Existence of Stationary Pulses for Nonlocal

Reaction-Diffusion Equations

Vitaly Volpert, Vitali Vougalter

Received: August 23, 2014 Revised: September 14, 2014

Communicated by Heinz Siedentop

Abstract. A nonlocal reaction-diffusion equation and a system of equations from population dynamics are considered on the whole axis. Existence of solutions in the form of stationary pulses is proved by a perturbation method. It is based on spectral properties of the linearized operators and on the implicit function theorem.

2010 Mathematics Subject Classification: 35K57, 35A16, 92D15 Keywords and Phrases: reaction-diffusion equation, existence of pulse solutions, perturbation methods

1 Introduction

Nonlocal reaction-diffusion equation

\frac{∂u}{∂t} = \frac{∂^{2}u}{∂x^{2}} + au^{2}(1 − J(u)) − σu,  (1.1)

where

J(u) = ∫_{−∞}^{∞} ϕ(x − y)u(y, t)dy

describes various biological phenomena such as emergence and evolution of biological species and the process of speciation in a more general context [17], [18]. In population dynamics, u(x, t) is the density of a population, the diffusion term describes its migration, the second term in the right-hand side is the reproduction rate and the last term is the mortality rate. In the case of sexual reproduction, the reproduction rate is proportional to the second power of the population density and to the available resources (K − J(u)). Here K = 1 is the rate of production of resources and J(u) is the rate of their consumption. The integral J(u) describes nonlocal consumption of resources which shows that an individual located at the space point y consumes resources at the space point x with the rate ϕ(x − y). Nonlocal consumption of resources corresponds to the intraspecific competition and allows the description of the emergence and evolution of biological species [7]. If we replace the integral J(u) by the function u, then we obtain the usual logistic term and the local consumption of resources. An important property of nonlocal reaction-diffusion equations equations is that they have solutions in the form of periodic travelling waves [3], [7], [8], [10]. Such solutions do not exist for the usual (scalar) reaction-diffusion equations. In this work we will prove the existence of a new type of solutions of this equation in the form of stationary pulses, stationary solutions of this equation decaying at infinity. We will consider equation (1.1) in the stationary case,

w′′ + aw^{2}(1 − J(w)) − σw = 0.  (1.2)

Here x ∈ ℝ, a and σ are positive constants,

J(w) = ∫_{−∞}^{∞} ϕ(x − y)w(y)dy, ϕ(x) = \{ \begin{matrix}1 & , & |x| ≤ N \\ 0 & , & |x| > N \end{matrix} ,

N is a positive number. We will prove that for N sufficiently large equation (1.2) has a positive solution w(x) ∈ C^{2}(ℝ) with the limits

w(±∞) = 0.  (1.3)

Instead of a step-wise constant function ϕ(x) we can consider any other bounded even non-negative integrable function such that it depends on a parameter and locally converges to 1 as the parameter tends to some given value. Nonlocal Fisher-KPP equation, which is similar to equation (1.1) with nonlinearity u(1 − J(u)), also has solutions in the form of simple and periodic travelling waves [1], [2], [4], [6] - [10], [12], [14], [19]. However solutions in the form of standing pulses, stationary solutions decaying at infinity, are unlikely to exist for this equation. Spike solutions are studied for some reaction-diffusion systems [11], [13], [15], [20], [21]. The main result of this work is given by the following theorem.

Theorem 1.1. Let a > 1/h_{0}, where h_{0} is defined below (Theorem 2.1). Then equation (1.2) has an even positive solution decaying at ±∞ for all N sufficiently large.

This theorem will be proved in Section 2. The method of proof is based on the perturbation technique. If we formally replace the integral J(w) in (1.2) by the integral I(w) = ∫_{∞}^{∞} w(y)dy, then the existence of solutions for this limiting equation can be easily proved. Hence we can expect that there exists a solution for sufficiently large values of N. We will use the implicit function theorem which implies invertibility of the linearized operator. We will prove it using the Fredholm property, index and solvability conditions of the operators under consideration [16], [17]. These properties of the operators will be also used in the last section to study existence of stationary pulses of a system of two equations arising in population dynamics.

