STABILITY AND INDEX OF PERIODIC SOLUTIONS

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume , Number 0,

Website: http://AIMsciences.org pp. 000–000

STABILITY AND INDEX OF PERIODIC SOLUTIONS OF A NONLINEAR TELEGRAPH EQUATION

Rafael Ortega Departamento de Matem´ atica Aplicada Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Abstract. For a class of abstract evolution equations, the topological index is sufficient to characterize the stability of periodic solutions. This fact can be applied to discuss the stability properties in the sine-Gordon equation with friction and periodic forcing.

1. Introduction. Consider the differential equation utt + cut − uxx = F (t, x, u)

(1.1)

with the boundary conditions u(t, x + `) = u(t, x), (t, x) ∈ R2 .

(1.2)

It can be interpreted as the equation of motion of a circular string of length ` > 0 with friction c > 0 and subjected to the force F . The function F : R2 × R → R is continuous, has a continuous partial derivative Fu (t, x, u) and satisfies the periodicity conditions F (t + T, x, u) = F (t, x + `, u) = F (t, x, u) for some T > 0. This is abbreviated by F ∈ C 0,1 [(R/T Z × R/`Z) × R]. The energy space associated to the model is X = V × H, V = H 1 (R/`Z), H = L2 (R/`Z) which is a Hilbert space with inner product Z ` ((φ1 , ψ1 ), (φ2 , ψ2 )) = {φ1 φ2 + φ01 φ02 + ψ1 ψ2 }dx. 0

The solutions of (1.1), (1.2) will be interpreted as functions u : t ∈ I ⊂ R 7→ u(t) ∈ V with u ∈ C 1 (I, H) ∩ C(I, V ). The theory of the initial value problem can be found in [3]. In this paper the interest will be in the doubly periodic solutions. These are solutions of (1.1), Φ(t, x) with periods T and `. The solution Φ will be called isolated (period T ) if there are no other doubly periodic solutions close to Φ. The index of an isolated Φ can be defined using degree theory. This index is an integer number which will be denoted by γ(Φ). The details on these definitions can be found in [11]. Also in [11] it was proved that γ(Φ) = 1 if Φ is asymptotically stable. In general the converse is not true and unstable solutions can have index one. In 2000 Mathematics Subject Classification. Primary: 37L15; Secondary: 35B10, 35B35, 35L70. Key words and phrases. Asymptotic stability, topological degree, periodic solution, sineGordon equation. Supported by MCYT, BFM2002-01308 (Spain). 1

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this paper it will be shown that the index is sufficient to characterize asymptotic stability when Fu is small. Theorem 1.1. Given c > 0 and ` > 0, there exists σ = σ(c, `) > 0 such that if Φ(t, x) is a doubly periodic solution of (1.1) which is isolated and satisfies |Fu (t, x, Φ(t, x))| ≤ σ, (t, x) ∈ R2 ,

(1.3)

then the index γ(Φ) can only take the values 1, −1 and 0. Moreover, Φ is asymptotically stable if γ(Φ) = 1 and unstable if γ(Φ) = −1 or 0. The proof will produce explicit estimates for σ, however they are far from being optimal. Characterizations of stability using degree theory have been discussed in the context of ordinary differential equations [8, 6, 16] and parabolic equations [2]. Recently I learned about an older paper by Kolesov [4] where he stated related results for fixed points of maps. Some of the results of [4] were rediscovered in [9]. The paper is organized looking towards the proof of Theorem 1.1. First, abstract evolution equations will be discussed in Section 2. They will be of the type U˙ = AU + N (t, U ), U ∈ X, (1.4) where X is a Hilbert space; conclusions similar to those of Theorem 1.1 will be obtained for (1.4). In this abstract context, the condition (1.3) will be replaced by a condition first introduced by R.A. Smith [14]. This condition, which was inspired by the theory of Lyapunov functions, consists in finding a quadratic form Q of index one in X and a number λ > 0 such that V (t) = e2λt Q(u(t)) is decreasing along solutions of the linearized equation. The proof of the result on (1.4) combines the techniques in [9] with a result on linear periodic equations which is presented in Section 3. Once the result for abstract equations has been obtained, the proof of Theorem 1.1 is reduced to verifying the Smith’s condition. To do this we discuss the dynamics of the linear telegraph equation (F = 0) in Section 4. This discussion motivates the construction of a quadratic form which applies when Fu is small. The paper is finished with an application of Theorem 1.1 to the sine-Gordon equation utt + cut − uxx + a sin u = p(t, x).

