in weighted spaces with exponents < λ_{c_{∗}}^{a}; therefore one must go through all the steps of the proof of Theorem 1 in [17].

Let us now give the main lines of the proof of Point 2. Let X_{r,δ} and C_{r,δ} be defined as in the above subsection; we want to prove that C_{r,δ} is closed in X_{r,δ}. On way to do it is as follows.

1. Prove an inequality of the type

ϕ_{c_{∗}}(x − M, y) − q_{1}e^{−st} ≤ u(x, y, t) ≤ ϕ_{c_{∗}}(x + M, y) + q_{2}e^{−st}.

This may be done as in Proposition 4.

2. Prove a uniform estimate as in Proposition 5. □

Remark 4 This method yields an extension to the second author’s Theorem 1.2 in [18] to initial data having an arbitrary exponential decay as x → −∞.

Remark 5 The assumption c_{∗} > c_{0} is less artificial than it seems to be. Indeed, in combustion theory, in the framework of large activation energies, the so-called ZFK source term satisfies this assumption; see [4]. As such, the corresponding model should share many features with the ”ignition temperature” model, investigated in [18]. This is exactly what we have just proved.

Acknowledgements It is our pleasure to thank Prof. H. Berestycki for the kind interest that he always gave to our work, for his helpful advice and useful remarks.

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