Mama Abdelli
email: abdelli-mama@yahoo.fr

María Anguiano
email: anguiano@us.es

Alain Haraux
email: haraux@ann.jussieu.fr

Existence, uniqueness and global behavior of the solutions to some nonlinear vector equations in a finite dimensional Hilbert space

Keywords: AMS classification numbers: 34A34, 34C10, 34D05, 34E99 Existence of the solution, Uniqueness of the solution, Decay rate

The initial value problem and global properties of solutions are studied for the vector equation:

) = 0 in a finite dimensional Hilbert space under suitable assumptions on g.

Introduction

Let H be a finite dimensional real Hilbert space, with norm denoted by . . We consider the following nonlinear equation

u ′ l u ′ ′ + A 1 2 u β Au + g(u ′ ) = 0, (1.1)
where l and β are positive constants, and A is a positive and symmetric linear operator on H. We denote by (•, •) the inner product in H. The operator A is coercive, which means : ∃λ > 0, ∀u ∈ D(A), (Au, u) ≥ λ u 2 .

We also define ∀u ∈ H, A

1 2 u := u D(A 1 2 )
, a norm equivalent to the norm in H. We assume that g : H → H is locally Lipschitz continuous.

When l = 0 and g(u ′ ) = c|u ′ | α u ′ Haraux [START_REF] Haraux | Sharp decay estimates of the solutions to a class of nonlinear second order ODE's[END_REF] studied the rate of decay of the energy of non-trivial solutions to the scalar second order ODE. In addition, he showed that if α > β β+2 all non-trivial solutions are oscillatory and if α < β β+2 they are non-oscillatory. In the oscillatory case he established that all non-trivial solutions have the same decay rates, while in the non-oscillatory case he showed the coexistence of exactly two different decay rates, calling slow solutions those which have the lowest decay rate, and fast solutions the others. [START_REF] Abdelli | Global behavior of the solutions to a class of nonlinear second order ODE's[END_REF] studied the scalar second order ODE where g(u ′ ) = c|u ′ | α u ′ , they proved the existence and uniqueness of a global solution with initial data (u 0 , u 1 ) ∈ R 2 . They used some modified energy function to estimate the rate of decay and they used the method introduced by Haraux [START_REF] Haraux | Sharp decay estimates of the solutions to a class of nonlinear second order ODE's[END_REF] to study the oscillatory or non-oscillatory of non-trivial solutions. If α > β(l+1)+l β+2 all non-trivial solutions are oscillatory and if α < β(l+1)+l β+2 they are non-oscillatory. In the non-oscillatory rate, as in the case l = 0, the coexistence of exactly two different decay rates was established.

Abdelli and Haraux

In this article, we use some techniques from Abdelli and Haraux [START_REF] Abdelli | Global behavior of the solutions to a class of nonlinear second order ODE's[END_REF] to establish a global existence and uniqueness result of the solutions, and under some additional conditions on g (typically g(s) ∼ c s α s), we study the asymptotic behavior as t → ∞. A basic role will be played by the total energy of the solution u given by the formula

E(t) = l + 1 l + 2 u ′ (t) l+2 + 1 β + 2 A 1 2 u(t) β+2 . (1.
2)

The plan of this paper is as follows: In Section 2 we establish some basic preliminary inequalities, and in Section 3 we prove the existence of a solution u ∈ C 1 (R + , H) with u ′ l u ′ ∈ C 1 (R + , H) for any initial data (u 0 , u 1 ) ∈ H × H under relevant conditions on g and the conservation of total energy for each such solution. In Section 4 we establish the uniqueness of the solution in the same regularity class under additional conditions on g. In Section 5 we prove convergence of all solutions to 0 under more specific conditions on g and we estimate the decay rate of the energy. Finally, in Section 6, we discuss the optimality of these estimates when g(s) = c s α s and l < α < β(1+l)+l β+2 ; in particular, by relying on a technique introduced by Ghisi, Gobbino and Haraux [START_REF] Ghisi | Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations[END_REF], we prove the existence of a open set of initial states giving rise to slow decaying solutions. In our last result, by relying on a technique introduced by Ghisi, Gobbino and Haraux [START_REF] Ghisi | Finding the exact decay rate of all solutions to some second order evolution equation with dissipation[END_REF], we prove that all non-zero solutions are either slow solutions or fast solutions.

Some basic inequalities

In this section, we establish some easy but powerful lemmas which generalize Lemma 2.2 and Lemma 2.3 from [START_REF] Aloui | Haraux Compactness of trajectories to some nonlinear second order evolution equations and applications[END_REF] and will be essential for the existence and uniqueness proofs of the next section. Troughout this section, H denotes an arbitrary (not necessary finite dimensional) real Hilbert space with norm denoted by . .

Proposition 2.1. Let (u, v) ∈ H × H and (α, β) ∈ IR + × IR + . Then the condition (α -β)( u -v ) ≥ 0, implies (αu -βv)(u -v) ≥ 1 2 (α + β) u -v 2 .
Proof. An easy calculation gives the identity

(αu -βv)(u -v) = 1 2 (α -β)( u 2 -v 2 ) + 1 2 (α + β) u -v 2 . Since (α -β)( u 2 -v 2 ) = ( u + v )(α -β)( u -v ),
the result follows immediately.

For the next results, we consider a number R > 0 and we set

J R = [0, R]; B R := {u ∈ H, u ≤ R} Proposition 2.2.
Let f : J R -→ IR + be non increasing and such that for some positive numbers p, c:

∀s ∈ J R , f (s) ≥ cs p .
Then we have

∀(u, v) ∈ B R × B R , (f ( u )u -f ( v )v)(u -v) ≥ c 2 ( u p + v p ) u -v 2 .
Proof. Applying the previous result with α = f ( u ) and β = f ( v ) the result follows immediately.

Corollary 2.3. Let f : J R -→ IR + be non increasing and such that for some positive numbers p, c:

∀s ∈ J R , f (s) ≥ cs p .
Then we have

∀(u, v) ∈ B R × B R , f ( u )u -f ( v )v) ≥ c 2 ( u p + v p ) u -v .
Proof. This inequality follows immediately from Cauchy-Schwarz inequality combined with the conclusion of the previous proposition.

Proposition 2.4. Let f : J R -→ IR + be non increasing and such that for some positive numbers p, c:

∀s ∈ J R , f (s) ≥ cs p .
Then we have

∀(u, v) ∈ B R × B R , (f ( u )u -f ( v )v)(u -v) ≥ δ u -v p+2 , with δ = c 2 max{p,1} .
Proof. It is sufficient to apply the previous result combined with the inequality

u -v p ≤ 2 max{p-1,0} ( u p + v p ).
Corollary 2.5. Let f : J R -→ IR + be non increasing and such that for some positive numbers p, c:

∀s ∈ J R , f (s) ≥ cs p .
Then we have for some constant C = C(c, p) > 0,

∀(u, v) ∈ B R × B R , u -v ≤ C f ( u )u -f ( v )v) 1 p+1 .
Proof. This inequality follows immediately from Cauchy-Schwarz inequality combined with the conclusion of the previous proposition.

Lemma 2.6. Assume that A is a positive, symmetric, bounded operator on H. for some constant D > 0 we have

∀(w, v) ∈ H × H ∀(w, v) ∈ H × H, A 1 2 w β Aw -A 1 2 v β Av ≤ D max( A 1 2 v , A 1 2 w ) β w -v .
Proof. We can write

A 1 2 w β Aw -A 1 2 v β Av = A 1 2 ( A 1 2 w β A 1 2 w -A 1 2 v β A 1 2 v) ≤ A 1 2 A 1 2 w β A 1 2 w -A 1 2 v β A 1 2 v .
Direct calculations (see also [START_REF] Aloui | Haraux Compactness of trajectories to some nonlinear second order evolution equations and applications[END_REF], Lemma 2.2) yield

A 1 2 w β A 1 2 w -A 1 2 v β A 1 2 v ≤ C 1 max( A 1 2 v , A 1 2 w ) β A 1 2 w -A 1 2 v ≤ C 2 max( A 1 2 v , A 1 2 w ) β w -v ,
and the result is proved.

3 Global existence and energy conservation for equation (1.1) In this section, we study the existence of a solution for the initial value problem associated to equation (1.1) where g : H → H is a locally Lipschitz continuous function which satisfies the following hypothesis:

∃k 1 > 0, k 2 > 0, ∀v, (g(v), v) ≥ -k 1 -k 2 v l+2 . ( 3
u ∈ C 1 (R + , H), u ′ l u ′ ∈ C 1 (R + , H) and u(0) = u 0 , u ′ (0) = u 1 .
Proof. To show the existence of the solution for (1.1), we consider the auxiliary problem

(ε + u ′ ε 2 ) l/2 u ′′ ε + l(u ′ ε , u ′′ ε )(ε + u ′ ε 2 ) l/2-1 u ′ ε + A 1 2 u ε β Au ε + g(u ′ ε ) = 0, u ε (0) = u 0 , u ′ ε (0) = u 1 . (3.2)
Here, ε > 0 is a small parameter, devoted to tend to zero. For simplicity in the sequel we shall write l/2 := m > 0.