(1.5)

It requires the smallness of a but allows forcings p of arbitrary size. We refer to [1] for previous results about the stability of doubly periodic solutions of this equation. To finish this introduction we notice that the abstract result could also be applied to obtain extensions of Theorem 1.1 to equations like (1.1) but in more space dimensions (with periodic or Neumann boundary conditions). The results about (1.5) are more delicate and the proofs in Section 6 use strongly that x is one-dimensional. 2. Abstract Evolution Equations. In this section we use some basic facts from the theory of linear semigroups and semi-linear differential equations. The reader is referred to [12] for the basic definitions and to [13] for more advanced material. We shall work on a real Hilbert space X. The inner product of vectors ξ, η ∈ X is denoted by (ξ, η) and the norm will be ||ξ|| = (ξ, ξ)1/2 . The space of bounded linear operators L : X → X will be denoted by L(X), with associated norm |||L||| = sup{||Lξ|| : ||ξ|| ≤ 1}. Consider a closed linear operator A : D ⊂ X → X, D = X

STABILITY AND INDEX OF PERIODIC SOLUTIONS

3

which is the infinitesimal generator of a c0 -group of operators in L(X). Given ξ ∈ X and b ∈ C(R, X), the initial value problem U˙ = AU + b(t), U (0) = ξ

(2.1)

has a unique solution in C(R, X). This solution is defined as Z t U (t) = S(t)ξ + S(t − s)b(s)ds 0

where {S(t)}t∈R is the c0 -group. The solution U (t) satisfies (2.1) in a classical sense when ξ ∈ D and b ∈ C 1 (R, X). Under these assumptions U is in C 1 (R, X) and U (t) remains in D for each t. The norm |||S(t)||| has exponential growth and this can be employed to prove the continuous dependence of U with respect to b. Next we consider the nonlinear initial value problem U˙ = AU + N (t, U ), U (0) = ξ

(2.2)

where N : R × X → X satisfies • N (t + T, ξ) = N (t, ξ), (t, ξ) ∈ R × X • N is continuous and for each t ∈ R the map ξ ∈ X 7→ N (t, ξ) ∈ X is Fréchet differentiable. Moreover the partial derivative with respect to ξ is such that the map (t, ξ) ∈ R × X 7→ ∂ξ N (t, ξ) ∈ L(X) is uniformly continuous on bounded sets • there exists k > 0 such that |||∂ξ N (t, ξ)||| ≤ k for each (t, ξ) ∈ R × X. These assumptions imply that N is Lipschitz-continuous in ξ and also that N maps bounded sets on bounded sets. The unique solution of (2.2), U ∈ C(R, X), satisfies the integral equation Z t U (t) = S(t)ξ + S(t − s)N (s, U (s))ds. (2.3) 0

Sometimes, to emphasize the dependence on initial conditions, the solution will be written as U (t, ξ). A standard argument shows that (t, ξ) ∈ R × X 7→ U (t, ξ) ∈ X is Fréchet differentiable with respect to ξ. Given δ ∈ X, the directional derivative V (t) = ∂ξ U (t, ξ)δ is the solution of V˙ = AV + ∂ξ N (t, U (t, ξ))V, V (0) = δ.

(2.4)

The periodicity in t of the nonlinear equation leads to the definition of the Poincaré map P : X → X, P (ξ) = U (T, ξ). From the previous discussions we know that P is C 1 and the derivative can be computed from (2.4). The fixed points of P coincide with the initial conditions of the periodic solutions (period T ) of (2.2). For this reason we would like to apply degree theory to study the roots of I −P . The Poincaré map is a homeomorphism and so it cannot be compact if X has infinite dimensions. For this reason one cannot employ the Leray-Schauder degree. Instead we shall impose conditions which guarantee that P is the sum of a contraction and a compact operator. This is sufficient to apply the theory of degree developed by Nussbaum [5].

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Lemma 2.1. Assume that (A1) there exist an orthogonal projection Π in L(X) with dim(Im Π) < ∞ and a number α > 0 such that for each ξ ∈ D Πξ ∈ D,

(A(I − Π)ξ, ξ) ≤ −α||(I − Π)ξ||2 ,

ΠAξ = AΠξ,

(A2) the map (t, ξ) ∈ R × X 7→ N (t, ξ) ∈ X is compact (this means that the image of a bounded set is relatively compact). Then the Poincaré map can be decomposed as P = L + K where L is a linear contraction (L ∈ L(X), |||L||| < 1) and K is compact. Proof. It is similar to the proof of Lemma 2.4.2 in [3]. Given ξ ∈ D we observe that U (t) = ΠS(t)ξ is a solution of the initial value problem U˙ = AU , U (0) = Πξ. This is so because A and Π commute. By uniqueness we deduce that U (t) = S(t)Πξ. Once we know that ΠS(t)ξ = S(t)Πξ for ξ ∈ D, we use the density of D in X to conclude that Π and S(t) commute. From this fact one deduces that Ker Π and Im Π are invariant under S(t). Next we prove the estimate |||S(t) ◦ (I − Π)||| ≤ e−αt , t ≥ 0. In fact if ξ ∈ D we use (A1) to obtain d ||S(t)(I − Π)ξ||2 = 2((I − Π)S(t)ξ, A(I − Π)S(t)ξ) ≤ dt −2α||S(t)(I − Π)ξ||2 . This differential inequality leads to ||S(t)(I − Π)ξ|| ≤ e−αt ||(I − Π)ξ||. Again the density of D in X is employed to finish the proof of the estimate. From the integral equation (2.3) we observe that Z T P (ξ) = S(T )ξ + S(T − s)N (s, U (s, ξ))ds. 0

