EXISTENCE AND STABILTY OF MULTIDIMENSIONAL TRAVELLING WAVES IN THE MONOSTABLE CASE

V.A. Volpert Laboratoire d’analyse numérique, Université Lyon I , UMR 5585 CNRS 69622 Villeurbanne Cédex, France

A.I. Volpert Department of Mathematics, Technion, 32000 Haifa, Israel

Abstract. The paper is devoted to existence and stability of travelling waves described by monotone parabolic systems in the monostable case. The first initial-boundary value problem is considered. Existence of waves is proved for all values of the velocity greater or equal to the minimal velocity. A minimax representation for the minimal velocity is obtained. Stability of monotone waves in a weighted norm is proved.

1. Introduction

We consider the parabolic system of equations

\frac{∂u}{∂t} = a(x′)Δu + ∑_{j=1}^{m} b_{j}(x′)\frac{∂u}{∂x_{j}} + F (u, x′),  (1.1)

in a cylinder Ω = Ω′ × R. Here u(x) = (u_{1}(x), ..., u_{n}(x)), x = (x_{1}, ..., x_{m}), x_{1} is the variable along the axis of the cylinder, x′ = (x_{2}, ..., x_{m}) is the variable in the section Ω′ of the cylinder, the domain Ω′ is bounded and has a boundary of the class C^{2+δ} with some positive δ, a(x′), b_{j}(x′) are smooth diagonal matrices,

a_{i}(x′) ≥ a_{0} > 0, x′ ∈ Ω′, i = 1, ..., n, j = 1, ..., m,

where a_{i} are diagonal elements of the matrix a, F = (F_{1}, ..., F_{n}) is a smooth vector-valued function. On the boundary ∂Ω of the cylinder we consider the boundary conditions

u(x) = ϕ(x′), x ∈ ∂Ω,  (1.2)

where ϕ(x′) ∈ C^{2+δ}(Ω̄) depends only on the variable in the section of the cylinder. Travelling wave solution of the problem (1.1), (1.2) is a solution of the form

u(x, t) = w(x_{1} − ct, x_{2}, ..., x_{m}),

where c is an unknown constant, the wave velocity. This function is a solution of the problem

a(x′)Δw + c\frac{∂w}{∂x_{1}} + ∑_{j=1}^{m} b_{j}(x′)\frac{∂w}{∂x_{j}} + F (w, x′) = 0,  (1.3)

w(x) = ϕ(x′), x ∈ ∂Ω.

We assume that for n > 1 the nonlinearity F satisfies the following condition

\frac{∂F_{i}}{∂u_{j}} ≥ 0, i ≠ j, i, j = 1, ..., n.

This condition means that we can use comparison theorems for the systems under consideration. For n = 1 there are no additional conditions and the comparison theorems are also applicable. The systems of this type arise in numerous applications (see [15]).

We look for the travelling waves having limits at infinity:

lim_{x_{1}→±∞} w(x) = w_{±}(x′),  (1.4)

where the functions w_{±} are solutions of the problem in the section of the cylinder

a(x′)Δ′w_{±} + ∑_{j=2}^{m} b_{i}(x′)\frac{∂w_{±}}{∂x_{j}} + F (w_{±}, x′) = 0,  (1.5)

w_{±}(x′) = ϕ(x′), x′ ∈ ∂Ω′,  (1.6)

where Δ′ is the Laplace operator with respect to the variables in the section of the cylinder.

We recall the classification of the problems according to stability of solutions w_{+} and w_{−}. We consider the eigenvalue problem for the corresponding linearized equation

L^{±}u ≡ a(x′)Δ′u + ∑_{j=2}^{m} b_{j}(x′)\frac{∂u}{∂x_{j}} + F′(w_{±}(x′), x′)u = λu  (1.7)

u = 0, x′ ∈ ∂Ω′.  (1.8)

If all eigenvalues of both operators L^{+} and L^{−} are in the left half-plane, then it is so-called bistable case. If for one of them there are eigenvalues in the right half-plane and for another one all eigenvalues have negative real parts, it is the monostable case. Finally, in the unstable case both operators have eigenvalues in the right half-plane. As it is well known (see [15]) properties of travelling waves are different in these three cases.

In this work we study the monostable case. We suppose that all eigenvalues of the operator L^{−} have negative real parts, and there are eigenvalues of the operator L^{+} with positive real parts. For one-dimensional scalar equation (m = n = 1) this problem was first studied in [7] and then followed by a large number of other works (see [15] for the references). For the multidimensional scalar equation the problem was considered in [2]-[4], [6], [14] and for one-dimensional systems of equations in [15], [17]. In this work we consider multidimensional systems of equations and develop the methods and results suggested in our previous works for one-dimensional systems.

Proof of the existence of travelling waves is based on the comparison theorems which are valid for the systems under consideration. We first prove existence of solutions of some auxiliary problems in half-cylinders and then, passing to the limt, we obtain existence of solutions in the whole cylinder.

Stability of travellig waves with respect to small perturbations is based on the analysis of the spectrum of the linearized operator which can have an independent interest. Consider the linear operator

Lu = a(x)Δu + ∑_{j=1}^{m} b_{j}(x)\frac{∂u}{∂x_{j}} + c(x)u

assuming that the matrices a, b_{j}, c are sufficiently smooth, a and b_{j} are diagonal, c has nonnegative off-diagonal elements and is functionally irreducible. We consider this operator on functions u ∈ C^{2+δ}(Ω̄) satisfying the boundary condition u = 0 on ∂Ω. Denote σ_{e} the supremum of the real parts of the points of the essential spectrum of this operator, and σ_{d} of its discrete spectrum. In [18] we showed that if σ_{d} > σ_{e}, then the principal eigenvalue of the operator is real, simple, and the corresponding eigenfunction is positive. Since the derivative of the monotone in x_{1} wave is an eigenfunction corresponding to zero eigenvalue, then it allows to conclude that all other eigenvalues are in the left half-plane and to prove the stability of the monotone wave.

However we can make this conclusion only in the case if the essential spectrum lies in the left half-plane, i.e. σ_{e} < σ_{d}. This inequality holds in the bistable case, but not in the monostable case considered in this work. In the monostable case σ_{e} > σ_{d} and to move the essential spectrum to the left half-plane we introduce a weighted norm. If the derivative of the wave belongs to this weighted space, we can still use the result above. So the question is what happens if it does not belong to this space. In the other words we have a positive solution v of the equation Lu = 0 which should be understood as a local equality because the function v may not belong to C^{2+δ}(Ω̄) and may be even unbounded. We show that in this case all eigenvalues of the operator lie in the left half-plane. It is a generalization of the results obtained in our previous work [17] for one-dimensional systems.

The contents of the paper are as follows. In Section 2 we prove existence of waves for all values of velocities greater or equal to the minimal velocity c_{0}. We obtain a miminax representation for the minimal velocity and show that there are no waves with the velocity less than c_{0}. In Section 3 we prove stability of waves in a weighted norm.