2 Existence of pulses for the scalar equation with nonlocal consumption

2.1 Existence in the case of the global consumption

In order to prove the existence of solutions of problem (1.2), (1.3) we will consider the equation

w′′ + aw^{2}(1 − I(w)) − σw = 0,  (2.1)

where

I(w) = ∫_{−∞}^{∞} w(y)dy.

By the change of variables w → w/a, h = 1/a we can reduce it to the equation

w′′ + w^{2}(1 − hI(w)) − σw = 0.  (2.2)

We will analyze the existence of classical solutions w(x) which satisfy the following properties:

w(x) > 0, x ∈ ℝ, w(x) → 0, x → ±∞, w(x) = w(−x).  (2.3)

Set

c = 1 − h ∫_{−∞}^{∞} w(y)dy  (2.4)

and consider the equation

w′′ + cw^{2} − σw = 0.  (2.5)

For each fixed positive c, there exists a unique solution of this equation satisfying (2.3). Its existence can be easily proved by the analysis of the phase plane of the system of two first-order equations,

w′ = p, p′ = −cw^{2} + σw

or by the explicit integration of the equation

\frac{dp}{dw} = \frac{1}{p} (−cw^{2} + σw)

(see below). Let us note that since σ > 0, then this solution exponentially decays at infinity. Denote this solution by w_{c}(x). Substituting it into (2.4), we obtain the equation

c = 1 − h ∫_{−∞}^{∞} w_{c}(y)dy.  (2.6)

Denote by w_{1} the solution of (2.5) with c = 1. Then w_{c} = w_{1}/c and we can write (2.6) as

c^{2} − c + h ∫_{−∞}^{∞} w_{1}(y)dy = 0.  (2.7)

This equation has two solutions if

h ∫_{−∞}^{∞} w_{1}(y)dy < \frac{1}{4} .  (2.8)

We note that for every σ fixed, solution w_{1}(x) of (2.5) with c = 1 exists and it is independent of h. Let us take a positive value of h which satisfies condition (2.8). Then equation (2.7) has two solutions, c_{1} and c_{2}, such that 0 < c_{1} < 1/2 < c_{2} < 1. If h → 0, then c_{1} → 0, c_{2} → 1. Therefore,

w_{c_{1}}(x) → ∞, w_{c_{2}}(x) → w_{1}(x), h → 0.

The first convergence occurs uniformly on every bounded interval, the second uniformly in ℝ. Denote h_{0} = 1/(4 ∫_{−∞}^{∞} w_{1}(y)dy). Then condition (2.8) is satisfied for h < h_{0}, and there are two solutions of equation (2.2).

Theorem 2.1. For any value of h such that 0 < h < h_{0}, there are two positive solutions of equation (2.2) exponentially decaying at infinity.

In the case of a > 24\sqrt{σ} we have the two pulse solutions of equation (2.1) given by the formula

w_{1,2}(x) = \frac{3σ}{2ac_{1,2} cosh^{2}(\frac{\sqrt{σ}}{2} x)} , c_{1,2} = \frac{1}{2} ± \sqrt{\frac{1}{4} − \frac{6}{a}\sqrt{σ}} .

If a = 24\sqrt{σ}, then the two solutions coincide. Finally, for 0 < a < 24\sqrt{σ}, there are no real valued pulse solutions.

2.2 Operators and spaces

Consider Hölder spaces E = C^{2+α}(ℝ) and F = C^{α}(ℝ), 0 < α < 1 and weighted spaces E_{μ} and F_{μ} defined as follows:

E_{μ} = {u : uμ ∈ E, ∥u∥_{E_{μ}} = ∥uμ∥_{E}}, F_{μ} = {u : uμ ∈ F, ∥u∥_{F_{μ}} = ∥uμ∥_{F} }.

As a weight function we take μ(x) = 1 + x^{2}. Set

A_{ϵ}(w) = w′′ + aw^{2}(1 − J_{ϵ}(w)) − σw,

A_{0}(w) = w′′ + aw^{2}(1 − J_{0}(w)) − σw,

where

J_{ϵ}(w) = ∫_{x−1/ϵ}^{x+1/ϵ} w(y)dy, J_{0}(w) = ∫_{−∞}^{∞} w(y)dy.