In view of the previous estimate the operator L := (I − Π) ◦ S(T ) is a linear contraction. If we define K := P − L, it can be split as K = K1 + T K2 with Z 1 T K1 = Π ◦ S(T ), K2 (ξ) = S(T − s)N (s, U (s, ξ))ds. T 0 The linear operator K1 has finite rank and therefore it is compact. To prove that K2 is also compact we first observe that the general solution of (2.2), (t, ξ) 7→ U (t, ξ), maps bounded sets on bounded sets. Given B bounded in X, the set {N (t, U (t, ξ)) : t ∈ [0, T ], ξ ∈ B} is contained in C, a compact subset of X. Here we have employed (A2). The map (t, ξ) ∈ R × X 7→ S(t)ξ ∈ X is continuous because {S(t)} is a c0 -group. In consequence C] = {S(t)ξ : t ∈ [0, T ], ξ ∈ C} is also a compact subset of X. The mean value theorem for vector valued integrals implies that K2 (B) is contained in co(C] ), the closed convex hull of C] . The proof is complete because co(C] ) is also compact. Let ϕ(t) be a T -periodic solution of (2.2) which is isolated (period T ). This means that there exists an open set W in X with ϕ(0) ∈ W and such that u(t, ξ) is not T -periodic if ξ ∈ W \ {ϕ(0)}. This is equivalent to say that ϕ(0) is isolated


5

in the set of fixed points of P . If we are in the conditions of Lemma 2.1 the index of ϕ is defined as γ(ϕ) := deg (I − P, W ). The standard notions in the theory of Lyapunov stability will be employed. The solution ϕ(t) of the differential equation and the fixed point ϕ(0) of P have the same stability properties. We shall connect these properties with the index γ(ϕ). We need a preliminary definition. A continuous and symmetric bilinear form L B : X × X → R has index n ≥ 1 if there is a splitting of the space X as X = X+ X− , X+ and X− closed subspaces of X with dimX− = n and there exists α > 0 such that B(ξ, ξ) ≥ α||ξ||2 , B(η, η) < 0, B(ξ, η) = 0 if ξ ∈ X+ , η ∈ X− , η 6= 0. Theorem 2.1. Assume that (A1), (A2) hold and let ϕ(t) be an isolated T -periodic solution of (2.2). In addition assume that there exists a continuous and symmetric bilinear form B with index one and two numbers λ > 0, ² > 0 such that B(ξ, Aξ + ∂ξ N (t, ϕ(t))ξ + λξ) ≤ −²||ξ||2

(2.5)

if ξ ∈ D, t ∈ R. Then γ(ϕ) can only take the values 1, −1, 0 and • ϕ is asymptotically stable if γ(ϕ) = 1 • ϕ is unstable if γ(ϕ) = −1, 0. This result can be seen as an extension to infinite dimensions of the main result in [7]. The condition (2.5) was first introduced by R.A. Smith, see [14] and the references therein. The proof of the theorem will be obtained as an application of the results in [9]. Indeed we shall employ an extension of the results of that paper to non-compact operators. The proofs in [9] (see also [2]) remains valid and we briefly describe the setting. Given L ∈ L(X) with spectrum σ(L), it is said that µ ∈ σ(L) is a simple eigenvalue if there exists ξ ∈ XCL \ {0} and X1 , closed subspace of XC , such that Lξ = µξ, L(X1 ) ⊂ X1 , XC = X1 Cξ and µ 6∈ σ(L1 ), where L1 : X1 → X1 is the restriction of L. Here XC is the complex Hilbert space {ξ +iη : ξ, η ∈ X}. It is clear that simple eigenvalues are isolated in the spectrum. Indeed σ(L) = σ(L1 ) ∪ {µ} and σ(L1 ) is compact. Let us consider the open unit disk D = {z ∈ C : |z| < 1}. We shall say that L ∈ L(X) satisfies the property Σ if the spectrum of L is in one of the following situations: (i) σ(L) ⊂ D or (ii) σ(L) = σ1 ∪ {µ} with σ1 ⊂ D and µ ∈ [1, ∞[ a simple eigenvalue. Consider now a nonlinear map F : X → X of class C 1 having an isolated fixed point ξ? . In addition F 0 (ξ? ) satisfies the property Σ and F = L + K with L a linear contraction and K compact. The index of ξ? is defined as deg (I − F, U) where U is a small open neighborhood of ξ? . The conclusions of [9] say that this index satisfies the same properties as those stated in Theorem 1.1 for γ(ϕ). Incidentally, we notice that in this context the asymptotic stability of a fixed point is uniform. Now we can be more precise about the proof of Theorem 2.1. It will consist in applying the previous discussions with F = P and ξ? = ϕ(0). In view of Lemma 2.1 we only