2. Existence of waves

2.1. Main theorem. We make the following assumptions on the solutions w_{+} and w_{−} of the problem (1.5), (1.6).

Assumption 1. The functions w_{±} belong to C^{(2+δ)}(Ω̄′) and the following inequality

w_{+}(x′) < w_{−}(x′), x′ ∈ Ω′  (2.1)

holds.

Assumption 2. There is no other solutions of the problem (1.5), (1.6) satisfying the inequality

w_{+}(x′) ≤ w ≤ w_{−}(x′), x′ ∈ Ω′  (2.2)

Assumption 3. There exists a sequence of functions {v_{n}(x′)} uniformly bounded in C^{(2+δ)}(Ω̄′) satisfying the inequality (2.2) and the following conditions:

v_{n}(x′) → w_{+}(x′) as n → ∞  (2.3)

uniformly in x′,

a(x′)Δ′v_{n} + ∑_{j=2}^{m} b_{j}(x′)\frac{∂v_{n}}{∂x_{j}} + F (v_{n}, x′) ≥ 0  (2.4)
v_{n}(x′) = ϕ(x′), x′ ∈ ∂Ω′.  (2.5)

The last assumption can be formulated in terms of eigenvalues of the linearized problem. We discuss it below.

We prove first the theorem on wave existence under Assumptions 1-3. We will show below that Assumptions 2 is not always necessary and can be replaced by a more weak assumption.

Theorem 2.1 Let Assumptions 1-3 be satisfied. Then there exists a constant c_{0} such that for every c ≥ c_{0} there exists a monotone in x_{1} solution of the problem (1.3), (1.4). The constant c_{0} is given by the minimax representation

c_{0} = inf_{ρ∈K} sup_{x∈Ω, i=1,...,n} B_{i},  (2.6)

where

B_{i} = (a_{i}(x′)Δρ_{i} + ∑_{j=1}^{m} b_{ij}(x′)\frac{∂ρ_{i}}{∂x_{j}} + F_{i}(ρ, x′)) / (−\frac{∂ρ_{i}}{∂x_{1}} ) ,  (2.7)

b_{ij} are diagonal elements of the matrix b_{j}, K is a class of sufficiently smooth vector-valued functions ρ(x) such that

lim_{x_{1}→±∞} ρ(x) = w_{±}(x′), ρ(x) = ϕ(x′) for x ∈ ∂Ω,  (2.8)

the derivative ∂ρ/∂x_{1} is negative and the normal derivative ∂(ρ − w_{+}(x))/∂ν, x ∈ ∂Ω in the direction of the outer normal is also negative.

For c < c_{0} such solutions do not exist.

Remark. We show below that there exist a function ρ(x) such that B_{i} is bounded and that the minimal value of the velocity given by (2.6) is finite.

Proof. We take c > c_{0}. Then there exists a function ρ for which sup_{x,i} B_{i} < c. It means that

a(x′)Δρ + c\frac{∂ρ}{∂x_{1}} + ∑_{j=1}^{m} b_{j}(x′)\frac{∂ρ}{∂x_{j}} + F (ρ, x′) < 0  (2.9)

for all x ∈ Ω. For any given N we can choose a function v_{k}(x′) which satisfies Assumption 3 and such that

w_{+}(x′) ≤ v_{k}(x′) ≤ ρ(x) |_{x_{1}=N} .  (2.10)

Indeed, since ρ → w_{+} as x_{1} → ∞ and the derivative ∂ρ/∂x_{1} is negative, then for all internal points x′ we have the inequality

w_{+}(x′) < ρ(x) |_{x_{1}=N} .  (2.11)

On the other hand, v_{k} → w_{+} as k → ∞. Then for k sufficiently large the inequality (2.10) holds in any closed domain Ω0^{′} ⊂ Ω′.

From the convergence (2.3) and the uniform boundedness of the second derivatives of the functions v_{k} follows the convergence of the first derivatives:

\frac{∂v_{k}}{∂x_{i}} − \frac{∂w_{+}}{∂x_{i}} → 0, i = 1, ..., m  (2.12)

uniformly in x′. Since the derivative ∂(ρ − w_{+}(x′))/∂ν, x_{1} = N is negative, we obtain the inequality (2.10) for all x′ ∈ Ω′ and sufficiently large k.

We introduce a domain Ω_{N} ⊂ Ω located in the half-space {x_{1} ≤ N} and such that Ω_{N} = Ω for x_{1} ≤ N − 1, and the boundary ∂Ω_{N} bellongs to C^{(2+δ)}. We consider now the initial-boundary value problem for the equation

\frac{∂u}{∂t} = Δu + a^{−1}(x′) (c\frac{∂u}{∂x_{1}} + ∑_{j=1}^{m} b_{j}(x′)\frac{∂u}{∂x_{j}} + F (u, x′) − f(x′)e^{−t})  (2.13)

in the domain Ω_{N} with the initial condition

u(x, 0) = v_{k}(x′),  (2.14)

and the boundary conditions

u(x, t) |_{∂Ω_{N}} = v_{k}(x′).  (2.15)

Here

f(x′) = a(x′)Δ′v_{k} + ∑_{j=2}^{m} b_{j}(x′)\frac{∂v_{k}}{∂x_{j}} + F (v_{k}, x′).

To show existence of the classical solution of this problem, we construct a sequence of bounded domains Ω_{N}^{i}, i = 1, 2, ... with sufficiently smooth boundary and such that

Ω_{N}^{i} ⊂ Ω_{N}^{i+1}, ∪_{i=1}^{∞}Ω_{N}^{i} = Ω_{N}

and consider equation (2.13) with the initial condition (2.14) in the domain Ω_{N}^{i} with the boundary condition

u(x, t) |_{∂Ω_{N}^{i}} = v_{k}(x′).  (2.16)

We note that the compatibility condition for the problem (2.13), (2.14), (2.16) is satisfied. So the classical solution exists (see [8]).

We show that the solution of the problem (2.13), (2.14), (2.16) satisfies the inequality

u(x, t) ≤ ρ(x).

Indeed, the function

z(x, t) = u(x, t) − ρ(x)

satsifes the equation

\frac{∂z}{∂t} = Δz + a^{−1}(x′) (c\frac{∂z}{∂x_{1}} + ∑_{j=1}^{m} b_{j}(x′)\frac{∂z}{∂x_{j}} + F (u, x′) − F (ρ, x′) − f(x′)e^{−t} − g) ,

where

g(x) = − (a(x′)Δρ + c\frac{∂ρ}{∂x_{1}} + ∑_{j=1}^{m} b_{j}(x′)\frac{∂ρ}{∂x_{j}} + F (ρ, x′)) > 0.

Since z(x, 0) ≤ 0, z(x, t) |_{∂Ω_{N}^{i}} ≤ 0, f(x′) and g(x) are nonnegative, then z(x, t) ≤ 0 for all t ≥ 0 and x ∈ Ω_{N}^{i}.