Assuming the existence of such a solution u ε , multiplying (3.2) by u ′ ε , we find

[(ε + u ′ ε 2 ) m + 2m(ε + u ′ ε 2 ) m-1 u ′ ε 2 ](u ′ ε , u ′′ ε ) + A 1 2 u ε β (Au ε , u ′ ε ) + (g(u ′ ε ), u ′ ε ) = 0, then (u ′ ε , u ′′ ε ) = - A 1 2 u ε β (Au ε , u ′ ε ) + (g(u ′ ε ), u ′ ε ) (ε + u ′ ε 2 ) m-1 (ε + (l + 1) u ′ ε 2 ) . ( 3.3)
From (3.2), we obtain that u ε is a solution of

   (ε + u ′ ε 2 ) m u ′′ ε -l A 1 2 u ε β (Au ε , u ′ ε ) + (g(u ′ ε ), u ′ ε ) ε + (l + 1) u ′ ε 2 u ′ ε + A 1 2 u ε β Au ε + g(u ′ ε ) = 0, u ε (0) = u 0 , u ′ ε (0) = u 1 . (3.4)
Conversely, (3.4) implies (3.3). Then, replacing (3.3) in (3.4), we obtain (3.2). Therefore (3.2) is equivalent to (3.4).

i) A priori estimates:

Next, we introduce

F ε (u ε , u ′ ε ) = l A 1 2 u ε β (Au ε , u ′ ε ) + (g(u ′ ε ), u ′ ε ) (ε + u ′ ε 2 ) m (ε + (l + 1) u ′ ε 2 ) u ′ ε - A 1 2 u ε β Au ε + g(u ′ ε ) (ε + u ′ ε 2 ) m .
Then (3.4), can be rewritten as

u ′′ ε = F ε (u ε , u ′ ε ), u ε (0) = u 0 , u ′ ε (0) = u 1 .
(3.5)

Since the vector field F ε is locally Lipschitz continuous, the existence and uniqueness of u ε in the class C 2 ([0, T ), H), for some T > 0 is classical. Multiplying (3.2) by u ′ ε , we obtain by a simple calculation the following energy identity:

d dt E ε (t) + (g(u ′ ε (t)), u ′ ε (t)) = 0,
where

E ε (t) = l + 1 l + 2 (ε + u ′ ε 2 ) m+1 -ε(ε + u ′ ε 2 ) m + 1 β + 2 A 1 2 u ε (t) β+2 .
Indeed, for any function v ∈ C 1 ([0, T ), H) we have the following sequence of identities

(((ε + v 2 ) m v) ′ , v) = 2m(ε + v 2 ) m-1 v 2 (v, v ′ ) + (ε + v 2 ) m (v, v ′ ) = 2m(ε + v 2 ) m-1 (ε + v 2 )(v, v ′ ) -2mε(ε + v 2 ) m-1 (v, v ′ ) + (ε + v 2 ) m (v, v ′ ) = (2m + 1)(ε + v 2 ) m (v, v ′ ) -2mε(ε + v 2 ) m-1 (v, v ′ ) = d dt l + 1 l + 2 (ε + v 2 ) m+1 -ε(ε + v 2 ) m .
Moreover for some constant C > 0 independent of ε, we have

E ε (t) + C ≥ l + 1 l + 2 (ε + u ′ ε 2 ) m+1 -ε(ε + u ′ ε 2 ) m + C ≥ 1 4 u ′ ε (t) l+2
and then as a consequence of (3.1)

d dt E ε (t) = -(g(u ′ ε (t)), u ′ ε (t)) ≤ k 1 + k 2 u ′ ε (t) l+2 ≤ k 3 + k 4 E ε (t). By Gronwall's inequality, this implies ∀t ∈ [0, T ), u ε (t) ≤ M 1 , u ′ ε (t) ≤ M 2 , (3.6)
for some constants M 1 , M 2 independent of ε. Hence, u ε and u ′ ε are uniformly bounded and u ε is a global solution, in particular T > 0 can be taken arbitrarily large.

ii) Passage to the limit:

In order to pass to the limit as ε → 0 we need to know that u ε and u ′ ε are uniformly equicontinous on (0, T ) for any T > 0. For u ε it is clear, since u ′ ε is bounded. Moreover we have

(ε + u ′ ε 2 ) l/2 u ′′ ε + l(u ′ ε , u ′′ ε )(ε + u ′ ε 2 ) l/2-1 u ′ ε = ((ε + u ′ ε 2 ) l/2 u ′ ε ) ′ ,
hence by the equation

((ε + u ′ ε 2 ) l/2 u ′ ε ) ′ ≤ A 1 2 u ε (t) β Au ε (t) + g(u ′ ε (t)) ,
and since g is locally Lipschitz continuous, hence bounded on bounded sets, we obtain

((ε + u ′ ε 2 ) l/2 u ′ ε ) ′ ≤ M 3 . Therefore the functions (ε + u ′ ε 2 ) m u ′
ε are uniformly Lipschitz continuous on R+. We claim that u ′ ε is a uniformly (with respect to ε) locally Hölder continuous function of t. Indeed by applying Corollary 2.5 with f (s) = (ε + s 2 ) m and p = 2m = l we find for some C > 0

u ′ ε (t 1 )-u ′ ε (t 2 ) ≤ C (ε+ u ′ ε 2 ) m u ′ ε (t 1 )-(ε+ u ′ ε 2 ) m u ′ ε (t 2 ) 1 l+1 ≤ C ′ t 1 -t 2 1 l+1 .
As a consequence of Ascoli's theorem and a priori estimate (3.6) combined with (3.5), we may extract a subsequence which is still denoted for simplicity by (u ε ) such that for every T > 0

u ε → u in C 1 ((0, T ), H),
as ε tends to 0. Integrating (3.2) over (0, t), we get

(ε + u ′ ε 2 ) m u ′ ε (t) -(ε + u ′ ε 2 ) m u ′ ε (t)(0) (3.7) = - t 0 A 1 2 u ε (s) β Au ε (s) ds - t 0 g(u ′ ε (s)) ds.
From (3.7), we then have, as ε tends to 0

(ε+ u ′ ε 2 ) m u ′ ε (t) → - t 0 A 1 2 u(s) β Au(s) ds- t 0 g(u ′ (s)) ds+ u ′ (0) l u ′ (0) in C 0 ((0, T ), H).
Hence

u ′ l u ′ = - t 0 A 1 2 u(s) β Au(s) ds - t 0 g(u ′ (s)) ds + u ′ (0) l u ′ (0), (3.8)
and u ′ l u ′ ∈ C 1 ((0, T ), H). Finally by differentiating (3.8) we conclude that u is a solution of (1.1).

Remark 3.2. It is not difficult to see that the solution u constructed in the existence theorem satisfies the energy identity

d dt E(t) = -(g(u ′ (t)), u ′ (t)).
The following stronger result shows that this identity is true for any solution even if uniqueness is not known. For infinite dimensional equations such as the Kirchhoff equation, both uniqueness and the energy identity for general weak solutions are old open problems.

Theorem 3.3. Let (u 0 , u 1 ) ∈ H × H. Then any solution u of (1.1) such that u ∈ C 1 (R + , H), u ′ l u ′ ∈ C 1 (R + , H) and u(0) = u 0 , u ′ (0) = u 1 , satisfies the formula d dt E(t) = -(g(u ′ (t)), u ′ (t)). (3.9)
The proof of this new result relies on the following simple lemma Lemma 3.4. Let J be any interval, assume that v ∈ C(J, H) and

v l v ∈ C 1 (J, H) then v(t) l+2 ∈ C 1 (J, H), and
(( v(t) l v(t)) ′ , v(t)) = d dt l + 1 l + 2 v(t) l+2 , ∀t ∈ J.
Proof. Case 1. For any t 0 ∈ J, if v(t 0 ) = 0 then v(t) = 0 in the neighborhood of t 0 and in this neighborhood we have v ∈ C 1 (J, H) with

d dt v(t) l+2 = d dt v(t) 2 l/2+1 = l 2 + 1 ( v(t) 2 ) l/2 d dt v(t) 2 = 2 l 2 + 1 v(t) l (v(t), v ′ (t)).
(3.10)

On the other hand, we find

(( v(t) l v(t)) ′ , v(t)) = ( v(t) l v ′ (t), v(t)) + ( v(t) l ) ′ (v(t), v(t)) = v(t) l (v ′ (t), v(t)) + l v(t) l (v ′ (t), v(t)) = (l + 1) v(t) l (v ′ (t), v(t)), (3.11)
and from (3.10)-(3.11), we obtain

(( v(t) l v(t)) ′ , v(t)) = d dt l + 1 l + 2 v(t) l+2 , for t = t 0 . Case 2. If v(t 0 ) = 0 then since v(t) l v(t) ∈ C 1 (J, H) , we have clearly (( v l v) ′ (t 0 ), v(t 0 )) = 0.
Moreover in the neighborhood of t 0 we have

v(t) l v(t) = 0(|t -t 0 |), in other terms v(t) l+1 ≤ C|t -t 0 |, therefore v(t) l+2 ≤ C l+2 l+1 |t -t 0 | 1+ 1 l+1 , then ( v(t) l+2 ) ′ = 0, for t = t 0 ,
and finally

(( v(t) l v(t)) ′ , v(t)) = d dt l + 1 l + 2 v(t) l+2 = 0 for t = t 0 .
We now give the proof of Theorem 3.3.