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have to check that the property Σ holds for P 0 (0). This linear operator can be seen as the Poincaré map associated to the variational equation V˙ = AV + ∂ξ N (t, ϕ(t))V. (2.6) In the next section we show that the property Σ holds for a class of equations including (2.6). 3. A Result of R.A. Smith on Floquet Theory. Consider the linear periodic equation V 0 = AV + L(t)V, (3.1) where {L(t)}t∈R is a family of operators in L(X) satisfying t ∈ R 7→ L(t) ∈ L(X) is continuous and, for each t, L(t + T ) = L(t). The operator A is the infinitesimal generator of a c0 -group, as in the previous section. The equation (2.6) is in this class. Our goal is to obtain a partial extension of Theorem 1 in [14] to the infinite dimensional context. It will be convenient to admit complex valued solutions. These are functions V ∈ C(R, XC ) satisfying the integral equation Z t V (t) = S(t)V (0) + S(t − s) ◦ L(s)V (s)ds. (3.2) 0

A non-trivial solution satisfying V (t + T ) = µV (t), t ∈ R for some µ ∈ C will be called a Floquet solution. The number µ is a Floquet multiplier. The Poincaré map M : V (0) 7→ V (T ) associated to (3.1) will be called the monodromy operator. Floquet multipliers are eigenvalues of M but in infinite dimensions there are cases where the set of Floquet multiplers is strictly contained in σ(M ). Proposition 3.1. Assume that there exists a continuous and symmetric bilinear form B with index one and such that for some λ > 0 and ² > 0, B(ξ, Aξ + L(t)ξ + λξ) ≤ −²||ξ||2 ∀ξ ∈ D, t ∈ R.

(3.3)

Then the monodromy operator M satisfies the property Σ. Moreover, there exists at most one multiplier satisfying |µ| ≥ e−λT and this multiplier lies in ]0, ∞[ and is simple. In the terminology of [17] this result implies that the instability index of (3.1) is at most one. We prepare the proof with two lemmas. Lemma 3.1. Let V (t) be a real solution of (3.1). Then, if t > τ , Z t e2λs ||V (s)||2 ds. e2λt B(V (t), V (t)) ≤ e2λτ B(V (τ ), V (τ )) − 2²

(3.4)

τ

Proof. Define

Z t+δ 1 L(s)V (s)ds, δ > 0. 2δ t−δ The functions bδ (t) converge to b(t) := L(t)V (t) as δ ↓ 0. This convergence is uniform on compact intervals. In addition bδ ∈ C 1 (R, X) and we consider the initial value problem V˙ δ = AVδ + bδ (t), Vδ (0) = ξδ , bδ (t) =


7

where ξδ ∈ D and ξδ → V (0). Then Vδ is C 1 and we can apply the chain rule and (3.3) to obtain d 2λt [e B(Vδ (t), Vδ (t))] = 2e2λt [B(Vδ , AVδ + bδ ) + λB(Vδ , Vδ )] dt 2e2λt [B(Vδ , AVδ + L(t)Vδ + λVδ ) + B(Vδ , bδ − L(t)Vδ )] ≤

−2e2λt [²||Vδ ||2 − B(Vδ , bδ − L(t)Vδ )].

By integrating this inequality between τ and t and letting δ ↓ 0 we arrive at the desired inequality. Notice that, by continuous dependence of (2.1), Vδ converge to V. Lemma 3.2. Assume that L ∈ L(X), µ ∈ σ(L) and L = L1 +L2 with L1 , L2 ∈ L(X), |||L1 ||| < |µ| and L2 compact. Then Ker(L − µI) 6= {0}. Proof. The map L1 − µI is an isomorphism and so the equation Lx − µx = f is equivalent to x + (L1 − µI)−1 L2 x = (L1 − µI)−1 f . The operator (L1 − µI)−1 ◦ L2 is compact and Riesz theory can be applied to the last equation. From there one deduces that µ is not in the spectrum when Ker(L − µI) = {0}. Proof of Proposition 3.1. We divide the proof in five steps. Step 1: Geometry of B. We mention some properties associated to the bilinear form which will be employed in subsequent steps. The proofs are very basic. Define the sets C+ = {ξ ∈ X : B(ξ, ξ) ≥ 0}, C− = {ξ ∈ X : B(ξ, ξ) ≤ 0}. Notice that X+ ⊂ C+ and X− ⊂ C− . Let π− ∈ L(X) be the projection satisfying Imπ− = X− , Kerπ− = X+ , then there exists a > 0 such that ||ξ|| ≤ a||π− ξ||, ∀ξ ∈ C− .

(3.5)

Also, given ξ ∈ X \ X− and η ∈ X− \ {0}, there exists a compact interval I = [σ− , σ+ ], σ− < σ+ such that ξ + ση ∈ C+ if and only if σ ∈ I. Moreover the products (π− (ξ + σ− η), η) and (π− (ξ + σ+ η), η) have opposite signs. Step 2: An equivalent norm. Define Z T ||ξ||? = [ e2λt ||V (t, ξ)||2 dt]1/2 , ξ ∈ X, 0

where V (t, ξ) is the solution of (3.1) with V (0) = ξ. The continuous dependence of (3.1) with respect to initial conditions is obtained from (3.2) and Gronwall’s Lemma. It can be employed to show that || · || and || · ||? are equivalent norms. From (3.4) we deduce the inequality e2λT B(M ξ, M ξ) ≤ B(ξ, ξ) − 2²||ξ||2? , ξ ∈ X.