We note that the solution of the problem (2.13), (2.14), (2.16) increases in time. Indeed, the function v = ∂u/∂t satisfies the problem

\frac{∂v}{∂t} = Δv + a^{−1} (c\frac{∂v}{∂x_{1}} + ∑_{k=1}^{n} b_{k} \frac{∂v}{∂x_{k}} + F′(u, x′)v + fe^{−t}) ,

v |_{t=0}= 0, v |_{∂Ω_{N}^{i}} = 0.

Then v(x, t) ≥ 0 and u(x, t) ≥ v_{k}(x′).

So the solution u_{N}^{i}(x, t) of the problem (2.13), (2.14), (2.16) is bounded independently of i in C and, consequently, in the C^{(2+δ)}-norm. Choosing a converging subsequence, we obtain a classical solution u_{N}(x, t) of the problem (2.13)-(2.15). As before we show that

w_{+}(x′) ≤ v_{k}(x′) ≤ u_{N}(x, t) ≤ ρ(x)

and u_{N}(x, t) is increasing in t. Passing to the limit as t → ∞, we obtain a solution w_{N}(x) of the equation (1.3) in the domain Ω_{N} with the boundary condition

w_{N} |_{∂Ω∩∂Ω_{N}} = ϕ(x′).

It is bounded in the C^{(2+δ)}-norm independently of N. Moreover

w_{+}(x′) ≤ w_{N}(x) ≤ ρ(x).

We note also that ∂w_{N}/∂x_{1} ≤ 0. Indeed, the function z = ∂u_{N}/∂x_{1} satisfies the problem

\frac{∂z}{∂t} = Δz + a^{−1} (c\frac{∂z}{∂x_{1}} + ∑_{k=1}^{n} b_{k} \frac{∂z}{∂x_{k}} + F′(u, x′)z) ,

z|_{∂Ω} = 0, z|_{t=0} = 0,

and z|_{∂Ω_{N}} ≤ 0 since u_{N}(x, t) ≥ v(x′). Then z ≤ 0.

Since the function w_{N}(x) is monotone in x_{1}, then there exists the limit

w_{N}^{−}(x′) = lim_{x_{1}→−∞} w_{N}(x).

It satisfies the problem (1.5), (1.6) and the inequality

v_{k}(x′) ≤ w_{N}^{−}(x′) ≤ w_{−}(x′).

Then w_{N}^{−} = w_{−}.

Thus we have constructed the sequence of functions {w_{N}}, N = 1, 2, ... defined for x_{1} ≤ N. We show now how to construct the solution in the whole cylinder Ω.

We choose an arbitrary interior point in Ω. Without loss of generality we can assume that it is x = 0. Denote x_{N} = (x_{1}_{N}, 0^{n−1}), where 0^{n−1} is a n − 1-vector with zero elements, x_{1}_{N} is a solution of the equation

w_{N}_{1}(x_{1}, 0^{n−1}) = ρ_{1}(0, 0^{n−1}).

Here the index 1 shows the number of the component of the vector-valued function. Existence and uniqueness of the solution follows from the monotonicity of the function w_{N} in x_{1}. Denote further w̃_{N}(x) = w_{N}(x + x_{N}) so that w̃_{N}_{1}(0) = ρ_{1}(0).

Since w_{N}_{1}(0) ≤ ρ_{1}(0), then x_{N} ≤ 0 and the functions w̃_{N} are defined at least for x_{1} ≤ N. Since the derivatives of this functions are uniformly bounded, then for any given M from the sequence {w̃_{N}} we can chose a subsequence converging to some limiting function uniformly on the set Ω ∩ {[−M, M]}. If we consider now a sequence of M going to infinity and choosing each time a converging subsequence, we obtain the limiting function w(x) defined in the whole Ω. It is a solution of equation (1.3) as a limit of solutions of this equation, and it satisfies the boundary conditions (1.2). The function w(x) has a nonengative derivative with respect to x_{1}. Hence there exist the limits of w(x) as x_{1} → ±∞. We have

w_{+}_{1}(0) < w_{N}_{1}(0) = ρ_{1}(0) < w_{−}_{1}(0).

Since the limits lim_{x_{1}→±∞} w(x) satisfy the problem (1.5), (1.6), then we necessarily obtain (1.4).

We have already mentioned that the derivative ∂w/∂x_{1} is not positive. We show now that it is strictly negative assuming that the matrix ∂F_{i}/∂w_{j} is functionally irreducible. Indeed, the function y = ∂w/∂x_{1} is a solution of the problem

a(x′)Δy + c\frac{∂y_{i}}{∂x_{1}} + ∑_{j=1}^{m} b_{ij}(x′)\frac{∂y_{i}}{∂x_{j}} + ∑_{j=1}^{n} \frac{∂F_{i}(w(x)))}{∂w_{j}}y_{i} = 0,

y|_{∂Ω} = 0,

and y ≥ 0. By the positivity theorems y_{k} > 0 in Ω or y_{k} ≡ 0. The latter cannot take place because of w(x) has different values at x_{1} = ±∞.

Thus the theorem is proved for c > c_{0}.

To show existence of a travelling wave for c = c_{0} we consider a sequence {c_{n}}, c_{n} > c_{0}, converging to c_{0}. From the corresponding sequence of waves {w^{(n)}} with the velocities c_{n} we can choose a subsequence converging to some limiting function w_{0}. It satisfies equation (1.3) with the c = c_{0} and the boundary conditions.

Consider finally the case c < c_{0}. Suppose that there is a solution w(x) of the problem (1.3), (1.4) monotone in x_{1}. By virtue of the maximum principle the normal derivative ∂(w − w_{+})/∂n on the boundary of the cylinder is negative, and w ∈ K. Then

c_{0} > inf_{ρ∈K} sup_{x,i} B_{i}.

This contradiction proves the theorem.

We note that Assumption 3 means that the principal eigenvalue of the operator L^{+} given by (1.7), (1.8) is nonnegative. If we suppose that it is positive, then we can put

v_{n}(x′) = w_{+}(x′) + τ_{n}u_{0}(x′),

where τ_{n} > 0 and τ_{n} → 0 as n → ∞, u_{0} is an eigenfunction corresponding to the principal eigenvalue. Since u_{0}(x′) > 0, x′ ∈ Ω′, then the functions v_{n} defined in this way satisfy Assumption 3.

2.2. Example of test function. We present here an example of test function for which the expressions B_{i} given by (2.7) are bounded. It will show that the minimal velocity is bounded and the waves exist.

We put

ρ(x) = w_{+}(x′) + ψ(x_{1})(w_{−}(x′) − w_{+}(x′)),

where ψ′(x_{1}) < 0, ψ(−∞) = 1, ψ(∞) = 0. Then

−B_{i} = a_{1}(x′)\frac{ψ′′}{ψ′} + b_{1}_{i}(x′)+

\frac{ψ[F_{i}(w_{+}, x′) − F_{i}(w_{−}, x′)] + F_{i}(w_{+} + ψ(w_{−} − w_{+}), x′) − F_{i}(w_{+}, x′)}{ψ′(w_{−} − w_{+})_{i}},  (2.17)

We can choose the function ψ(x) such that ψ′′/ψ′ is bounded for all x, ψ/ψ′ is bounded at +∞ and (1 − ψ)/ψ′ at −∞. It means that it approaches exponentially its limits at infinity. We show that the third summand in the right hand side of (2.17) remains bounded.