Proof. Setting v = u ′ , from Lemma 3.4, we deduce

(( u ′ (t) l u ′ (t)) ′ , u ′ (t)) = d dt l + 1 l + 2 u ′ (t) l+2 , ∀t ∈ J.
By multiplying equation (1.1) by u ′ , we obtain easily

d dt E(t) = -(g(u ′ (t)), u ′ (t)).
4 Uniqueness of solution for (u 0 , u 1 ) given

In this section we suppose that

∀R ∈ IR + , ∀(u, v) ∈ B R × B R , g(u) -g(v) ≤ k 3 (R) u -v , (4.1)
for some positive constant k 3 (R).

Theorem 4.1. Let (u 0 , u 1 ) ∈ H × H, J an interval of R and t 0 ∈ J. Then (1.1) has at most one solution

u ∈ C 1 (J, H), u ′ l u ′ ∈ C 1 (J, H) and u(t 0 ) = u 0 , u ′ (t 0 ) = u 1 .
Remark 4.2. The uniqueness of solutions of (1.1) will be proved under conditions on the initial data (u 0 , u 1 ). The next proposition concerns the uniqueness result for u 1 = 0.

Proposition 4.3. Let τ ∈ R + and J = (τ, T ), T > τ . Then there is at most one solution of (1.1) with u(τ ) = u 0 and u ′ (τ ) = u 1 for T -τ small enough such that u ∈ C 1 (J, H), and

u ′ l u ′ ∈ C 1 (J, H).
Proof. Since u ′ (τ ) = 0, the second derivative u ′′ (τ ) exists and u ′′ (t) also exists for τ ≤ t < τ + ε with ε small enough. On (T, T + ε), (1.1) reduces to

u ′′ = l u ′ 2( l 2 -1) u ′ A 1 2 u β (Au, u ′ ) + (g(u ′ ), u ′ ) (l + 1) u ′ l u ′ l - A 1 2 u β Au + g(u ′ ) u ′ l ,
the existence and uniqueness of u in the class C 2 (J, H) for this equation is classical.

Proposition 4.4. Let a = 0. Then for J an interval containing 0 and such that |J| is small enough, equation (1.1) has at most one solution satisfying

u ∈ C 1 (J, H), u ′ l u ′ ∈ C 1 (J, H) and u 0 = a, u 1 = 0.
Proof. From (1.1), we obtain

( u ′ l u ′ ) ′ (0) = -A 1 2 a β Aa = ξ,
where ξ = 0. We set u ′ l u ′ = ψ(t), then ψ(0) = 0. For any t = 0 we have

ψ(t) t = 1 t t 0 ψ ′ (s) ds.
It follows that ψ(t) t → ξ as t → 0, therefore for |t| small enough we have,

ψ(t) ≥ t 2 ξ .
Hence, u ′ l ≥ ηt l l+1 for |t| small enough and some η > 0.

Integrating (1.1) over (0, t), we have since u ′ (0) = 0

u ′ (t) l u ′ (t) = - t 0 A 1 2 u(τ ) β Au(τ ) dτ - t 0 g(u ′ (τ )) dτ.
Let u(t) and v(t) be two solutions, then

w(t) = u(t) -v(t) satisfies u ′ (t) l u ′ (t) -v ′ (t) l v ′ (t) = - t 0 ( A 1 2 u(τ ) β Au(τ ) -A 1 2 v(τ ) β Av(τ )) dτ - t 0 (g(u ′ (τ )) -g(v ′ (τ ))) dτ. (4.2)
Applying Corollary 2.3 with f (s) = s l and p = l , we get

u ′ (t) l u ′ (t) -v ′ (t) l v ′ (t) ≥ ηt l l+1 u ′ (t) -v ′ (t) ,
and applying Lemma 2.6, (4.1) and from (4.2), we now deduce

w ′ (t) ≤ C t l l+1 t 0 τ 0 w ′ (s) ds dτ + C t l l+1 t 0 w ′ (τ ) dτ ≤ C(T )t -l l+1 t 0 w ′ (τ ) dτ. (4.3) Setting φ(t) = t 0 w ′ (τ ) dτ , by solving (4.3) on [δ, t] we obtain φ(t) ≤ φ(δ)e C(T ) t δ s -l/(l+1) ds ,
and by letting δ → 0 we conclude that w(t) = 0. A similar argument gives the uniqueness for t negative with |t| small enough.

Proposition 4.5. For any interval J and any t 0 ∈ J if a solution u of (1.1) satisfies

u ∈ C 1 (J, H), u ′ l u ′ ∈ C 1 (J, H) and u(t 0 ) = u ′ (t 0 ) = 0, then u ≡ 0.
Proof. From theorem 2.3 we know that

d dt E(t) = -(g(u ′ (t)), u ′ (t)).
Using (4.1), we have

| d dt E(t)| = |(g(u ′ (t)), u ′ (t))| ≤ k 3 u ′ l+2 ≤ k 3 E(t).
Now, let t 0 ∈ J such that E(t 0 ) = 0. By integration we get

|E(t)| ≤ |E(t 0 )|e k 3 |t-t 0 | = 0.
5 Energy estimates for equation (1.1)

In this section, we suppose that

∃η 1 > 0, ∀v, g(v) ≤ η 1 v α+1 , (5.1)
and

∃η 2 > 0, ∀v, (g(v), v) ≥ η 2 v α+2 , (5.2)
for some α > 0.

Theorem 5.1. Assuming α > l, there exists a positive constant η such that if u is any solution of (1.1) with E(0) = 0 lim inf t→+∞ t l+2 α-l E(t) ≥ η.

(5.3)

(i) If α ≥ β(1+l)+l β+2 , then there is a constant C(E(0)) depending on E(0) such that ∀t ≥ 1, E(t) ≤ C(E(0))t -l+2 α-l . (ii) If α < β(1+l)+l β+2
, then there is a constant C(E(0)) depending on E(0) such that

∀t ≥ 1, E(t) ≤ C(E(0))t -(α+1)(β+2) β-α .
Proof. From the definition of E(t) we have

u ′ (t) α+2 ≤ C(l, α)E(t) α+2 l+2 ,
where C(l, α) is a positive constant, hence from (3.9) and (5.1) we deduce

d dt E(t) ≥ -C(l, α, η 1 )E(t) α+2 l+2 .
Assuming α > l we derive

d dt E(t) -α-l l+2 = - α -l l + 2 E ′ (t)E(t) -α+2 l+2 ≤ α -l l + 2 C(l, α, η 1 ) = C 1 .
By integrating, we get

E(t) ≥ (E(0) -α-l l+2 + C 1 t) -l+2 α-l , implying lim inf t→+∞ t l+2 α-l E(t) ≥ η = C -l+2 α-l 1 .
Hence (5.3) is proved. Now, we show (i) and (ii), we consider the perturbed energy function

E ε (t) = E(t) + ε( u 2γ u, u ′ l u ′ ), (5.4)
where l > 0, γ > 0 and ε > 0.