(3.6)

Step 3: The manifold W+ . The subspace of X, W+ = {ξ ∈ X :

lim eλt ||V (t, ξ)|| = 0}

t→+∞

can be characterized as the set of points which remain in C+ forever. More precisely, W+ coincides with {ξ ∈ X : V (t, ξ) ∈ C+ ∀t ∈ R}.

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In fact, given ξ ∈ W+ \ {0} and τ ∈ R we let t → +∞ in formula (3.4). It leads to Z ∞ B(V (τ, ξ), V (τ, ξ)) ≥ 2² e2λ(s−τ ) ||V (s, ξ)||2 ds > 0. τ

Conversely, given ξ ∈ X such that V (t, ξ) remains in C+ for all t, we apply (3.4) again with τ = 0 and t → +∞ and obtain Z ∞ 2² e2λs ||V (s, ξ)||2 ≤ B(ξ, ξ). P∞

0 2λnT

||M n ξ||2? < ∞ and, from the equivalence of the norms, This implies n=0 e λnT n e ||M ξ|| → 0. From here one uses the continuous dependence and deduces that ξ ∈ W+ . After this characterization we observe that W+ is closed in X and invariant under M , M (W+ ) ⊂ W+ . From (3.6) we deduce that B(ξ, ξ) ≥ 2²||ξ||2? , ξ ∈ W+ , and so B is coercive on W+ . This fact allows us to introduce, on the subspace W+ , a third equivalent norm, ||ξ||B = B(ξ, ξ)1/2 , ξ ∈ W+ . Going back to (3.6) we obtain ||M ξ||B ≤ k||ξ||B , ξ ∈ W+ , −λT

for some k < e . This says that the restriction M+ : W+ → W+ is a linear contraction in the new norm and σ(M+ ) ⊂ {µ ∈ C : |µ| < e−λT }.

(3.7)

L Step 4: W+ has codimension one. We are going to prove that X = W+ X− . Given ξ ∈ W+ ∩X− we use the characterization of W+ and observe that B(ξ, ξ) = 0. This implies that ||ξ||B = 0 and so W+ ∩ X− = {0}. Now it is sufficient to prove that X = W+ + X− . First of all we fix η ∈ X− with ||η|| = 1 and consider the functional φ : X → R, φ(ξ) = (π− ξ, η). We claim that φ(V (t, ξ)) 6= 0, if ξ ∈ C− \ {0}, t ≥ 0.

(3.8)

To prove this we apply (3.4) to deduce that V (t, ξ) ∈ C− \ {0} if ξ ∈ C− \ {0} and t ≥ 0. Since ||π− ξ|| = |φ(ξ)| we deduce from (3.5) that φ(V (t, ξ)) does not vanish in [0, ∞[. Once this claim is proved we invoke the geometry described in Step 1 and, given ξ ∈ X \ X− , we find the interval I = [σ− , σ+ ] described there. Since φ(ξ + σ+ η) and φ(ξ + σ− η) have opposite sign, the same happens to φ(V (t, ξ + σ+ η)) and φ(V (t, ξ + σ− η)). The intermediate value property for continuous functions allows us to find σ(t) ∈ I such that φ(V (t, ξ+σ(t)η)) = 0. Thus V (t, ξ+σ(t)η) ∈ X+ ⊂ C+ . Take a sequence tn → +∞ such that σ(tn ) → σ ˆ ∈ I. From (3.4) we know that V (t, ξ + σ(tn )η) ∈ C+ , t ∈] − ∞, tn ]. Letting n → ∞ we find that V (t, ξ + σ ˆ η) remains in C+ forever and so ξ + σ ˆ η ∈ W+ .