We note first that the expression (w_{−} − w_{+})_{i}/(w_{−} − w_{+})_{j} is bounded for x′ ∈ Ω̄′. Indeed, in the interior points of Ω′ it follows from the inequality w_{−} > w_{+}. On the boundary

\frac{∂w_{−}}{∂ν} > \frac{∂w_{+}}{∂ν} .

As x′ → ∂Ω the numerator of the last term in (2.17) has the form

ψ(B(x)(w_{+} − w_{−}))_{i} + (B̃(x)ψ(w_{−} − w_{+}))_{i},

where

B(x) = ∫_{0}^{1} F′(w_{+} + t(w_{−} − w_{+}), x′)dt, B̃(x) = ∫_{0}^{1} F′(w_{+} + tψ(w_{−} − w_{+}), x′)dt.

When x_{1} → +∞, we represent it in the form

ψ[F_{i}(w_{+}, x′) − F_{i}(w_{−}, x′)] + ψ(B̃(x)(w_{+} − w_{−}))_{i}.

Finally, as x_{1} → −∞, we can consider it in the form

(1 − ψ)[−F_{i}(w_{+}, x′) + F_{i}(w_{−}, x′)] + (1 − ψ)(B̂(x)(w_{+} − w_{−}))_{i},

where

B̂(x) = ∫_{0}^{1} F′(w_{+} + ψ(w_{−} − w_{+}) + t(1 − ψ)(w_{+} − w_{−}))dt.

In all three case devided by ψ′(w_{−} − w_{+}) it remains bounded.

Thus we have shown that c_{0} in the theorem is bounded from above, and the waves exist. Its boundedness from below follows from Theorem 2.2 (see also (2.19)).

2.3. Sign of velocity. We show that if the principal eigenvalue of the operator L^{+} is positive, then the wave velocity is positive.

Consider the operator

A(v) = a(x′)Δ′v + ∑_{k=2}^{m} b_{k}(x′)\frac{∂v}{∂x_{k}} + F (v, x′)

acting on the functions from C^{(2+δ)}(Ω̄′) with the boundary condition

v(x′) = ϕ(x′), x′ ∈ ∂Ω′.

Let

v(x′) = w_{+}(x′) + u(x′),

where

u(x′) ≥ 0, x′ ∈ Ω̄′, u(x′) = 0, x′ ∈ ∂Ω′.

Then

A(v) = L^{+}u + B(x′)u,

where

B(x′) = ∫_{0}^{1} Fv^{′}(w_{+} + tu, x′)dt − Fv^{′}(w_{+}, x′).

Denote λ_{0} the principal eigenvalue of the operator L^{+} and u^{∗}(x′) the corresponding eigenfunction of the adjoint operator L^{+}^{∗}. Then

J ≡ ∫_{Ω′}(u^{∗}, A(v))dx′ = λ_{0} ∫_{Ω′}(u^{∗}, u)dx′ + ∫_{Ω′}(u^{∗}, B(x′)u)dx′.

By assumption, λ_{0} is positive. The eigenfunction u^{∗} is positive in Ω′ if F′(w_{+}(x′), x′) is functionally irreducible. If we assume that the derivative Fv^{′} satisfies Lipschitz condition, then for any positive in Ω′ and sufficiently small in C-norm function u, J > 0.

Equation (1.3) can be represented in the form

a\frac{∂^{2}w}{∂x_{1}^{2}} + (c + b_{1})\frac{∂w}{∂x_{1}} + A(w) = 0.

Denote

z(x) = w(x) − w_{+}(x′).

Then

a\frac{∂^{2}z}{∂x_{1}^{2}} + (c + b_{1})\frac{∂z}{∂x_{1}} + A(w) = 0.

Multiplying this equation by u^{∗} and integrating over Ω′, we obtain

∫_{Ω′}(u^{∗}, a\frac{∂^{2}z}{∂x_{1}^{2}} )dx′ + ∫_{Ω′}(u^{∗}, (c + b_{1})\frac{∂z}{∂x_{1}} )dx′ + ∫_{Ω′}(u^{∗}, A(w))dx′ = 0.

Since w(x) → w_{+}(x′) as x_{1} → +∞ uniformly in x′, then there exists x_{1}^{0} such that the last integral in the left hand side of this equality is positive for all x_{1} ≥ x_{1}^{0}. Let x_{1}^{1} ≥ x_{1}^{0}. We integrate the last equality from x_{1}^{0} to x_{1}^{1}:

∫_{Ω′}(u^{∗}, a\frac{∂z}{∂x_{1}} )|_{x_{1}^{1}}dx′ − ∫_{Ω′}(u^{∗}, a\frac{∂z}{∂x_{1}} )|_{x_{1}^{0}}dx′ + ∫_{Ω′}(u^{∗}, (c + b_{1})z)|_{x_{1}^{1}}dx′ − ∫_{Ω′}(u^{∗}, (c + b_{1})z)|_{x_{1}^{0}}dx′

(2.18)

+ ∫_{x_{1}^{0}}^{x_{1}^{1}} ∫_{Ω′}(u^{∗}, A(w))dx′ = 0.

We note that the second derivative ∂^{2}z/∂x_{1}^{2} is bounded. Since z(x) → 0 as x_{1} → +∞, then the first integral in the left hand side of (2.18) tends to zero, and the last integral increases and tends to a finite limit. Passing to the limit in (2.18) as x_{1} → +∞, we obtain

∫_{Ω′}(u^{∗}, (c + b_{1})z)|_{x_{1}^{0}}dx′ = − ∫_{Ω′}(u^{∗}, a\frac{∂z}{∂x_{1}} )|_{x_{1}^{0}}dx′ + ∫_{x_{1}^{0}}^{∞} ∫_{Ω′}(u^{∗}, A(w))dx′.  (2.19)

The right hand side of the last equality is positive. Then c + b_{1}(x′) ≢ 0 and if c + b_{1}(x′) ≠ 0, then c + b_{1}(x′) > 0.

We have proved the following theorem.

Theorem 2.2 Suppose that the principal eigenvalue of the operator L^{+} is positive. Then for any monotone in x_{1} wave c + b_{1}(x′) ≢ 0 and if c + b_{1}(x′) ≠ 0 for all x′ ∈ Ω′, then c + b_{1}(x′) > 0. If b_{1} ≡ 0, then c is positive.

Corollary 1 Let c_{0}(b_{1}) denote the value of the minimal velocity as a functional determined by the coefficient b_{1}(x′). If

c_{0}(0) ≥ max_{x} b_{1}(x′) − min_{x} b_{1}(x′),

then

c_{0}(b_{1}) + b_{1}(x′) > 0.