We will consider the operator A_{ϵ} for ϵ > 0 and ϵ = 0 as defined above. We can extend it for negative ϵ by symmetry, A_{ϵ} = A_{−ϵ}, ϵ < 0. It is a bounded and continuous operator acting from E_{μ} into F_{μ}. We will show that it is continuous with respect to ϵ.

Lemma 2.2. For any δ > 0 there exists ϵ_{0} such that

∥A_{ϵ}(w) − A_{0}(w)∥_{F_{μ}} < δ, ∀ϵ, w, 0 < ϵ ≤ ϵ_{0}, ∥w∥_{E_{μ}} ≤ M,  (2.9)

where ϵ_{0} can depend on M.

Proof. We have

A_{0}(w) − A_{ϵ}(w) = aw^{2}(J_{ϵ}(w) − J_{0}(w)).

Set

g(x) = μ(x)w^{2}(x)(J_{ϵ}(w) − J_{0}(w)).

We should estimate the Hölder norm of the function g. Let us begin with the uniform norm. Since

|μ(x)w(x)| ≤ M, |w(x)| ≤ \frac{M}{μ(x)} ,

then we have the estimate

|J_{ϵ}(w)|, |J_{0}(w)| ≤ M_{1}

with some positive constant M_{1}. Hence for any δ > 0 we can choose x_{0} > 0 such that

|g(x)| ≤ δ for |x| ≥ x_{0}.

We will now obtain a similar estimate for |x| < x_{0}. We have

g(x) = −μ(x)w^{2}(x) (∫_{−∞}^{x−1/ϵ} w(y)dy + ∫_{x+1/ϵ}^{∞} w(y)dy) .  (2.10)

We can choose ϵ_{0} such that for any ϵ ≤ ϵ_{0} the estimates

|μ(x)w^{2}(x) ∫_{x+1/ϵ}^{∞} w(y)dy| < \frac{δ}{2} , |μ(x)w^{2}(x) ∫_{−∞}^{x−1/ϵ} w(y)dy| < \frac{δ}{2} , ∀|x| ≤ x_{0}

hold. Hence we have the estimate

sup_{x∈ℝ} |g(x)| ≤ δ.  (2.11)

Next, we should estimate the expression

H = sup_{x_{1},x_{2}∈ℝ} \frac{|g(x_{1}) − g(x_{2})|}{|x_{1} − x_{2}|^{α}} .

Obviously, it is sufficient to consider the case where |x_{1} − x_{2}| < 1. We can proceed as before and consider either |x| > x^{∗} for some x^{∗} sufficiently large or |x| ≤ x^{∗}. In the first case, H is small since μ(x)w(x) and the integrals in (2.10) are bounded in the Hölder norm while w(x) → 0 in the Hölder norm as |x| → ∞. In the second case, we use the fact that the integrals converge to 0 in the Hölder norm as ϵ → 0. The lemma is proved.

□

In what follows we will also consider the subspaces of the spaces E_{μ} and F_{μ} which consist of even functions:

E_{μ}^{0} = {u ∈ E_{μ}, u(x) = u(−x), ∀x ∈ ℝ},
F_{μ}^{0} = {u ∈ F_{μ}, u(x) = u(−x), ∀x ∈ ℝ}.

If w ∈ E_{μ}^{0}, then J_{ϵ}(w) is also an even function and A_{ϵ}(w) ∈ F_{μ}^{0}. Therefore we can consider this operator acting from E_{μ}^{0} into F_{μ}^{0}.

2.3 Linearized operator

Consider the linearized operator to the operator A_{0}(w),

Lu = u′′ + 2auw_{0}(1 − I(w_{0})) − aw_{0}^{2}I(u) − σu

for some w_{0} ∈ E_{μ}, and the formally adjoint operator

L^{∗}v = v′′ + 2avw_{0}(1 − I(w_{0})) − aI^{∗}(v) − σv,

where

I^{∗}(v) = ∫_{−∞}^{∞} w_{0}^{2}(x)v(x)dx.

We will consider the operator L acting from E_{μ} into F_{μ} and the operator L^{∗} from H_{∞}^{2}(ℝ) into L_{∞}^{2}(ℝ) [16].