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Step 5: Conclusion. We can assume that σ(M ) ∩ {µ ∈ C : |µ| ≥ e−λT } contains at least one number µ1 , for otherwise the result is immediate. Let p ∈ L(X) be the projection with Im p = W+ , Ker p = X− and renorm X by ||ξ||p = ||pξ||B + ||(I − p)ξ||. If we apply Lemma 3.2 with L = M , L1 = M ◦ p, L2 = M ◦ (I − p) we deduce that Ker (M − µ1 I) 6= 0. From (3.7) we know that µ1 6∈ σ(M+ ). If µ1 were not real, also the conjugate would beL an eigenvalue and W+ should have codimension at least two. We have X = W+ Rη? where η? ∈ Ker (M − µ1 I) \ {0}. This implies that µ1 is simple. Also σ(M ) = σ(M+ ) ∪ {µ1 }. Finally we prove that µ1 > 0. From (3.6) it follows that e2λT µ21 B(η? , η? ) ≤ B(η? , η? ) − 2²||η? ||2? < B(η? , η? ) and therefore η? ∈ C− . Now we can apply (3.8) and deduce that the sign of φ(V (t, η? )) is constant if t ≥ 0. Since V (t, η? ) is a Floquet solution, φ(V (T, η? )) = µ1 φ(η? ) and so µ1 must be positive. Notice that related arguments were employed in [14] for the study of parabolic equations. 4. The Linear Telegraph Equation. We shall work with the system ut = v, vt = uxx − cv in the phase space X = H 1 (R/`Z) × L2 (R/`Z). This system generates a c0 -group with infinitesimal generator A(φ, ψ) = (ψ, φ00 − cψ), ξ = (φ, ψ) ∈ D = H 2 (R/`Z) × H 1 (R/`Z). Consider the projection Π : X → X, Π(φ, ψ) = (φ, ψ) where φ and ψ are the mean values. The operators A and Π commute because ΠA(φ, ψ) = AΠ(φ, ψ) = (ψ, −cψ). From here it follows that Π commutes with S(t) and the manifolds Ker Π = {(φ, ψ) ∈ X : φ = ψ = 0} and Im Π = R2 are invariant under S(t). At this moment we describe the dynamics of the telegraph equation at an intuitive level. The subspace Ker Π has codimension two and, on this subspace, the origin is asymptotically stable. In the plane Im Π the equation becomes u˙ = v, v˙ = −cv and there is a continuum of equilibria attracting the remaining solutions. The quantity W (t) = v(t)2 − γ(v(t) + cu(t))2 , γ > 0 can be thought as a ”modified energy”. It satisfies ˙ + λW = −(2c − λ)v 2 − λγ(v + cu)2 , W which is negative if 2c > λ and (u(t), v(t)) is not the trivial solution. The phase space is sketched as

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Ker Π ? D

D D D@ @ R@ R@ @ @ R@ R@ @ @ R D @ D Im Π @ @ @ @ @r@r@r@r@r D D @ @@ @@ @@ @@ @ D D I @ I I I I @@ @@ @@ @@ @D D D D 6

We go back to rigorous discussions. The first step will be to introduce a new inner product in X which is well adapted to the geometry of the group {S(t)}. Given ² > 0 and ξ1 = (φ1 , ψ1 ), ξ2 = (φ2 , ψ2 ) in X we define Z ` ² (ξ1 , ξ2 )² = {ψ˜1 ψ˜2 + φ01 φ02 + (φ˜1 ψ˜2 + φ˜2 ψ˜1 )}dx + φ1 φ2 + ψ 1 ψ 2 , 2 0 where φ˜i = φi − φi , ψ˜i = ψi − ψ i . This is a continuous and symmetric bilinear form and we shall prove that it is coercive when ² is small. Indeed, ˜ 2 2 + ||φ0 ||2 2 − ²` ||φ0 ||L2 ||ψ|| ˜ L2 + φ2 + ψ 2 , (ξ, ξ)² ≥ ||ψ|| L L 2π where we have employed the inequality ˜ L2 ≤ ` ||φ0 ||L2 , ||φ|| 2π valid for functions in H 1 (R/`Z). From these computations we µ deduce that (·, ·)² ¶ is 1 −²`/4π an inner product equivalent to the usual one when the matrix −²`/4π 1 is definite positive. This happens if 4π . (4.1) 0 ¶ ²c`

µ

² , 2

²(

c` 2 c ² ) < − 8π 2 4

(4.2)

c − 2² − 8π is definite positive. We can conclude that if ² satisfies ² − ²c` 8π 2 (4.1) and (4.2) then (A1) holds and there exists Γ² > 0 such that

the matrix

(Aξ, ξ)² ≤ −Γ² ||ξ||2² , ξ ∈ D ∩ Ker Π.

(4.3)

The inequality (4.3) explains why the origin is asymptotically stable on Ker Π. Now we go to Im Π and, inspired by the modified energy, we consider the bilinear form in R2 Σ[(x1 , y1 ), (x2 , y2 )] = y1 y2 − γ(y1 + cx1 )(y2 + cx2 ), where γ > 0. The lines y + cx = 0 and y = 0 are Σ-orthogonal and the associated quadratic form is positive along the first line and negative along the second. Finally we define the bilinear form in X B(ξ1 , ξ2 ) = ((I − Π)ξ1 , (I − Π)ξ2 )² + Σ[Πξ1 , Πξ2 ]. It has index one with X+ = {ξ = (φ, ψ) : ψ + cφ = 0}, X− = {ξ = (x, 0) : x ∈ R}. We conclude this section with a computation which will be employed later. Given ξ ∈ D and λ > 0, B(ξ, Aξ + λξ) = ((I − Π)ξ, (I − Π)Aξ)² + λ((I − Π)ξ, (I − Π)ξ)² + 2

Σ[Πξ, AΠξ + λΠξ] ≤ −(Γ² − λ)||(I − Π)ξ||2² − (c − λ)ψ − λγ(ψ + cφ)2 , where we have used (4.3). From here we conclude that if λ < min(Γ² , c) then there exists β = β(λ, γ, ²) > 0 such that B(ξ, Aξ + λξ) ≤ −β||ξ||2² , ξ ∈ D.