Corollary 2 Suppose that b_{1} ≡ 0 and the principal eigenvalues of both operators L^{+} and L^{−} are positive. Then monotone waves do not exist.

Proof. If a monotone wave existed, we could repeat the same construction for x_{1} → −∞ and obtain c < 0.

Remarks. 1. The last assertion shows that monotone waves do not exist in the unstable case. The sign of the wave velocity is determined by local properties of the operators in small neighbourhoods of w_{+} and w_{−}. So this results remains valid for wider classes of equations than we consider in this work (cf. [15], [17]).

2. In the case b_{1} ≡ 0 Theorem 2.1 remains valid if instead of Assumption 2 we require that there is a finite number of solutions of the problem (1.5), (1.6) satisfying (2.2) and for all of them except for w_{−}, the corresponding linearized operators have eigenvalues in the right half-plane.

3. Stability of waves

3.1. Spectrum and stability. We begin with some general results on stability of stationary solutions in Banach spases. Their proofs are given in [15], [16]. In the next subsection we use them to prove stability of travelling waves.

We consider the equation

\frac{du}{dt} = Au + f(u)  (3.1)

in a Banach space E. Here u(t) ∈ E for all t ∈ [0, ∞) , A is a linear and f nonlinear operators, acting in E.

Let the stationary equation

Au + f(u) = 0  (3.2)

have a family of solutions u = ϕ_{α} ∈ E. Here α is a real parameter, α ∈ (−ᾱ, ᾱ). We present the results on the stability of the stationary solutions with respect to small perturbations in an arbitrary Banach space H ⊂ E.

Let the following conditions on ϕ_{α}, f(u), and A be satisfied:

Assumption 1. a). There exists the derivative ϕα^{′} of the stationary solution with respect to α, α ∈ (−ᾱ, ᾱ), taken in the norm of the space H, ϕα^{′} ∈ H (it is not assumed here that ϕ_{α} ∈ H).

b). ϕα^{′} satisfies Lipchitz condition in α, α ∈ (−ᾱ, ᾱ) in the norm of the space H.

Assumption 2. a). Nonlinear operator f(u) is defined on the whole E, is bounded and has the first Gâteaux differential f′(u) in an arbitrary direction v in the space E, u, v, f′(u)v ∈ E.

b). The Gâteaux differential f′(u)v is continuous in u ∈ E for any fixed v ∈ E.

c). The operator f′(ϕ_{α} + v) is bounded in H for v ∈ H, α ∈ (−ᾱ, ᾱ) and satisfies the Lipschitz condition in v for ∥v∥ ≤ 1. Here ∥ · ∥ is the norm in H.

Assumption 3. a). There is a restriction of the operator A, acting in H, which is a generator of an analytical semi-group.

b). The spectrum σ(L) of the operator L = A + f′(ϕ_{0}) is as follows: zero is a simple eigenvalue; all other points of the spectrum lie in a closed angle lying in the left half of the complex plane. This means that there are positive numbers a_{1} and b_{1} such that

Reλ + a_{1} | Imλ | +b_{1} ≤ 0  (3.3)

for λ ∈ σ(L), λ ≠ 0.

Theorem 3.1 Let Assumptions 1-3 be satisfied. Then there is a positive ϵ such that for any ū ∈ E, satisfying the condition ∥ū − ϕ_{0}∥ ≤ ϵ, the solution u(t) of the equation (3.1) with the initial condition u(0) = ū exists in the space E for any t ∈ [0, ∞). This solution is unique, and for some α ∈ (−ᾱ, ᾱ) the estimation

∥u(t) − ϕ_{α}∥ ≤ Me^{−bt}

takes place. Here b and M do not depend on ū, α, and t, b > 0.

Remark. It is asssumed in the formulation of the Theorem that ū − ϕ_{0} ∈ H but it is not assumed that ū, ϕ_{0} ∈ H.

We need also the stability theorem for the case when the stationary equation (3.2) has an isolated solution ϕ_{0}. Let Assumption 2 with α = 0 and Assumption 3 a) be satisfied, and (3.3) be satisfied for all λ ∈ σ(L). Then the following well known result takes place.

Theorem 3.2 There is a positive ϵ such that for any ū ∈ E, satisfying the condition ∥ū − ϕ_{0}∥ ≤ ϵ, the solution u(t) of the equation (3.1) with the initial condition u(0) = ū exists in the space E for any t ∈ [0, ∞). This solution is unique and the estimation

∥u(t) − ϕ(t)∥ ≤ Me^{−bt}

takes place. Here b is an arbitrary number less than b_{1}, M does not depend on ū and t.

3.2. Local stability of waves. In Section 2 we have shown existence of travelling waves, i.e. stationary solutions of the equation

\frac{∂u}{∂t} = a(x′)Δu + c\frac{∂u}{∂x_{1}} + ∑_{j=1}^{m} b_{j}(x′)\frac{∂u}{∂x_{j}} + F (u, x′),  (3.4)

in the cylinder Ω with a bounded section Ω′, with the boundary condition

u(x) = ϕ(x′), x ∈ ∂Ω.  (3.5)

We have proved also that these waves are monotone in x_{1}. We apply now the results of Section 3.1 to prove local stability of monotone in x_{1} waves.

We introduce spaces E and H. We put E = C(Ω̄). Let L_{μ}^{2}(Ω) denote the weighted L^{2} space with the norm

∥u∥_{L_{μ}^{2}} = ∥μu∥_{L^{2}}.

C_{0}(Ω̄) is a space of continuous functions which satisfy the boundary condition u = 0, x ∈ ∂Ω and tend to 0 as |x| → ∞ and C_{0,μ}(Ω̄) is a weighted space, u ∈ C_{0,μ}(Ω̄) if μu ∈ C_{0}(Ω̄). Consider the spaces

H^{0} = L^{2}(Ω) ∩ C_{0}(Ω̄) and H = L_{μ}^{2}(Ω) ∩ C_{0,μ}(Ω̄)

with the norms

∥u∥_{H^{0}} = ∥u∥_{L^{2}} + ∥u∥_{C_{0}} and ∥u∥_{H} = ∥u∥_{L_{μ}^{2}} + ∥u∥_{C_{0,μ}},

respectively. Domains of linear operators L acting in H^{0} or in H are, respectively,

D^{0} = W^{2,2}(Ω) ∩ D_{0}^{q} and D = W_{μ}^{2,2}(Ω) ∩ D_{0,μ}^{q},

where

D_{0}^{q} = {u, u ∈ C_{0}(Ω̄), Lu ∈ C_{0}(Ω̄), u ∈ W_{loc}^{2,q}(Ω), q > m, }
D_{0,μ}^{q} = {u, u ∈ C_{0,μ}(Ω̄), Lu ∈ C_{0,μ}(Ω̄), u ∈ W_{loc}^{2,q}(Ω), q > m.}

The weight function μ(x_{1}) is a sufficiently smooth function equal 1 for x_{1} ≤ 0 and exp(νx_{1}) for x_{1} ≥ 1. Here ν > 0.