By Young's inequality, with the conjugate exponents l + 2 and l + 2 l + 1 , we get

|( u 2γ u, u ′ l u ′ )| ≤ C 1 A 1 2 u (2γ+1)(l+2) + c 2 u ′ l+2 .
We choose γ so that (2γ + 1)(l + 2) ≥ β + 2, which reduces to

γ ≥ β -l 2(l + 2) . (5.5)
Then, for some C 1 > 0, M > 0, we have

|( u 2γ u, u ′ l u ′ )| ≤ C 1 A 1 2 u β+2 + c 2 u ′ l+2 ≤ M E(t).
(5.6)

By using (5.6), we obtain from (5.4)

(1 -M ε)E(t) ≤ E ε (t) ≤ (1 + M ε)E(t). Taking ε ≤ 1 2M , we deduce ∀t ≥ 0, 1 2 E(t) ≤ E ε (t) ≤ 2E(t).
(5.7)

On the other hand, we have

E ′ ε (t) = -(g(u ′ ), u ′ ) + ε(( u 2γ u) ′ , u ′ l u ′ ) + ε( u 2γ u, ( u ′ l u ′ ) ′ ).
We observe that

( u 2γ u) ′ = (( u 2 ) γ ) ′ u + u 2γ u ′ = 2γ u 2(γ-1) (u, u ′ )u + u 2γ u ′ ,
and we have

(( u 2γ u) ′ , u ′ l u ′ ) = 2γ u 2(γ-1) u ′ l (u, u)(u ′ , u ′ ) + ( u 2γ u ′ , u ′ l u ′ ) = 2γ u 2γ u ′ l+2 + ( u 2γ u ′ , u ′ l u ′ ).
This is equivalent to the condition

γ ≥ β -α 2(α + 1)
, (5.13) and taking δ small enough, we have

-ε( u 2γ u, g(u ′ )) ≤ ε 4 A 1 2 u 2γ+β+2 + P ′ ε u ′ α+2 .
By replacing in (5.12), we have

E ′ ε (t) ≤ (-η 2 + Qε) u ′ α+2 - ε 2 A 1 2 u 2γ+β+2 ,
where Q = P + P ′ . By choosing ε small, we get

E ′ ε (t) ≤ - ε 2 u ′ α+2 + A 1 2 u 2γ+β+2 ≤ - ε 2 ( u ′ l+2 ) α+2 l+2 + ( A 1 2 u β+2 ) 2γ+β+2 β+2
.

(5.14) This inequality will be satisfied under the assumptions (5.5), (5.10) and ( 5.13) which lead to the sufficient condition

γ ≥ γ 0 = max β -l 2(l + 2) , (β + 2)(α -l) 2(l + 2) , β -α 2(α + 1)
.

(5.15)

We now distinguish 2 cases.

(i) If α ≥ β(1 + l) + l β + 2 , then clearly (β + 2)(α -l) 2(l + 2) ≥ β -l 2(l + 2) . Moreover β -α 2(α + 1) = 1 2 β + 1 α + 1 -1 ≤ 1 2   β + 1 β(1+l)+l β+2 + 1 -1   = β -l 2(l + 2)
.

In this case γ 0 = (β + 2)(α -l) 2(l + 2) and choosing γ = γ 0 , we find

2γ + β + 2 = α + 2 l + 2 (β + 2), since 2γ + β + 2 β + 2 = 1 + α -l l + 2
, replacing in (5.14), we obtain for some ρ > 0

E ′ ε (t) ≤ -ρE(t) 1+ α-l l+2 ≤ -ρ ′ E ε (t) 1+ α-l l+2 ,
where ρ and ρ ′ are positive constants.

(ii) If α < β(l + 1) + l β + 2 , then (β + 2)(α -l) l + 2 < β -l l + 2 . Moreover β -α 2(α + 1) - β -l 2(l + 2) = (β -α)(l + 2) -(β -l)(α + 1) 2(α + 1)(l + 2) = β(l + 1) + l -α(β + 2) 2(α + 1)(l + 2) > 0.
In this case

γ 0 = β -α 2(α + 1)
and choosing γ = γ 0 , we find

2γ + β + 2 = (β + 2) 1 + 2γ β + 2 = (β + 2) 1 + β -α (α + 1)(β + 2) , (5.16) since γ > (β + 2)(α -l) l + 2 , we have 2γ + β + 2 β + 2 = 1 + 2γ β + 2 > 1 + α -l l + 2 = α + 2 l + 2 ,
replacing in (5.14), we obtain

E ′ ε (t) ≤ -δε( u ′ l+2 + A 1 2 u β+2 ) 2γ+β+2 β+2
, for some δ > 0. Using (5.16), we have

E ′ ε (t) ≤ -ρE (1+ β-α (α+1)(β+2) ) ≤ -ρ ′ E ε (t) (1+ β-α (α+1)(β+2) ) ,
where ρ and ρ ′ are positive constants.

6 Slow and fast solutions for equation (1.1)

In the case where α < β(1+l)+l β+2 , Theorem 5.1 gives two different decay rates for the lower and the upper estimates of the energy. In the scalar case, this fact was explained in [START_REF] Abdelli | Global behavior of the solutions to a class of nonlinear second order ODE's[END_REF] by the existence of two (and only two) different decay rates of the solutions, corresponding precisely to the lower and upper estimates. The solutions behaving as the lower estimate were called "fast solutions" and those behaving as the upper estimate were called "slow solutions." Moreover in the scalar case, it was shown that the set of initial data giving rise to "slow solutions" has non-empty interior in the phase space IR 2 .

In the general case, by reducing the problem to a related scalar equation, it is rather immediate to show the coexistence of slow and fast solutions in the special case of power nonlinearities. More precisely we have Proposition 6.1. Let α < β(1+l)+l β+2 and c > 0. Then the equation

( u ′ l u ′ ) ′ + A 1 2 u β Au + c u ′ α u ′ = 0
has an infinity of "fast solutions" with energy comparable to t -l+2 α-l and an infinity of "slow solutions" with energy comparable to t

- (α+1)(β+2) β-α
as t tends to infinity.

Proof. Let λ > 0 be any eigenvalue of A and Aϕ = λϕ with ϕ = 1. Let (v 0 , v 1 ) ∈ IR 2 and v be the solution of

(|v ′ | l v ′ ) ′ + C 1 |v| β v + C 2 |v ′ | α v ′ = 0,
where C 1 , C 2 are positive constants to be chosen later. Then

u(t) = v(t)ϕ satisfies ( u ′ l u ′ ) ′ + A 1 2 u β Au + c u ′ α u ′ = (|v ′ | l v ′ ) ′ ϕ l ϕ + |v| β v A 1 2 ϕ β Aϕ + c|v ′ | α v ′ ϕ α ϕ = [(|v ′ | l v ′ ) ′ + λ β 2 +1 |v| β v + c|v ′ | α v ′ ]ϕ. Choosing C 1 = λ β 2 +1 and C 2 = c , u(t) = v(t)
ϕ becomes a solution of the vector equation. The existence of an infinity of "fast solutions" and an infinity of "slow solutions" are then an immediate consequence of the same result for the scalar equation proven in [START_REF] Abdelli | Global behavior of the solutions to a class of nonlinear second order ODE's[END_REF]. Remark 6.2. In the special case A = λI, we can take for ϕ any vector of the sphere ϕ = 1. We obtain in this way an open set of slow solutions in R N +1 corresponding to the initial data of the form (v 0 ϕ, v 1 ϕ). Actually, in the general case, by generalizing a modified energy method introduced in [START_REF] Ghisi | Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations[END_REF] , under the additional condition l < 1 we shall now prove the existence of an open set of slow solutions. Theorem 6.3. Assume that g satisfies (5.2) and l < 1, l < α < β(1+l)+l β+2 . Then, there exist a nonempty open set S ⊂ H ×H and a constant M such that, for every (u 0 , u 1 ) ∈ S, the unique global solution of equation (1.1) with initial data (u 0 , u 1 ) satisfies

u(t) ≥ M (1 + t) α+1 β-α ∀t ≥ 0. ( 6.1)
Proof. Assuming (u 0 , u 1 ) ∈ H × H and u 0 = 0, we consider the following constants

σ 0 := (β + 2)(l + 1) l + 2 u 1 l+2 + A 1 2 u 0 β+2 1 β+2
, and

σ 1 := u 1 2 A 1 2 u 0 2( β-α α+1 +1)
.

For any ε 0 > 0, ε 1 > 0, the set S ⊂ H × H of initial data such that σ 0 < ε 0 , σ 1 < ε 1 is clearly a nonempty open set which contains at least all pairs (u 0 , u 1 ) with u 1 = 0 and u 0 = 0 with u 0 small enough. We claim that if ε 0 and ε 1 are small enough, for any (u 0 , u 1 ) ∈ S, the global solution of (1.1) satisfies (6.1).