Let us recall the definition of the function spaces. Denote by D the space of infinitely differentiable functions with bounded supports and by D′ its dual. Let E ⊂ D′ be a Banach space. Then E_{loc} is defined as ensemble of all u ∈ D′ such that fu ∈ E for all f ∈ D. Let, further, ϕ_{i} ∈ D be a partition of unity. Then the space E_{∞} is defined as ensemble of all u ∈ E_{loc} such that sup_{i} ∥uϕ_{i}∥_{E} < ∞. The norm in this space is given by the equality

∥u∥_{∞} = sup_{i} ∥uϕ_{i}∥_{E}.

The definition of the space E_{∞} is applicable for any Banach space E. If E = L^{2}(ℝ), then the corresponding space is denoted by L_{∞}^{2}(ℝ). For E = H^{2}(ℝ), it becomes H_{∞}^{2}(ℝ). The norms in these spaces are given by the equalities:

∥u∥_{L_{∞}^{2}(ℝ)} = sup_{i} ∥ϕ_{i}u∥_{L^{2}(ℝ)}, ∥u∥_{H_{∞}^{2}(ℝ)} = sup_{i} ∥ϕ_{i}u∥_{H^{2}(ℝ)}.

The operator L and L^{∗} are linear bounded operators in the corresponding spaces. They satisfy the relation

∫_{−∞}^{∞} v(x)(Lu)dx = ∫_{−∞}^{∞} u(x)(L^{∗}v)dx.

Let w_{0}(x) be an even positive solution of equation (2.1). Set u_{0} = −w′_{0}. Differentiating this equation, we obtain

L_{0}u_{0} ≡ u′′_{0} + 2au_{0}w_{0}(1 − I(w_{0})) − σu_{0} = 0.

Since I(u_{0}) = I^{∗}(u_{0}) = 0, then Lu_{0} = L^{∗}u_{0} = 0. Hence u_{0} is the eigenfunction corresponding to the zero eigenvalue of both operators. The eigenvalue λ = 0 of the operators L_{0} : E → F is simple. Indeed, if there are two linearly independent bounded eigenfunctions, then all solutions of the equation L_{0}u = 0 are bounded as their linear combination. However it has exponentially growing solutions. We can now summarize the spectral properties of the operator L_{0}. Its essential spectrum lies in the left-half plane. Its principal eigenvalue is simple and positive, and the corresponding eigenfunction is positive, according to the standard Sturm-Liouville theory. It has a simple zero eigenvalue with the eigenfunction u_{0}(x) = −w′_{0}(x) which is positive for positive x and negative for negative x. It can be verified that it does not have other positive eigenvalues except for the principal eigenvalue since u_{0}(x) has a unique zero at the origin. These properties remain true for more general nonlinearities.

Lemma 2.3. If I(w_{0}) ≠ 1/2 (a > 24\sqrt{σ}), then the equation L^{∗}v = 0 has a unique, up to a constant factor, solution u_{0}.

Proof. Suppose that v_{0} is an eigenfunction corresponding to the zero eigenvalue of the operator L^{∗}. Then

v′′_{0} + 2av_{0}w_{0}(1 − I(w_{0})) − aI^{∗}(v_{0}) − σv_{0} = 0.

Multiplying this equality by w_{0} and integrating, we obtain

− ∫_{−∞}^{∞} v′_{0}w′_{0}dx + 2aI^{∗}(v_{0})(1 − I(w_{0})) − aI(w_{0})I^{∗}(v_{0}) − σ ∫_{−∞}^{∞} v_{0}w_{0}dx = 0.

(2.12)

Since w_{0} is a solution of equation (2.1), we multiply the equation

w′′_{0} + aw_{0}^{2}(1 − I(w_{0})) − σw_{0} = 0  (2.13)

by v_{0} and integrate:

− ∫_{−∞}^{∞} v′_{0}w′_{0}dx + aI^{∗}(v_{0})(1 − I(w_{0})) − σ ∫_{−∞}^{∞} v_{0}w_{0}dx = 0.  (2.14)

Subtracting this equation from equation (2.12), we get

I^{∗}(v_{0})(1 − I(w_{0})) − I(w_{0})I^{∗}(v_{0}) = 0.

If I(w_{0}) ≠ 1/2, then I^{∗}(v_{0}) = 0. Hence v_{0} is an eigenfunction of the operator L_{0} corresponding to the zero eigenvalue. Since this eigenfunction is unique up to a constant factor, we get v_{0} = u_{0}.