We specify assumptions on F (u, x′). We suppose that it is a bounded continuous vectorvalued function defined for u ∈ R^{n}, x′ ∈ Ω′. Its derivative Fu^{′}(u, x′) with respect to u is a uniformly continuous matrix-valued function satisfying Lipschitz condition with respect to u uniformly in x′ in any bounded domain in R^{n}.

Consider the operator linearized about the stationary solution:

Lv = a(x′)Δv + c\frac{∂v}{∂x_{1}} + ∑_{j=1}^{m} b_{j}(x′)\frac{∂v}{∂x_{j}} + F′(w(x), x′)v,

v(x) = 0, x ∈ ∂Ω

acting in H with the domain D. We note that the operator T of multiplication by μ is a bounded operator from H to H^{0} with a bounded inverse. Hence the operator L̃ = T LT^{−1} acts in H^{0}.

We consider the following eigenvalue problems:

a(x′)Δ′u + ∑_{i=2}^{n} b_{i}(x′)\frac{∂u}{∂x_{i}} + (−ξ^{2}a(x′) + iξ(c + b_{1}(x′)) + c_{−}(x′))u = λu  (3.6)

u(x′) = 0, x′ ∈ ∂Ω′,  (3.7)

and

a(x′)Δ′u + ∑_{i=2}^{n} b_{i}(x′)\frac{∂u}{∂x_{i}} + [−ξ^{2}a(x′) + iξ(c + b_{1}(x′) − 2νa(x′))  (3.8)
+ν^{2}a(x′) − ν(c + b_{1}(x′)) + c_{+}(x′)]u = λu
u(x′) = 0, x′ ∈ ∂Ω′,  (3.9)

where

c_{±}(x′) = F′(w_{±}(x′), x′).

We denote them P^{−} and P^{+}, respectively.

If the operator L̃ is considered as acting in L^{2} with the domain W^{2,2} or in C^{δ} with the domain C^{2+δ} with the corresponding boundary conditions, then its essential spectrum is given by all eigenvalues of these problems for all real ξ (see [15], [16], [19]).

We note that the spectrum of the problem P^{−} is connected with the spectrum of the operator L^{−} given by (1.7). Since we assume that all eigenvalues of L^{−} are in the left halfplane, then the spectrum of the problem P^{−} is also in the left half-plane if b_{1}(x′) ≡ b_{0}I, where b_{0} is a constant, I is the identity matrix. If the matrix b_{1} is not scalar, it is probably possible that a part of the spectrum of the problem P^{−} is in the right half-plane. So we assume that the whole spectrum of this problem is in the half-plane Reλ < 0 for all real ξ.

We recall that the operator L^{+} has eigenvalues in the right half-plane. In the weighted space the essential spectrum corresponding to the problem P^{+} is moved to the left (cf. [12]). We assume that a positive number ν can be chosen such that all eigenvalues of the problem P^{+} are in the left half-plane for all real ξ.

Lemma 3.1 Consider the operators Au = L̃u with the domain D(A) = {u : u ∈ W^{2,2}(Ω), u = 0, x ∈ ∂Ω} and Âu = L̃u with the domain D(Â) = {u : u ∈ C_{σ}^{2+δ}(Ω̄), u = 0, x ∈ ∂Ω}, where C_{σ}^{2+δ}(Ω̄) is a weighted Hölder space with the weight σ(x_{1}) = 1 + x_{1}^{2}. Suppose that both operators A and Â are Fredholm. If

Au = f, u ∈ D(A), f ∈ C_{σ}^{δ}(Ω̄),

then u ∈ D(Â) and Âu = f.

Proof. Let u_{k}, k = 1, ..., p be a complete system of solutions of the equation Au = 0 and v_{k} ∈ L^{2}, k = 1, ..., p be a complete system of functions orthogonal to the image of A. Let g_{k} and h_{k}, k = 1, ..., p be smooth finitary functions biorthogonal to u_{k} and v_{k}, respectively.

Consider the finitedimensional operator

Ky = (g_{1}, y)h_{1} + ... + (g_{p}, y)h_{p},

where ( , ) denotes the inner product in L^{2}. We set B = A + K. Then the equation Bu = 0 has only zero solution.

Consider the equation By = f + Ky. By the Fredholm property it has a solution y = ỹ ∈ D(Â). On the other hand y = u is also its solution. Then by uniqueness u = ỹ. The lemma is proved.

Lemma 3.2 Let G be the maximal region of the complex plane which contains positive real half-axis and does not contain eigenvalues of the problems P^{+} and P^{−} for any real ξ.

If λ ∈ G is a regular point of the operator L̃ as acting in L^{2}(Ω), then it is also a regular point of this operator as acting in H^{0}.

Proof. Let λ be a regular point of the operator L̃ as acting in L^{2}(Ω). Consider the equation

L̃u − λu = f.  (3.10)

If f ∈ H^{0} then u ∈ W^{2,2}(Ω). We should show that u ∈ C_{0}(Ω̄).

Let f_{n} ∈ L^{2}(Ω) ∩ C_{σ}^{δ}(Ω̄), where C_{σ}^{δ}(Ω̄) is a weighted Hölder space, the weight function σ can be taken for example as σ(x_{1}) = x_{1}^{2} + 1. Since λ ∈ G, then equation (3.10) is uniquely solvable in W^{2,2}(Ω). The operator L̃ − λ is Fredholm as acting in C_{σ}^{δ}(Ω̄) with the domain C_{σ}^{2+δ}(Ω̄) and its index is zero [19]. The dimension of its kernel is zero because C_{σ}^{2+δ}(Ω̄) ⊂ W^{2,2}(Ω). So (3.10) is uniquely solvable in C_{σ}^{2+δ}(Ω̄). Hence its solutions u_{n} ∈ W^{2,2}(Ω) ∩ C_{σ}^{2+δ}(Ω̄). Then u_{n} ∈ W^{2,p}(Ω), p ≥ 2.

The sequence {f_{n}} can be chosen in such a way that it converges to f in L^{2}(Ω) and in C(Ω̄). Then this convergence occurs also in L^{p}(Ω), p ≥ 2 and this sequence is uniformly bounded in L^{p}(Ω). We show that the sequence of solutions {u_{n}} is uniformly bounded in W^{2,p}(Ω). For this we use the Agmon-Douglis-Nirenberg estimates [1]:

∥u_{n}∥_{W^{2,p}} ≤ c(∥u_{n}∥_{L^{p}} + ∥f_{n}∥_{L^{p}}).

Since the functions f_{n} are uniformly bounded in L^{2} and λ is a regular point of the operator L̃ as acting in L^{2}, then the functions u_{n} are uniformly bounded in W^{2,2}. Starting with p = 2 and using embedding theorems, by the standard arguments after a finite number of steps we obtain that u_{n} is uniformly bounded in W^{2,p}(Ω) for any given p. Hence u_{n} ∈ C^{1+δ}(Ω̄). From this and convergence u_{n} → u in W^{2,2} follows that u ∈ C^{1}(Ω̄).