First of all, from (3.9) and (5.2), we see that

d dt E(t) ≤ -η 2 u ′ (t) α+2 < 0,
hence E(t) ≤ E(0) for every t ≥ 0, and from (1.2), we deduce that

A 1 2 u(t) ≤ [(β + 2)E(0)] 1 β+2 = σ 0 ∀t ≥ 0. (6.2)
Then, we shall establish that if ε 0 and ε 1 are small enough, we have

u(t) = 0 ∀t ≥ 0, (6.3)
and for some C > 0

u ′ (t) 2 A 1 2 u(t) 2( β-α α+1 +1) ≤ C ∀t ≥ 0. (6.4)
Assuming this inequality, it follows immediately that u is a slow solution. Indeed let us set y(t) := u(t) 2 . We observe that

y ′ (t) = 2 u ′ (t), u(t) ≤ 2 u ′ (t) • u(t) ≤ 2 u ′ (t) A 1 2 u(t) β-α α+1 +1 • A 1 2 u(t) β-α α+1 +1 u(t) ≤ 2 √ C u(t) β-α α+1 +2 ≤ 2 √ C |y(t)| β-α 2(α+1) +1 ,
and in particular

y ′ (t) ≥ -2 √ C |y(t)| β-α 2(α+1) +1 ∀t ≥ 0.
Taking into account that y(t)

-β-α 2(α+1) ′ = - β -α 2(α + 1) y(t) -β-α 2(α+1) -1 y ′ (t),
we have

y(t) -β-α 2(α+1) ′ ≤ √ C β -α α + 1 .
Integrating between 0 and t and since y(0) > 0, we deduce that there exists a constant M 1 such that y(t)

-β-α 2(α+1) ≤ M 1 (t + 1) ∀t ≥ 0.
This inequality concludes the proof. So we are left to prove (6.3) and (6.4).

where

δ 1 := ρ η 2 1 -l α + 2 -l and δ 2 := ρ η 2 α + 1 α + 2 -l
, and taking into account that γ = β+1 α+1 , we deduce

|H 2 | ≤ δ 1 u ′ (t) 2 A 1 2 u(t) 2γ α+2-l 2 A 1 2 u(t) γ(α+2-l) A 1 2 u(t) 2γ + δ 2 A 1 2 u(t) γ(α-l) = δ 1 H α+2-l 2 (t) A 1 2 u(t) γ(α-l) + δ 2 A 1 2 u(t) γ(α-l) . (6.8)
For simplicity in the sequel we shall write l/2 := m > 0.

In order to estimate H 3 , we use (6.5) and we have that

H 3 = - d dt A 1 2 u(t) 2γ(1+m) H(t) m A 1 2 u(t) 2γ u ′ (t) l H(t) = -mH(t) m H ′ (t) A 1 2 u(t) 2γm u ′ (t) l -2γ(1 + m)H(t) m+1 A 1 2 u(t) 2(γm-1) u ′ (t) l u ′ (t), Au(t) =: H 4 + H 5 .
We observe that taking into account (6.5), we have

H 4 = -mH ′ (t), (6.9)
and

|H 5 | ≤ 2γ(1 + m)H(t) m+1 A 1 2 u(t) 2γm u ′ (t) l • u ′ (t) A 1 2 u(t) = 2γ(1 + m)H(t) 3 2 A 1 2 u(t) γ-1 .
We observe that the hypothesis

α < β(1 + l) + l β + 2 , implies γ -1 > γ(α -l),
and taking into account (6.2), we obtain

|H 5 | ≤ 2γ(1 + m)σ γ-1-γ(α-l) 0 H(t) 3 2 A 1 2 u(t) γ(α-l) . (6.10)
Then, taking into account (6.7)-(6.10) in (6.6), we deduce that Here we observe that (6.11) is still valid if u ′ (t) = 0, since then H(t) = 0 = H ′ (t) and the RHS is non-negative. Now, let

H ′ (t) ≤ A 1 2 u(t) γ(α-l) -ρ η 2 H α+2-l 2 (t) + ρ η 2 + 2γσ γ-1-γ(α-l) 0 H(t)
h(s, Γ) = -ρ η 2 s α+2-l 2 + ρ η 2 + 2γΓs 3 2
,

where Γ := σ γ-1-γ(α-l) 0

.

We observe that h

((2/η 2 2 ) 2/(α+2-l) , Γ) = - ρ η 2 + 2γΓ 2/η 2 2 3 α+2-l . In particular h (2/η 2 2 ) 2/(α+2-l) , Γ → - ρ η 2 < 0, if Γ → 0.
Let us therefore assume that σ 0 is sufficiently small to achieve l) , then H(t) is bounded for all t ∈ (0, T ). Indeed if H is not bounded for all t ∈ (0, T ), then there exists T ∈ (0, T ) such that H( T ) = ε 1 and H(t) < ε 1 for all t ∈ (0, T ). By (6.11) and taking into account that u ∈ C 1 (R + , H) and u ′ l u ′ ∈ C 1 (R + , H), we have

h((2/η 2 2 ) 2/(α+2-l) , σ γ-1-γ(α-l) 0 ) < 0. We claim that if H(0) = σ 1 < ε 1 := (2/η 2 2 ) 2/(α+2-
H ′ ( T ) ≤ A 1 2 u( T ) γ(α-l) h(ε 1 , σ γ-1-γ(α-l) 0 ) ≤ u( T ) γ(α-l) h(ε 1 , σ γ-1-γ(α-l) 0
) < 0, then since T ∈ (0, T ) implies u( T ) > 0, H decreases near T . This contradicts the definition of T and we can claim that H is bounded for all t ∈ (0, T ).

Finally, we claim that T = +∞. Let us assume by contradiction that this is not the case. Then, taking into account that H is bounded for all t ∈ [0, T ), we observe that

u ′ (t) 2 ≤ C A 1 2 u(t) 2γ ≤ C u(t) 2γ ∀t ∈ [0, T ).
Therefore, from the continuity of the vector (u ′ , u) with values in H × H it now follows that u(T ) = 0 implies u ′ (T ) = 0, hence u ≡ 0 by backward uniqueness, a contradiction. Hence, T = +∞, we obtain (6.3) and (6.4), and the conclusion follows.

Our next result is the generalization to the vector case of the slow-fast alternative established in [START_REF] Abdelli | Global behavior of the solutions to a class of nonlinear second order ODE's[END_REF] for the scalar equation. By relying on a technique introduced by Ghisi, Gobbino and Haraux [START_REF] Ghisi | Finding the exact decay rate of all solutions to some second order evolution equation with dissipation[END_REF], we prove that all non-zero solutions to (1.1) are either slow solutions or fast solutions. Theorem 6.4 (The slow-fast alternative). Assume that g is locally Lipschitz continuous and satisfies (5.2) and l < 1, l < α < β(1+l)+l β+2 . Let u(t) be a global solution to (1.1). Then, one and only one of the following statements apply:

1. (Fast solutions) There exist some positive constants η, C and T such that

u ′ (t) 2 ≥ η A 1 2 u(t) 2( β+2 l+2 ) , ∀t ≥ T, and
E(t) ≤ C (t + 1) l+2 α-l
, ∀t ≥ 0. (6.12)

(Slow solutions)

There exist some positive constants M , c and T such that

u ′ (t) 2 ≤ M A 1 2 u(t) 2( β+2 l+2 ) , ∀t ≥ T, and
u(t) ≥ c (1 + t) α+1 β-α
, ∀t ≥ 0. (6.13)

Proof. Fast solutions:

We first establish that if for some η > 0 we have

u ′ (t) 2 ≥ η A 1 2 u(t) 2( β+2 l+2 ) , ∀t ≥ T, (6.14)
then (6.12) holds.

Using (1.2), we obtain

l + 1 l + 2 u ′ (t) l+2 = - 1 β + 2 A 1 2 u(t) β+2 + E(t) ≥ E(t) -δ u ′ (t) l+2 ,
where δ := (β + 2)η From the energy identity, we deduce

E ′ (t) ≤ -CE(t) α-l l+2 +1 ,
where

C := η l + 1 l + 2 + δ -α+2 l+2
. Integrating between T and t and since E(t) ≥ 0, we deduce that there exists a constant C 1 such that

E(t) ≤ C 1 (t -T ) -l+2 α-l , ∀t ≥ T + 1.
The result follows immediately. (1+m)(α+2-l) . Now, we estimate the third term. We have that K is bounded for all t ≥ t n 0 . Then, using Young's inequality applied with the conjugate exponents α+2-l 2 and α+2-l α-l , 2γH(t)K .

Then, we obtain

H ′ (t) ≤ A 1 2 u(t) γ(α-l) - 1 2 ρ η 2 H α+2-l 2 (t) + K ,
where K := ρ η 2 + 4(α -l) ρ η 2 (α + 2 -l) 2 M α+2-l α-l . If u ′ (t) = 0, since as previously H is differentiable at t with H ′ (t) = 0 = H(t), the inequality is still valid.

It remains to prove that H(t) is bounded for all t ≥ t n 0 . Indeed if H is not bounded for all t ≥ t n 0 , then there exist T 2 ≥ t n 0 such that

T 2 := inf{t ≥ t n 0 : ∀τ ∈ [t n 0 , t], 1 2 ρ η 2 H α+2-l 2 (τ ) < 1 2 ρ η 2 H α+2-l 2
(t n 0 ) + K + 1}.

We have

H ′ (T 2 ) ≤ A 1 2 u(T 2 ) γ(α-l) - 1 2 ρ η 2 H α+2-l 2 (T 2 ) + K = -A 1 2 u(T 2 ) γ(α-l) 1 + 1 2 ρ η 2 H α+2-l 2
(t n 0 ) < 0, and since T 2 ≥ t n 0 , implies u(T 2 ) = 0, H decreases near T 2 . This contradicts the definition of T 2 and we can claim that H is bounded for all t ≥ t n 0 . The end of the proof is identical to that of Theorem 6.3.