□

Remark 2.4. We proved in Section 2.1 that I(w_{0}) = 1/2 corresponds to the bifurcation point where solutions of equation (2.1) appear due to a subcritical bifurcation. For these values of parameters, the zero eigenvalue of the operator L^{∗} is double, because of the bifurcation and of the invariance with respect to translation. The previous lemma affirms that outside of the bifurcation point this eigenvalue is simple.

Lemma 2.5. If I(w_{0}) ≠ 1/2 (a > 24\sqrt{σ}), then equation Lu = 0 has a unique solution u = u_{0} in E_{μ}.

Proof. The operators L : E_{μ} → F_{μ} and L^{∗} : H_{∞}^{2}(ℝ) → L_{∞}^{2}(ℝ) satisfy the Fredholm property and have the zero index. It follows from Lemma 2.3 that the equation L^{∗}v = 0 has a unique solution. Therefore, since the index equals zero, the nonhomogeneous equation L^{∗}v = f has a unique solvability condition. Suppose that equation Lu = 0 has two linearly independent solutions u_{0}, u_{1} ∈ E_{μ}. Then equation L^{∗}v = f has at least two solvability conditions. Indeed, we can choose a function f ∈ L_{∞}^{2}(ℝ) such that

∫_{−∞}^{∞} f(x)u_{0}(x)dx = 0, ∫_{−∞}^{∞} f(x)u_{1}(x)dx ≠ 0.

If equation L^{∗}v = f has a solution, then we multiply this equation by u_{1} and integrate over ℝ. We get

∫_{−∞}^{∞}(L^{∗}v)u_{1}dx = ∫_{−∞}^{∞} v(Lu_{1})dx = 0, ∫_{−∞}^{∞}(L^{∗}v)u_{1}dx = ∫_{−∞}^{∞} fu_{1}dx ≠ 0.

This contradiction proves the lemma.

□

It follows from the lemma that a real eigenvalue of the linearized operator cannot cross the origin and change stability of the solution.

2.4 Existence in the case of nonlocal consumption

We will prove existence of solutions of equation (1.2) by the implicit function theorem. We consider the operator A_{ϵ}(w) : E_{μ}^{0} → F_{μ}^{0}. It is bounded and continuous. We suppose that the equation A_{0}(w) = 0 has a solution w_{0}. Conditions of the existence of solutions are given in Section 2.1. Let us consider the Fréchet derivative of the operator A_{ϵ}(w):

A′_{ϵ}(w)u = u′′ + 2aw(x)(1 − J_{ϵ}(w))u − σu − aw^{2}(x)J_{ϵ}(u).

Lemma 2.6. The operator A′_{ϵ}(w) is continuous with respect to w and ϵ in the operator norm.

The proof of the lemma is standard and we omit it.

Lemma 2.7. If I(w_{0}) ≠ 1/2 (a > 24\sqrt{σ}), then the operator A′_{0}(w_{0}) : E_{μ}^{0} → F_{μ}^{0} is invertible.

Proof. Consider the equation

A′_{0}(w_{0})u = f  (2.15)

for an arbitrary f ∈ F_{μ}^{0}. Since f is an even function and u_{0} is odd, u_{0}(x) = −w′_{0}(x), then equality

∫_{−∞}^{∞} f(x)u_{0}(x)dx = 0

holds. It is the unique solvability condition for equation (2.15). Indeed, since it is a Fredholm operator with the zero index and its kernel has dimension 1 (Lemma 2.5), then the codimension of its image is also one-dimensional. Therefore equation (2.15) has a solution u_{1} ∈ E_{μ}. Since A′_{0}(w_{0})u_{0} = 0, then any function v_{k}(x) = u_{1}(x) + ku_{0}(x) is a solution of this equation for any real k. Let us verify that only one of them belongs to E_{μ}^{0}. Since f(x) and w_{0}(x) are even functions, then along with solution u_{1}(x), the function u_{1}(−x) is also a solution of this equation. Set z(x) = u_{1}(x) − u_{1}(−x). Since z(x) is a solution of the homogeneous equation, then

u_{1}(x) − u_{1}(−x) = k_{1}u_{0}(x),  (2.16)

where k_{1} is a constant. Then it is possible to choose a number k_{2} such that the function v_{k_{2}}(x) = u_{1}(x) + k_{2}u_{0}(x) is even. Indeed, from the equality v_{k_{2}}(x) = v_{k_{2}}(−x) we get

u_{1}(x) + k_{2}u_{0}(x) = u_{1}(−x) + k_{2}u_{0}(−x).