It remains to note that since u ∈ W^{2,2}(Ω) ∩ C^{1}(Ω̄), then u(x) → 0 as |x| → ∞ and u ∈ C_{0}(Ω̄). The lemma is proved.

Theorem 3.3 Suppose that the problem P^{−} has all eigenvalues in the left half-plane for all real ξ and there exists ν such that the problem P^{+} has all eigenvalues in the left half-plane also for all real ξ.

If the derivative of the wave ∂w(x)/∂x_{1} belongs to the space H, then it is asymptotically stable with shift in this space.

Proof. We should verify that Assumptions 1-3 of the previous subsection are satisfied. The derivative v = ∂w/∂x_{1} of the travelling wave w(x) satisfies the problem

Lv = 0, x ∈ Ω, v = 0, x ∈ ∂Ω.

Denote u = T v. Then u ∈ H^{0} and L̃u = 0. By estimates of solutions of elliptic problems we obtain u ∈ C^{1}(Ω̄). From this and the fact that u ∈ L^{2}(Ω), multiplying the equality L̃u = 0 by u and integrating by parts, we obtain that u ∈ W^{1,2}(Ω). By usual arguments we get u ∈ W^{2,2}(Ω). From Lemma 3.1, u ∈ C_{σ}^{2+δ}(Ω̄). Assumptions 1 and 2 follow from this.

Consider Assumption 3. It is known that the operator L̃ with the Dirichlet boundary conditions is a generator of analytic semigroup in L^{2} and C (see [13]). So it is a generator in the space H^{0}. It follows that the operator L with the Dirichlet conditions is generator in the space H.

Consider now location of spectrum. The spectrum of operator L̃ in the space L^{2}(Ω) satisfies conditions of Assumption 3b) (see [18]). From Lemmas 3.1 and 3.2 it follows that it is also true in the space H^{0}. Therefore it is true for operator L in the space H. The theorem is proved.

We study now the case where the derivative of the wave does not belong to the space H. We begin with some auxiliary results. We consider the operator

L̃u = ã(x)Δu + ∑_{j=1}^{m} b̃_{j}(x)\frac{∂u}{∂x_{j}} + c̃(x)u

in the domain Ω_{N} ⊂ Ω∩{x_{1} ≥ N} such that its boundary is sufficiently smooth and coincides with ∂Ω for x_{1} ≥ N + 1. We represent ∂Ω_{N} = Γ_{1} ∪ Γ_{2}, where Γ_{1} is the part of the boundary in ∂Ω and Γ_{2} is the remaining part.

We assume that the coefficients of the operator are Hölder continuous and have limits at infinity and use the superscript ± to denote them. The convergence to the limiting coefficients is in the Hölder norm with respect to the variables x′.

As above the matrices ã(x) and b̃_{j}(x) are diagonal, c̃(x) has nonnegative off-diagonal elements. We denote C_{0}^{2+δ}(Ω̄_{N}) the space of functions from C^{2+δ}(Ω̄_{N}) equal 0 on ∂Ω_{N}. We consider the operator L̃ not only on functions from this space but also on functions which do not belong to it. In this case we should understand L̃u = 0 as a local equality in a neighbourhood of each point of Ω.

Lemma 3.3 Let the operator L̃ as acting from C_{0}^{2+δ}(Ω̄_{N}) to C^{δ}(Ω̄_{N}) have the essential spectrum in the left half plane and be invertible.

Let further L̃u = 0, L̃v = 0 in Ω_{N}, u = v = 0 on Γ_{1}, v ≥ u on Γ_{2}, v > 0 in Ω_{N} and u → 0 as x_{1} → +∞. Then v ≥ u in Ω̄_{N}.

Proof. Essential spectrum of the operator L̃ is determined by the eigenvalues of the problem

ã^{+}(x′)Δ′u + ∑_{j=2}^{m} b̃_{j}^{+}(x′)\frac{∂u}{∂x_{j}} + (−ã^{+}(x′)ξ^{2} + b̃_{1}^{+}(x′)iξ + c̃^{+}(x′))u = λu,

u = 0, x′ ∈ ∂Ω′

in the section of the cylinder for all real ξ. In particular, for ξ = 0 all eigenvalues of this problem lie in the left half-plane. Then the problem

ã^{+}(x′)Δ′w + ∑_{j=2}^{m} b̃_{j}^{+}(x′)\frac{∂w}{∂x_{j}} + c̃^{+}(x′)w = 0,

w = −ϵ, x′ ∈ ∂Ω′

is uniquely solvable and by virtue the positiveness theorem its solutions is negative in Ω̄′. We denote it w_{ϵ}(x′).

Consider further the problem

L̃w = 0, w = −ϵ on ∂Ω_{N}.

Existence of solution for it follows from the invertibility of the operator L̃. We denote it w̃_{ϵ}(x). Then

w̃_{ϵ}(x) → w_{ϵ}(x′), x_{1} → +∞

and this convergence is uniform in x′. We put u_{ϵ} = u + w̃_{ϵ}. We have

L̃u_{ϵ} = 0 in Ω_{N}, u_{ϵ} = −ϵ on Γ_{1}, u_{ϵ} → w_{ϵ}(x′) < 0, x_{1} → +∞, u_{ϵ} < v on Γ_{2}.

We show that

u_{ϵ} < v, x ∈ Ω̄_{N}.  (3.11)

Indeed, this inequality is satisfied for x_{1} ≥ x_{1}^{∗} where x_{1}^{∗} is sufficiently large and in a neighbourhood of ∂Ω_{N} for x_{1} ≤ x_{1}^{∗}. Since v is positive in Ω_{N}, then for some τ_{0} ≥ 1, u_{ϵ} < τ_{0}v in Ω̄_{N}. We decrease now τ from τ_{0} to 1. If for some τ and x_{0} ∈ Ω̄_{N}, u_{ϵ}(x_{0}) = τv(x_{0}), then x_{0} ∈ Ω_{N} and we obtain a contradiction with the positiveness theorem. Thus (3.11) is proved.

In the limit as ϵ → 0 we obtain u ≤ v in Ω̄_{N}. The lemma is proved.

Remarks. 1. If the matrix c̃(x) is functionally irreducible and v(x) ≢ u(x), then v(x) > u(x) in Ω_{N}.

2. We do not require that v(x) ∈ C_{0}^{2+δ}(Ω̄_{N}). In particular it can unbounded. In this case the operator L̃ is considered not necessarily on functions from C_{0}^{2+δ}(Ω̄_{N}).

3. If the coefficients of the operator L̃ do not depend on x_{1}, then the invertibility condition can be omitted. Indeed, in this case we can put w̃_{ϵ}(x) ≡ w_{ϵ}(x′). This function satisfies equation L̃w = 0 in Ω_{N}, equals −ϵ on Γ_{1} and it is negative on Γ_{2}.

We introduce the following limiting operators

L̃^{±}u = ã^{±}(x)Δu + ∑_{j=1}^{m} b̃_{j}^{±}(x)\frac{∂u}{∂x_{j}} + c̃^{±}(x)u.