Remark 6.5. In the special case A = λI, by using the scalar equation we can prove the existence of an N -dimensional variety of fast solutions. Actually, by Theorem 5.4, (4) of [START_REF] Ghisi | Finding the exact decay rate of all solutions to some second order evolution equation with dissipation[END_REF] we know that the same result is true in general for the non-singular linearly damped equation u ′′ + u ′ + A 1/2 u β Au = 0.

The question now arises of whether the dimensionality of the variety of fast solutions is exactly equal to N or possibly larger. The special case of the equation u ′′ + u ′ + u β u = 0 (6.31)

shows it is exactly equal to N at least in some cases. Indeed we claim that in this case, all fast solutions have their N components proportional to the same fast solution of a related scalar equation.

  ρ := ρ(α+1) (1+m)(α+2-l) .

Slow solutions:

  In addition to the already defined number γ = β -

1 2 (t) A 1 2

 11 u(t) γ-1-γ(α-l) ≤ M H(t)

Acknowledgments

M. Anguiano has been supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466, and in part by European Commission, Excellent Science-European Research Council (ERC) H2020-EU.1.1.-639227.

Then, we can deduce that

2 u β Au) -ε( u 2γ u, g(u ′ )).

(5.8)

We now estimate the right side of (5.8), The fourth term:

-ε( u 2γ u, A

(5.9)

The second term:

The third term:

Applying Young's inequality, with the conjugate exponents α+2 α-l and α+2 l+2 , we have

We assume (α + 2)2γ α -l ≥ 2γ + β + 2, which reduces to the condition γ ≥ (β + 2)(α -l) 2(l + 2) , (5.10) and taking δ small enough, we have for some P > 0, therefore the second and the third terms becomes

(5.11) Using (5.2), (5.8), (5.9) and (5.11), we have

(5.12) Applying Young's inequality, with the conjugate exponents α + 2 and α+2 α+1 and using (5.1), we have

This term will be dominated by the negative terms assuming

Let us set

Since u(0) = 0, we have that T > 0 and if T < +∞, then u(T ) = 0.

Let us consider, for all t ∈ (0, T ), the energy

where

If u ′ (t) = 0, then we claim that H is differentiable at t with H ′ (t) = 0. Indeed by the equation

Taking into account (1.2) and (3.9), we deduce

where ρ := l + 2 l + 1 .

Let us estimate H 1 , H 2 and H 3 . Using (5.2) and (6.5), we observe that

In order to estimate H 2 , we use Young's inequality applied with the conjugate exponents α+2-l 1-l and α+2-l α+1 ,

In this section, we consider the energies H(t) and

defined whenever u(t) = 0. In this part of the proof, it remains to consider the case where (6.14) is false for every η. Then, assuming the solution to be non-trivial there exists a sequence t n → +∞ such that u(t n ) = 0 and K(t n ) -→ 0. Therefore, for any ε 1 > 0 there exists

Under this assumption, we prove that a similar estimate holds true for all sufficiently large times, namely for some n 0 ∈ N u(t) = 0 and K(t) ≤ M, ∀t ≥ t n 0 , (6.17) for a suitable constant M ≥ ε 1 2 . In order to achieve this property, we select another positive number, ε 2 , to be fixed later in the proof, and we consider n 0 ∈ N large enough to achieve the additional condition

By (6.16), we deduce that u(t n 0 ) = 0 and we have that T > 0. If T < +∞, then u( T ) = 0. Let us consider the energy K, which is defined by (6.15) for every t ∈ [t n 0 , T ).

If u ′ (t) = 0, then we claim that K is differentiable at t with K ′ (t) = 0. Indeed by the equation we find that u ′ l u ′ (s) = O(s -t) for s close to t, then since l < 1 we deduce u ′ (t) 2 = O(s -t), thereby proving the claim.

If u ′ (t) = 0, then again K is differentiable at t with:

Taking into account (1.2) and the energy identity, we deduce

where ρ := l + 2 l + 1 .

Let us estimate K 1 , K 2 and K 3 . Using (6.15), we observe

In order to estimate K 2 , we use Young's inequality applied with the conjugate exponents α+2-l 1-l and α+2-l α+1 ,

,

, and taking into account that γ = β+1 α+1 , we deduce

where

By simple calculations, we deduce from α < β(1+l)+l β+2 that γ(α -l) < µ, therefore

For simplicity in the sequel we shall write l/2 := m > 0. (6.21)

In order to estimate K 3 , we use (6.15) and we have that

We observe that taking account of (6.15), we have

and

We observe that our condition on α can be written in the form

Using (6.2) we therefore obtain

Then, taking into account (6.19)-(6.23) in (6.18), we deduce that

where ρ :=

Here we observe that (6.24) is still valid if u ′ (t) = 0, since then K(t) = 0 = K ′ (t) and the RHS is non-negative. Now, let

Since α < β(1+l)+l β+2 , we have

We observe that h(s) < 0, if s → 0.

Fixing ε 1 sufficiently small to achieve h(ε 1 ) < 0 , we can then adjust ε 2 so that h(ε 1 ) < 0. Now, if K is not bounded for all t ∈ (t n 0 , T ), then there exists T 1 ∈ (t n 0 , T ) such that K(T 1 ) = ε 1 and K(t) < ε 1 for all t ∈ (t n 0 , T 1 ). By (6.24) and taking into account that u ∈ C 1 (R + , H) and u ′ l u ′ ∈ C 1 (R + , H), we have

This contradicts the definition of T 1 and we can claim that K is bounded for all t ∈ (t n 0 , T ).

We claim that T = +∞. Let us assume by contradiction that this is not the case. Then, taking into account that K is bounded for all t ∈ [t n 0 , T ), we observe that

Therefore, from the continuity of the vector (u ′ , u) with values in H × H it now follows that u( T ) = 0 implies u ′ ( T ) = 0, hence u ≡ 0 by backward uniqueness, a contradiction. Hence, T = +∞. Then, we obtain (6.17).

So we are left to prove that for some M > 0

Let us consider the energy H, which is defined by (6.5) for every t ≥ t n 0 .

Assuming first u ′ (t) = 0, we can differentiate H at t, and we find

Taking into account (1.2) and the energy identity, we deduce

where ρ := l + 2 l + 1 .

Let us estimate H 1 , H 2 and H 3 . Using (6.5), we observe that

In order to estimate H 2 , we use Young's inequality applied with the conjugate exponents α+2-l 1-l and α+2-l α+1 ,

,

where

, and taking into account that γ = β+1 α+1 , we deduce

In order to estimate H 3 , using (6.21) and (6.5), we have that

We observe that taking into account (6.5), we have

and using (6.15), we obtain

We observe, after some calculations, that

which reduces easily to the condition

Then, taking into account (6.27)-(6.30) in (6.26), we deduce that

Proof. Given a solution u of (6.31), any component y of u satisfies y ′′ + y ′ + u β y = 0.

Considering two different components (y, z) we obtain that the Wronskian function w(t) := (y ′ z -yz ′ )(t) is a solution of

Hence w(t) = w(0) exp(-t). Since on the other hand w, being a quadratic combination of 4 functions decaying like exp(-t), has to decay like exp(-2t), the only possibility is w(0) = 0, hence w = 0. Hence any two components are proportional. Let us set

where y is one of the non-zero components. Then we have

. Let us introduce φ := µy where µ > 0 will be chosen later.

Then φ satisfies φ ′′ + φ ′ + µm y β y = 0 or

so that the choice µ = m 1/β leads to

Finally we have y j = b j φ, with b j = m -1/β a j and φ a fast solution of the scalar equation. Since the set of such fast solutions φ is invariant by translation in t, for the moment we reduced the dimension to N + 1 instead of N and we know that this is not relevant since in the scalar case the dimension is 1. Actually the numbers b j are constrained by an additional condition: indeed we have for all j the equation y ′′ j + y ′ j + u β y j = b j (φ ′′ + φ ′ + u β φ) = 0, and since at least one of the b j is not 0, we obtain φ ′′ + φ ′ + u β φ = 0. Now

Since slow solutions do not vanish for t large we conclude that N j=1 b 2 j = 1, the additional constraint we were looking for, reducing the dimension to N . The set of slow solutions has the structure of a 1D curve fibrated by the (N -1)-dimensional unit sphere.

However, when either g is non-linear or A is not a multiple of the identity, we have no information on the dimension of this set. This is one of the last open problems of this theory.