Since u_{0}(x) is an odd function, from the last equality and (2.16) we obtain k_{2} = −k_{1}/2. Hence we proved that there exists an even solution of equation (2.15). Let us verify that it is unique. If there are two such solutions z_{1}(x) and z_{2}(x), then their difference satisfies the homogeneous equation. Hence z_{1}(x) − z_{2}(x) = k_{3}u_{0}(x). Since the difference of two even function is an even function, and u_{0}(x) is an odd function, then this equality can hold only for k_{3} = 0. Hence the two even solutions coincide. Thus equation (2.15) has a unique solution in E_{μ}^{0} for any f ∈ F_{μ}^{0}. By the Banach theorem, the operator A′_{0}(w_{0}) has a bounded inverse.

□

We can now proof Theorem 1.1.

Proof of Theorem 1.1. Consider the operator A_{ϵ}(w) : E_{μ}^{0} → F_{μ}^{0}. It is bounded, continuous and equation A_{0}(w) = 0 has a solution w_{0}. The Fréchet derivative A′_{ϵ}(w) is a bounded linear operator, continuous with respect to w and ϵ in the operator norm. Finally, the operator A′_{0}(w_{0}) is invertible. By the implicit function theorem equation A_{ϵ}(w) = 0 has a unique solution from E_{μ}^{0} in the vicinity of the function w_{0} for all ϵ sufficiently small.

□

Let us note that under the conditions of the theorem, equation (2.1) has two pulse solutions. Theorem 2.8 affirms the existence of pulse solutions of equation (1.2) in the vicinity of these solutions.

3 System of nonlocal equations

Existence of solutions of equation (1.2) is proved in Theorem 1.1 by the perturbation technique. We use the existence of solutions for equation (2.1) and spectral properties of the linearized operator L studied in Section 2.3. A similar method can be used in other applications. In this section we consider the system of equations

d_{1}u′′ + auv(1 − I(u) − I(v)) − σu = 0,  (3.1)

d_{2}v′′ + auv(1 − I(u) − I(v)) − σv = 0  (3.2)

which describes the distribution of a population in the space of phenotypes. Here u is the density of males, v is the density of females. The second term in the left-hand sides of these equations is the reproduction rate which is proportional to the product uv and to available resources (1 − I(u) − I(v)). The last terms are their mortality. It is assumed that both parents have the same phenotype. We will look for a positive solution of this system with the limits at infinities

u(±∞) = v(±∞) = 0.  (3.3)

Diffusion terms in these equations correspond to genetic variability which shows how the phenotypes of offsprings differ from the phenotype of parents. If d_{1} = d_{2}, then taking the difference of two equations, we get u = v. In this case we can reduce the system of equations to the scalar equation (2.1). However these two coefficients can differ from each other since genetic variability of males is usually greater than that of females. We will prove here the existence of solutions of problem (3.1)-(3.3) in the case where the difference between diffusion coefficients is sufficiently small. As before we will use the existence of solutions for equation (2.1) and spectral properties of the linearized operator. We write system (3.1), (3.2) in the form

u′′ + a_{0}uv(1 − I(u) − I(v)) − σ_{0}u = 0,  (3.4)

v′′ + a_{ϵ}uv(1 − I(u) − I(v)) − σ_{ϵ}v = 0,  (3.5)

where a_{0} = a/d_{1}, σ_{0} = σ/d_{1}, a_{ϵ} = a_{0}+ϵ, σ_{ϵ} = σ_{0}+ϵ. If ϵ = 0, then u = v = w/2, where w is a solution of the equation

w′′ + \frac{a_{0}}{2} w^{2}(1 − I(w)) − σ_{0}w = 0, w(±∞) = 0.  (3.6)

Consider next the system linearized about w for ϵ = 0:

ũ′′ + \frac{a_{0}}{2} ũw(1−I(w))+ \frac{a_{0}}{2} ṽw(1−I(w))− \frac{a_{0}}{4} w^{2}(I(ũ)+I(ṽ))−σ_{0}ũ = 0, (3.7)

ṽ′′ + \frac{a_{0}}{2} ũw(1−I(w))+ \frac{a_{0}}{2} ṽw(1−I(w))− \frac{a_{0}}{4} w^{2}(I(ũ)+I(ṽ))−σ_{0}ṽ = 0. (3.8)

Set z = ũ−ṽ. Taking the difference of these two equations, we get the equation for z:

z′′ − σ_{0}z = 0, z(±∞) = 0.

Therefore z ≡ 0 and ũ ≡ ṽ. Hence system (3.7), (3.8) can be reduced to the single equation:

ũ′′ + a_{0}ũw(1 − I(w)) − \frac{a_{0}}{2} w^{2}I(ũ) − σ_{0}ũ = 0.  (3.9)

It coincides with the equation obtained as the linearization of equation (3.6). Due to Lemma 2.5 it has a unique solution ũ_{0} ∈ E_{μ} if a_{0} > 2/h_{0}. Hence system (3.7), (3.8) also has a unique even solution ũ = ṽ = ũ_{0}. Similar to Lemma 2.7 we can now conclude that the corresponding operator is invertible on the subspace of even functions. We can now formulate the existence theorem.

Theorem 3.1. Let a_{0} > 2/h_{0}, where h_{0} is defined in Theorem 2.1. Then system (3.4), (3.5) has an even positive solution decaying at infinities for all ϵ sufficiently small.

The proof of this theorem is similar to the proof of Theorem 2.8.

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[9] S.A. Gourley. Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), no. 3, 272–284.

[10] S.A. Gourley, M.A.J. Chaplain, F.A. Davidson. Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation. Dyn. Syst., 16 (2001), no. 2, 173–192.

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[12] G. Nadin, L. Rossi, L. Ryzhik, B. Perthame. Wave-like solutions for nonlocal reaction-diffusion equations: a toy model. Math. Model. Nat.Phenom., 8 (2013), no. 3, 33–41.

[13] Y. Nec, M. J. Ward. The stability and slow dynamics of two-spike patterns for a class of reaction-diffusion system. Math. Model. Nat. Phenom., 8 (2013), no.5, 206–232.

[14] B. Perthame, S. Genieys. Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit. Math. Model. Nat. Phenom., 2 (2007), no. 4, 135–151.

[15] J.C. Tzou, A. Bayliss, B.J. Matkowsky, V.A. Volpert. Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model. European J. Appl. Math, 22 (2011), no. 5, 423-453.

[16] V. Volpert. Elliptic partial differential equations. Volume 1. Fredholm theory of elliptic problems in unbounded domains. Birkhäuser, 2011.

[17] V. Volpert. Elliptic partial differential equations. Volume 2. Reactiondiffusion equations. Birkhäuser, 2014.

[18] V. Volpert, S. Petrovskii. Reaction-diffusion waves in biology. Physics of Life Reviews, 6 (2009), 267—310.

[19] V. Volpert, V. Vougalter. Emergence and propagation of patterns in nonlocal reaction-diffusion equations arising in the theory of speciation. In: Dispersal, individual movement and spatial ecology. M. Lewis, Ph. Maini, S. Petrovskii. Editors. Springer Applied Interdisciplinary Mathematics Series. Lecture Notes in Mathematics, Volume 2071, 2013, 331-353.

[20] J. Wei, M. Winter. Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R^{1}. Methods Appl. Anal., 14 (2007), no. 2, 119-163.

[21] F. Zhang. Coexistence of a pulse and multiple spikes and transition layers in the standing waves of a reaction-diffusion system. J. Differential Equations, 205 (2004), no. 1, 77–155.

Vitaly Volpert

Institut Camille Jordan UMR 5208 CNRS University Lyon 1 69622 Villeurbanne France volpert@math.univ-lyon1.fr Vitali Vougalter University of Toronto Department of Mathematics Toronto, Ontario M5S 2E4 Canada vitali@math.toronto.edu 1154

Documenta Mathematica 19 (2014)