(4.4)

5. Proof of Theorem 1.1. We interpret (1.1), (1.2) as a first order evolution equation. Keeping the notations of the previous Section we define N : R × X → X, N (t, ξ) = (0, F (t, x, φ(x))). The results of Section 2 are not directly applicable because the function Fu can be unbounded and so we cannot guarantee that the partial derivative ∂ξ N is bounded. We are not imposing any growth condition of F at infinity but the compact immersion H 1 (R/`Z) ⊂ C(R/`Z) allows to show that the initial value problem for U˙ = AU + N (t, U ) is well posed, at least locally (see Example 4.1 in [11]). Also this immersion implies that if U = (u, v) is a solution in C(I, X) then u can be interpreted as a continuous function in the two variables, u = u(t, x) with (t, x) ∈ I × R. These facts, together with the continuous dependence with respect to initial conditions, imply that we can modify the function F outside a neighborhood of Φ. The index and stability are local properties and will not be affected by this change. Let r : R → R be a monotone and smooth function satisfying r(u) = u if |u| ≤ 1, |r(u)| = 2 if |u| ≥ 3, r0 (u) ≤ 1 for each u ∈ R. For δ > 0 we define rδ (u) = δr( uδ ) and Fδ (t, x, u) = F (t, x, Φ(t, x) + rδ (u − Φ(t, x))).

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RAFAEL ORTEGA

Since Φ belongs to C(R/T Z × R/`Z), the function Fδ is in C 0,1 and we can select δ small enough so that ∂Fδ | (t, x, u)| ≤ 2σ (t, x, u) ∈ R3 . ∂u Here we are assuming that F satisfies (1.3) for some σ > 0 which will be determined later. From now on we work with Fδ instead of F . The operator N is defined as N when F is replaced by Fδ . In order to apply the theory of Section 2 we fix ² > 0 such that (4.1) and (4.2) hold. The space X will be endowed with the inner product (·, ·)² . The operator N satisfies the general conditions of Section 2, in particular |||∂ξ N (t, ξ)||| ≤ k = k(σ). We must also check the conditions (A1) and (A2) of Lemma 2.1. The first condition was verified in the previous Section. To prove (A2) we observe that, for each R > 0, B = {φ ∈ H 1 (R/`Z) : ||φ||H 1 ≤ R} is an equicontinuous family in C(R/`Z). The same property holds for ˆ = {Fδ (t, ·, φ(·)) : t ∈ [0, T ], φ ∈ B}. B ˆ is relatively compact The Ascoli Theorem can be applied and one deduces that B 2 in C(R/`Z) and hence in L (R/`Z). This proves (A2). We are ready to complete the proof by an application of Theorem 2.1 to ϕ = ˙ The bilinear form defined in the previous Section will be employed for some (Φ, Φ). γ > 0. The number λ > 0 is chosen so that (4.4) holds. To verify (2.5) we observe that if t ∈ R and ξ ∈ X, B(ξ, ∂ξ N (t, ϕ(t))ξ) ≤ µσ||ξ||2² for some µ > 0. Finally we choose σ small enough so that the constant β appearing in (4.4) satisfies β > µσ. The proof of Theorem 1.1 is complete and, with some care, one could produce an explicit value for σ. It is interesting to observe that this value must satisfy 2π σ < ( )2 . ` Indeed, if we consider the linear equation 2π utt + cut − uxx − ( )2 u = 0, ` it has stationary solutions 2πx 2πx u(t, x) = c1 cos + c2 sin . ` ` This shows that µ = 1 is a non-simple Floquet multiplier, a situation which is not compatible with Proposition 3.1. 6. The Sine-Gordon Equation. In this Section we assume T = ` = 2π and employ the notation T = R/2πZ. As in Section 7 of [11] we consider the equation utt + cut − uxx + a sin(u + P (t, x)) = s 2

(6.1)

where a > 0, P ∈ C(T ) and s ∈ R. After a change of variables, the equation (1.5) mentioned in the Introduction can be expressed in this form. We want to apply


13

Theorem 1.1 to this equation and so we need to impose a ≤ σ. Also we want to apply the results in [11] and so we also need a ≤ ν, where ν is the constant related to the Maximum Principle and introduced in [10]. We shall impose a third restriction on a. It is linked to a result about linear equations. For convenience we state it now and postpone the proof to the end of the Section. Lemma 6.1. There exists σ1 = σ1 (c) > 0 such that if α ∈ L∞ (T2 ) satisfies ||α||L∞ ≤ σ1 2