Miguel Couceiro
email: miguel.couceiro@inria.fr

Michel Grabisch
email: michel.grabisch@univ-paris1.fr

On integer-valued means and the symmetric maximum

Keywords: Mathematics Subject Classification: 26E60, 08A99, 06A99 integer means, nonassociative algebra, symmetric maximum, decomposability k times , a k+1 , . . . , a n

Integer-valued means, satisfying the decomposability condition of Kolmogoroff/Nagumo, are necessarily extremal, i.e., the mean value depends only on the minimal and maximal inputs. To overcome this severe limitation, we propose an infinite family of (weak) integer means based on the symmetric maximum and computation rules. For such means, their value depends not only on extremal inputs, but also on 2nd, 3rd, etc., extremal values as needed. In particular, we show that this family can be characterized by a weak version of decomposability.

Introduction

In 1930, Kolmogoroff [START_REF] Kolmogoroff | Sur la notion de moyenne[END_REF] and Nagumo [START_REF] Nagumo | Über eine Klasse der Mittelwerte[END_REF] independently characterized the class of quasiarithmetic means (including the well-known arithmetic, geometric and harmonic means) through a property which was later called decomposability [START_REF] Grabisch | Aggregation functions[END_REF], barycentric associativity [START_REF] Marichal | Barycentrically associative and preassociative functions[END_REF], or simply associativity [START_REF] Bennett | Integer valued means[END_REF]. As associativity in the classical algebraic sense will play a central rôle in this paper, we prefer to adopt the term "decomposability".

Decomposability is a very natural property that amounts to saying that the average of a set of numbers should not be changed if a subset of it is replaced by its average. Denoting such an averaging function by µ, the latter notion can be expressed by the condition µ(a 1 , . . . , a n ) = µ(b, . . . , b

More recently, Bennett et al. [START_REF] Bennett | Integer valued means[END_REF] investigated the effect of this property on integervalued averaging functions, that is, averaging functions defined and valued in Z. This setting is both natural and relevant in practice since in many situations input values are integer, for instance, number of children, of items, etc., and in such cases mean values that are not integer bear no meaning. Surprisingly, by imposing decomposability in addition to the usual requirements of averaging functions (nondecreasingness, anonymity and internality), it turns out that such integer-valued averaging functions (called integer means or Z-means in [START_REF] Bennett | Integer valued means[END_REF]) are extremal, that is, the output depends only on the minimum and maximum input values. In symbols, µ(a 1 , . . . , a n ) = µ(a 1 , a n ) with a 1 ≤ a 2 ≤ • • • ≤ a n . Put otherwise, integer means ignore all input values but the minimum and the maximum ones.

The aim of this paper is to attempt to finding a class of integer-valued averaging functions that have a less drastic behavior and whose output retains more information of the input, by allowing to considering not only minimal and maximal inputs, but also, when necessary, second, third, etc., min/max values. This can be achieved through the use of the symmetric maximum, a symmetrization of the classical maximum operator proposed by Grabisch [START_REF] Grabisch | The Möbius function on symmetric ordered structures and its application to capacities on finite sets[END_REF], and denoted by . Roughly speaking, is a binary operator on Z yielding the extremal value of its inputs (i.e., the greatest in absolute value), except if these are opposite values, in which case the output is 0, for instance

a (-a) = 0, 2 3 = 3, (-2) 3 = 3, 2 (-3) = -3.
As it is easy to see, such an operator is not associative in the algebraic sense. Indeed, (3 (-3)) 2 = 0 2 = 2, 3 ((-3) 2) = 3 (-3) = 0.

(2)

Therefore, defining an averaging function by , i.e., writing (a 1 , . . . , a n ) for any sequence a 1 , . . . , a n of inputs is meaningless without further refinement of the definition. In fact, nonassociativity arises if and only if there exist extremal opposite terms in the input sequence (like 3, -3), otherwise (a 1 , . . . , a n ) is well defined. In order to overcome this drawback, Grabisch [START_REF] Grabisch | The Möbius function on symmetric ordered structures and its application to capacities on finite sets[END_REF] proposed systematic ways to aggregate all terms of a sequence using a single binary operator as in [START_REF] Couceiro | On the poset of computation rules for nonassociative calculus[END_REF]. These systematic ways of putting parentheses were called rules of computation, and were fully investigated by the authors in [START_REF] Couceiro | On the poset of computation rules for nonassociative calculus[END_REF]. A simple example of such a rule is the following: aggregate first all pairs of extremal opposite numbers in decreasing order, till no such pair exist. On the sequence of number 4, -4, 3, -3, 2, 1, this would yield:

(4, -4, 3, -3, 2, 1) = (4 (-4)) (3, -3, 2, 1) = 0 (((3 (-3)) (2, 1) = 0 (2, 1) = 2.
viewed as an integer-valued averaging function is clearly not extremal, but as said above, it satisfies extremality in a wider sense: in the above example we have (4, -4, 3, -3, 2, 1) = (2, 1) and (2,1) is the 3d pair of min/max inputs, the two first pairs having been eliminated because they are pairs of opposite values.

As shown in [START_REF] Couceiro | On the poset of computation rules for nonassociative calculus[END_REF], infinitely many rules of computation exist, yielding an infinite (and even uncountable) family of integer-valued averaging functions. Interestingly, none of them satisfy decomposability (Corollary 1), but they satisfy a weaker version of it, hence we call them weak integer means. More precisely, the decomposability property (1) is valid only for certain subsequences of a 1 , . . . , a n (Lemma 2).

We provide a complete characterization of this family of weak integer means, by using weak decomposability (Theorem 3), or a related property called the replacement property (Lemma 5). The various characterizations are summarized in Theorem 4.

Preliminaries

Z-means and the symmetric maximum

We consider a countable set C which is totally ordered by , and with minimal element 0. Let C -= {-c | c ∈ C} be its dually isomorphic copy, and consider C = C ∪ C -, setting -0 = 0.1 For convenience, we introduce the notation

C * = n∈N ( C) n ∪ {ε},
where ε is the empty string. Put differently, C * is the set of all words built over the alphabet C using concatenation.

Throughout the paper we deal with functions µ : C * → C, and always adopt the following convention: µ(ε) = 0 and µ(a) = a for all a ∈ C.

We recall the main result from [START_REF] Bennett | Integer valued means[END_REF]. A function

µ : C * → C
is a Z-mean or integer-valued mean if it satisfies the following properties:

(i) Anonymity: for every n ∈ N, every permutation π on [n],

µ(a 1 , . . . , a n ) = µ(a π(1) , . . . , a π(n) );

(ii) Internality:

for every n ∈ N, if a 1 • • • a n , a 1 µ(a 1 , . . . , a n ) a n ; (iii) Monotonicity: for every n ∈ N, a i a ′ i for i = 1, . . . , n implies µ(a 1 , . . . , a n ) µ(a ′ 1 , . . . , a ′ n );
(iv) Decomposability2 : for every k, n ∈ N with k ≤ n and µ(a 1 , . . . , a k ) = b, we have

µ(a 1 , . . . , a k , a k+1 , . . . , a n ) = µ(b, . . . , b k , a k+1 , . . . , a n ). A function µ : C * → C is extremal if for any n ∈ N and any a 1 ≤ • • • ≤ a n , we have µ(a 1 , . . . , a n ) = µ(a 1 , a n ) Theorem 1. All Z-means are extremal.
Hence, Z-means, unlike means defined on real numbers, do not consider all values to be aggregated but only the extreme ones. Our aim is to overcome this rather negative result by weakening the definition of decomposability, which is mainly responsible for extremality, and to propose a whole family of aggregation functions fulfilling decomposability in a weak sense. This will be achieved through the symmetric maximum [START_REF] Grabisch | The Möbius function on symmetric ordered structures and its application to capacities on finite sets[END_REF], which is a binary operation : C2 → C defined by

a b =      -(|a| ∨ |b|), if b = -a and |a| ∨ |b| = -a or -b 0, if b = -a |a| ∨ |b|, otherwise. (3)
In other words, except if a = -b, a b equals the element among a, b that is extremal, i.e., that has the largest absolute value (magnitude). It is not difficult to see that satisfies the following properties:

(i) (C1) coincides with the maximum on C 2 ;

(ii) (C2) a (-a) = 0 for every a ∈ C;

(iii) (C3) -(a b) = (-a) (-b) for every a, b ∈ C .
By (C1), (C3), coincides with the minimum on (C -) 2 . Also, it is monotone, internal, commutative (or symmetric), and 0 is its neutral element (by (C1) and (C3)). Hence, almost behaves like a commutative group operation on C, however it lacks associativity: it is not true in general that a

(b c) = (a b) c, (4)
as shown by the following example:

(-3 3) 1 = 0 1 = 1; -3 (3 1) = -3 3 = 0.
The following result gives an algebraic characterization of and shows that if one requires that (C1), (C2) and (C3) hold, then (3) is the best possible definition for .

Theorem 2. [3, Prop. 5] No binary operation satisfying (C1), (C2), (C3) fulfills (4) on a larger domain than .

It is easy to see that the domain where is associative is given by the set of triples (a, b, c) such that min(a, b, c) = -max(a, b, c).