Lemma 3.4 Let the operator L̃ as acting from C_{0}^{2+δ}(Ω̄) to C^{δ}(Ω̄) have the essential spectrum in the left half plane, the operator L̃^{+} be invertible from C_{0}^{2+δ}(Ω̄_{0}^{+}) to C^{δ}(Ω̄_{0}^{+}) and the operator L̃^{−} be invertible from C_{0}^{2+δ}(Ω̄_{0}^{−}) to C^{δ}(Ω̄_{0}^{−}). Here Ω_{0}^{+} = Ω_{N} for N = 0, Ω_{0}^{−} is a symmetric to Ω_{0}^{+} with respect to x_{1} = 0.

Suppose that L̃v = 0 in Ω, v > 0 in Ω, v = 0 on ∂Ω. Then there is no solutions of the equation L̃u = 0 in C_{0}^{2+δ}(Ω̄) such that u ≢ 0, u ≢ v and u → 0 as x_{1} → ±∞.

Proof. For N sufficiently large the operator L̃ is invertible from C_{0}^{2+δ}(Ω̄_{N}^{+}) to C^{δ}(Ω̄_{N}^{+}) and from C_{0}^{2+δ}(Ω̄_{N}^{−}) to C^{δ}(Ω̄_{N}^{−}). So we can apply for it and for these domain the previous lemma.

Suppose that there exists a solution u(x) ∈ C_{0}^{2+δ}(Ω̄) of the equation L̃u = 0, u ≢ 0. Without loss of generality we can assume that at least one of its components has positive values in the domain Ω^{0} = Ω/(Ω_{N}^{+} ∪ Ω_{N}^{−}). Otherwise we can change u by −u.

Denote z = τv − u. Then for τ sufficiently large z > 0 in Ω^{0} and ∂z/∂ν < 0 on ∂Ω^{0} ∩ ∂Ω. Denote τ_{0} the minimal value of τ for which z still satisfies these properties. Clearly τ_{0} > 0.

Thus for τ = τ_{0}, z(x) ≥ 0 in Ω̄^{0} and a) z(x_{0}) = 0 for some x_{0} ∈ Ω̄^{0} ∩ Ω or b) ∂z/∂ν = 0 at some point on ∂Ω^{0} ∩ ∂Ω. Then from the previous lemma z(x) ≥ 0 in Ω̄. In the case a) we obtain a contradiction with the positiveness theorem, in the case b) with the negativeness of the normal derivative on the boundary. The theorem is proved.

Remark. In the conditions of the lemma it can be shown similarly that the operator L̃ cannot have positive eigenvalues in C_{0}^{2+δ}(Ω̄).

Lemma 3.5 Let the operator L̃^{+} as acting from C_{0}^{2+δ}(Ω̄_{0}^{+}) to C^{δ}(Ω̄_{0}^{+}) have the essential spectrum in the left half-plane. Then it does not have zero eigenvalue.

Proof. We apply Remark 3 to Lemma 3.3. For this we construct a positive solution v_{+} of the inequality L̃^{+}v ≤ 0 which is an upper function and provides existence of a positive solution v(x) of the equation L^{+}v = 0. We put

v_{+}(x) = e^{−κx_{1}}ϕ_{+}(x′)

for some small positive κ. The function ϕ_{+} should be positive in Ω′ and satisfy the inequality

ã^{+}(x′)Δ′ϕ_{+} + ∑_{j=2}^{m} b̃_{j}^{+}(x′)\frac{∂ϕ_{+}}{∂x_{j}} + (ã^{+}(x′)κ^{2} − b̃_{1}^{+}(x′)κ + c̃^{+}(x′))ϕ_{+} ≤ 0.

If κ is sufficiently small, we can take as ϕ_{+} the principal eigenfunctions of the operator corresponding to the left hand side of these inequality.

Suppose that there is a solution u(x) ∈ C_{0}^{2+δ}(Ω̄_{0}^{+}) of the equation L̃u = 0. Then u(x) → 0 as x_{1} → +∞. Otherwise λ = 0 would be a point of essential spectrum of this operator [19]. Since v(x) > 0 in Ω_{0}^{+} and on Γ_{2}, we can apply Remark 3 to the functions τv and u for any positive τ. If τ → 0, we obtain in the limit u ≤ 0 in Ω̄_{0}^{+}. Changing u to −u we show that u ≥ 0. Thus u ≡ 0. The lemma is proved.

Similarly we show the invertibility of the operator L̃^{−} in C_{0}^{2+δ}(Ω̄_{0}^{−}).

The last theorem states the asymptotic stability without shift when the derivative of the wave does not belong to the weighted space.

Theorem 3.4 Suppose that the problem P^{−} has all eigenvalues in the left half-plane for all real ξ and there exists ν such that the problem P^{+} has all eigenvalues in the left half-plane also for all real ξ.

If the derivative of the wave ∂w(x)/∂x_{1} does not belong to the space H, then the wave is asymptotically stable in this space.

Proof. We should verify that all eigenvalues of the operator L in the space H lie in the left half-plane. Consider the operator L̂ = T^{−1}LT in the space H^{0}, where T is the operator of multiplication by μ. It is sufficient to verify that all its eigenvalues lie in the left half-plane.

The essential spectrum of this operator is determined by the problems (3.6), (3.7) and (3.8), (3.9) and lies in the left half-plane. Suppose that it has an eigenvalue with nonnegative real part. Then its principal eigenvalue being real is nonnegative and tends uniformly to zero as x_{1} → ±∞. We can apply Lemma 3.4 since the operators

L̂^{+}u = a^{+}(x′)Δu + ∑_{j=2}^{m} b_{j}^{+}(x′)\frac{∂u}{∂x_{j}} + (b_{1}^{+}(x′) + c − 2a^{+}(x′)ν)\frac{∂u}{∂x_{1}}

+(a^{+}(x′)ν^{2} − (b_{1}^{+}(x′) + c)ν + c^{+}(x′))u,

and

L̂^{−}u = a^{−}(x′)Δu + ∑_{j=2}^{m} b_{j}^{−}(x′)\frac{∂u}{∂x_{j}} + (b_{1}^{+}(x′) + c)\frac{∂u}{∂x_{1}} + c^{−}(x′)u

are invertible (i.e. do not have zero eigenvalue) as acting from C_{0}^{2+δ}(Ω̄_{N}^{+}) to C^{δ}(Ω̄_{N}^{+}) and from C_{0}^{2+δ}(Ω̄_{N}^{−}) to C^{δ}(Ω̄_{N}^{−}), respectively (Lemma 3.5).

Thus we have shown that the equation L̂u = λu with nonnegative λ cannot have nonzero solutions converging to zero at infinity and different from the function v = −∂w/∂x_{1} μ. Hence the corresponding eigenfunctions cannot belong to the space H^{0}. Otherwise it would converge to zero at infinity and, consequently, coincide with v. This contradicts to the assumption that the derivative of the wave does not belong to the weighted space. The theorem is proved.

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