and u ∈ C(T ) is a solution of

Z

utt + cut − uxx + α(t, x)u = µ

0

2

in D (T ),

u=0 T2

with µ a constant, then u = 0. Define σ? = min{σ, ν, σ1 } and observe that σ? is a positive constant which only depends on c. Theorem 6.1. Assume that a ≤ σ? and P ∈ C(T2 ) is a fixed function. Then there exist numbers s+ , s− with −a ≤ s− ≤ s+ ≤ a such that the equation (6.1) has • at least two doubly periodic solutions if s ∈]s− , s+ [, one of them asymptotically stable and another unstable • at least one doubly periodic solution if s = s+ or s = s− , which is not asymptotically stable • no doubly periodic solutions if s > s+ or s < s− . Proof. This result is a consequence of Theorem 7.1 in [11] excepting for the stability properties of the solutions for s ∈]s− , s+ [. A first important property for the case s ∈]s− , s+ [ is that doubly periodic solutions are isolated. The proof, based in the Lyapunov-Schmidt reduction, is similar to the well known proof for the pendulum equation (see [6]). Let us point out the differences with that proof. First we must observe that the notion of doubly periodic solution employed in this paper is equivalent to finding solutions in C(T2 ), which are understood in the sense of distributions. This is justified in page 440 of [11]. Next we apply the Lyapunov-Schmidt reduction to the doubly periodic problem for ˜ 2 ) L R, where C(T ˜ 2 ) is the subspace of func(6.1). The space will be C(T2 ) = C(T tions with mean value zero. The compactness of the generalized resolvent follows from Proposition 4.4 in [10]. The uniqueness and non-degeneracy of the solution of the auxiliary equation is a consequence of Lemma 6.1. The rest of the proof is identical to the case of the pendulum. Once we know that doubly periodic solutions are isolated we extract some information from the proof of Theorem 7.1 in [11]. There exist two solutions Φ1 and Φ2 with γ(Φ1 ) 6= γ(Φ2 ). Also, there exists a finite set S of doubly periodic solutions such that X γ(Φ) = 1. Φ∈S

The index can only take the values 1, −1 or 0 and the previous properties imply the existence of solutions Φ and Ψ with γ(Φ) = 1 and γ(Ψ) 6= 1. Then Φ is asymptotically stable and Ψ is unstable.

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RAFAEL ORTEGA

Proof of Lemma 6.1. By an indirect argument assume that there exist sequences αn , un and µn with Z ||αn ||L∞ → 0, un = 0, un 6= 0 T2

and such that un is a solution of utt + cut − uxx + αn (t, x)u = µn

in D0 (T2 ).

It is not restrictive to assume that either all µn vanish or µn = 1 for each n. Case 1. µn = 0 for all n. The function un is a solution of

Z

utt + cut − uxx = fn (t, x)

in D0 (T2 ),

u = 0, T2

where fn (t, x) = −αn (t, x)un (t, x). From Proposition 4.4 in [10] we deduce that ||un ||L∞ ≤ C||αn ||L∞ ||un ||L∞ where C is independent of n. This is not possible because ||αn ||L∞ tends to zero. Case 2. µn = 1 for all n Fix α ∈]0, 1[. The same proof as in Lemma 5.7 in [10] shows that un ∈ C 0,α (T2 ) and ||un ||C 0,α is bounded. We can extract a subsequence converging uniformly to some u ∈ C(T2 ). By a passage to the limit this function should satisfy utt + cut − uxx = 1 and this is absurd. REFERENCES [1] B. Birnir and R. Grauer, An explicit description of the global attractor of the damped and driven sine-Gordon equation, Comm. Math. Phys., 162 (1994), 539–590. [2] E.N. Dancer, Upper and lower stability and index theory for positive mappings and applications, Nonlinear Analysis TMA, 17 (1991), 205–217. [3] A. Haraux, Sist` emes dynamiques dissipatifs et applications, Masson, Paris, 1991. [4] J.S. Kolesov, Schauder’s principle and the stability of periodic solutions, Soviet Math. Dokl., 10 (1969), 1290–1293. [5] R.D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pur. Appl., 89 (1971), 217–258. [6] F.I. Njoku and P. Omari, Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions, App. Math. Comp., 135 (2003), 471–490. [7] R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc., 42 (1990), 505–516. [8] R. Ortega, Some applications of the topological degree to stability theory, in “Topological Methods in Differential Equations and Inclusions”, pp. 377–409, Kluwer Academic, Dordrecht, 1995. [9] R. Ortega, A criterion for asymptotic stability based on topological degree, in “Proceedings of the First World Congress of Nonlinear Analysts”, pp 383–394, Walter de Gruyter, Berlin, 1996. [10] R. Ortega and A. Robles-P´ erez, A Maximum Principle for periodic solutions of the telegraph equation, J. Math. Anal. Appl. 221 (1998), 625–651. [11] R. Ortega and A. Robles-P´ erez, A Duality Theorem for periodic solutions of a class of second order evolution equations, J. Diff. Equs., 172 (2001), 409–444. [12] F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Pub., New York, 1990. [13] G.R. Sell and Y. You, Dynamics of evolutionary equations, Springer, New York, 2002.


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[14] R.A. Smith, Certain differential equations have only isolated periodic orbits, Ann. Mat. Pura Appl., 137 (1984), 217–244. [15] R.A. Smith, Poincar´ e-Bendixon theory for certain reaction-diffusion boundary value problems, Proc. Roy. Soc. Edinburgh, 124 (1994), 33–69. [16] P.J. Torres, Existence and stability of periodic solutions of a Duffing equation using a new maximum principle, Mediterranean Journal of Mathematics, 1 (2004), 479–486. [17] S.V. Zelik, The Mathieu-Hill Operator Equation with dissipation and estimates on its instability index, Math. Notes, 61 (1997), 451–464.

Received December 2004; revised April 2005. E-mail address: [email protected]