(5)

Thorem 2 states that under (C1), (C2) and (C3), any binary operation µ on C is nonassociative as soon as (5) is not satisfied (i.e., for triples with opposite extremal elements). Moreover, if µ = , then there exists a triple satisfying [START_REF] Kolmogoroff | Sur la notion de moyenne[END_REF] for which µ is nonassociative.

For instance,

0 ∨ defined by 0 ∨(a 1 , . . . , a n ) =      a 1 , if a n 0 0, if a 1 < 0 < a n a n , if a 1 0 (6)
with

a 1 • • • a n , satisfies (C1), (C2) and (C3), but (-2 0 ∨ 2) 0 ∨ 3 = 0 0 ∨ 3 = 3; -2 0 ∨(2 0 ∨ 3) = -2 0 ∨ 3 = 0.

How to make associative: Rules of computation

The lack of associativity implies that it is not possible to extend the definition of to C * , since the result depends on the particular way that we apply the binary to the terms of a sequence with more than two terms. Therefore, no function like a Z-mean can be defined based on this operation. However, a way to overcome this difficulty, suggested in [START_REF] Grabisch | The Möbius function on symmetric ordered structures and its application to capacities on finite sets[END_REF] and fully developed in [START_REF] Couceiro | On the poset of computation rules for nonassociative calculus[END_REF], is to fix beforehand a systematic way of putting parentheses on any sequence of C * , procedure that was called a rule of computation.

Elements of C * are called sequences (of "integers"), denoted by σ = (a i ) i∈I for some finite index set I, and a i ∈ C for all i ∈ I (note that 0 is a possible value, contrarily to [START_REF] Couceiro | On the poset of computation rules for nonassociative calculus[END_REF]). Since we consider only anonymous functions on C * , the order is unimportant and so we consider only sequences of integers in decreasing order of their absolute value, positive integers first (e.g., 5, 5, -5, -3, 2, -2, 1, 0). The empty sequence is ε. The set of all such integer sequences, including the empty sequence, is denoted by S. Since sequences are ordered as described above, a convenient notation is the following. For an arbitrary sequence σ = (n 1 , . . . , n 1

p 1 times , -n 1 , . . . , -n 1 m 1 times
, . . . , n q , . . . , n q pq times , -n q , . . . , -n q mq times ) we introduce θ(σ) = (n 1 , . . . , n q ), the sequence of absolute values (magnitudes) of integers in σ, and ψ(σ) = ((p 1 , m 1 ), . . . , (p q , m q )) all numbers of occurrence of these integers. A sequence (a i ) i∈I fulfills associativity (or, simply, is associative) if min i∈I a i = -max i∈I a i (equivalently, if either p 1 = 0 or m 1 = 0). The set of sequences which do not fulfill associativity is denoted by S 0 . Note that by this definition the sequence (1, -1), although containing only 2 elements, is not associative.

The above notation permits to introduce rules of computation in a proper way. We define 5 elementary rules ρ i : S → S which operate on σ as follows:

(i) Elementary rule ρ 1 : if p 1 > 1 and m 1 > 0, then p 1 is changed to p 1 = 1;

(ii) Elementary rule ρ 2 : idem with p 1 , m 1 exchanged;

(iii) Elementary rule ρ 3 : if p 1 > 0, m 1 > 0, the pair (p 1 , m 1 ) is changed into (p 1 -c, m 1c), where c = p 1 ∧ m 1 ;

(iv) Elementary rule ρ 4 : if p 1 > 0, m 1 > 0, and if p 2 > 0, then p 2 is changed into p 2 = 0;

(v) Elementary rule ρ 5 : idem with m 2 replacing p 2 .

Hence, elementary rules delete terms in sequences which do not fulfill associativity, and do nothing otherwise. A (well-formed) computation rule R is a word built with the alphabet {ρ 1 , . . . , ρ 5 }, i.e., R ∈ L(ρ 1 , . . . , ρ 5 ), such that R(σ) ∈ S \ S 0 for all σ ∈ S. The set of (well-formed) computation rules is denoted by R. Examples of rules are (words are read from left to right) [START_REF] Couceiro | On the poset of computation rules for nonassociative calculus[END_REF] that any computation rule has the form R = T 1 T 2 • • • , where each T i has the form ωρ α 1 ρ β 2 ρ 3 , with ω ∈ L(ρ 4 , ρ 5 ) and α, β ∈ {0, 1} (factorization scheme). 3 Moreover, R is uncountable (the set of its maximal elements is isomorphic to 2 N ). We denote by R 123 the set of rules where ω = ε in each term T i (i.e., the rule uses only ρ 1 , ρ 2 , ρ 3 ). Also, we introduce Ker(R) = {σ ∈ S | R(σ) = ε}, the kernel of R.

(i) • 0 = ρ * 3 ; (ii) • = = (ρ 1 ρ 2 ρ 3 ) * ; (iii) • + -= (ρ 4 ρ 5 ) * ρ 1 ρ 2 ρ 3 . We use the notation R for • R. It is shown in
Two rules R, R ′ are equivalent (denoted by R ∼ R ′ ) if R = R ′ on S. For a given rule R and a sequence σ = (a i ) i∈I , we denote by J R σ the set of elements deleted by R in σ. We use the following notation: for a sequence (a i ) i∈I and any set K ⊆ I, we write a K for the subsequence (a i ) i∈K . Hence, we may write

(a i ) i∈I = (a K , a I\K ).
For convenience, we may also write σ K for a restriction of the sequence σ to K.

The basic mechanism and properties of R

It is important to analyse the effect of a rule R = T 1 T 2 • • • (not necessarily in R) on a sequence σ ∈ S 0 , with representation ψ(σ) = ((p 1 , m 1 ), . . . , (p r , m r )), θ(σ) = (n 1 , . . . , n r ).

Let us first consider a single term T 1 = ωρ, where ρ is a shorthand for ρ α 1 ρ β 2 ρ 3 . It is straightforward to verify that T 1 (σ) is another sequence with representation

ψ(T 1 (σ)) = ((p ′ 1 , m ′ 1 )
, (p ′ q , m ′ q ), (p q+1 , m q+1 ), . . . , (p r , m r )), θ(T 1 (σ)) = (n 1 , n q , . . . , n r ) with the following requirements:

0 p ′ 1 < p 1 , 0 m ′ 1 < m 1
, and at most one of them is positive 2 q r p ′ q = 0 or p q , m ′ q = 0 or m q This situation is pictured in Figure 1. The first hatched block is the part deleted by

J ρ σ J ω σ (p 1 , m 1 )
head tail

Figure 1: The sequences σ and T 1 (σ) (white part), with values from left to right in decreasing order of their magnitude. The hatched part is J T 1 σ ρ, while the second one is deleted by ω. The "head" is the undeleted part of (p 1 , m 1 ) between the two hatched blocks and it may be the case that it does not exist. It contains only n 1 or only -n 1 , with possible repetitions. The "tail" is the part left undeleted by ω and may not exist. The left hatched block always exist, while the second may not, in which case we say that the head is empty.

Applying successively the different terms T 2 , T 3 , . . . of R it is not difficult to see that the final result R(σ) has almost the same form. Indeed, if a head exists in T 1 (σ), then T 1 (σ) is associative, and the rule stops. If not, the next term applies (if nonassociative), yielding the same form as above. Hence the representation for R(σ) is

ψ(R(σ)) = ((p ′ k , m ′ k ), (p ′ q , m ′ q )
, (p q+1 , m q+1 ), . . . , (p r , m r )), θ(R(σ)) = (n k , n q , . . . , n r ) with the following requirements:

k 1 0 p ′ k p k , 0 m ′ k m k
, and at most one of them is positive k + 1 q r p ′ q = 0 or p q , m ′ q = 0 or m q This situation is pictured in Figure 2. We define the head and tail of R(σ) similarly: the head is the part left undeleted by R between the two deleted parts, while the tail is the part left undeleted which has no deleted part on its right side. If the head exists, it contains only value n k or only -n k , possibly with repetition. Note that when they both exist, the elements in the tail have strictly smaller magnitude than any element of the head.

(p 1 , m 1 ) (p k , m k ) head tail
We denote by H, T the subsets of I corresponding to the head and tail of a sequence σ = (a i ) i∈I , and therefore may use the notation σ H , σ T or a H , a T for the head and tail.

Based on this result, we make some useful observations. Lemma 1. Let σ = (a i ) i∈I ∈ S 0 . The following holds.

(i) R (σ) = R (a I\J R σ ) = (a I\J R σ ) = min i∈I\J R σ a i max i∈I\J R σ a i for any R ∈ R; (ii) R is odd for any R ∈ R, i.e., R (-σ) = -R (σ); (iii) Let R = T 1 T 2 • • • . If R(σ) has a (nonempty) head, then R (σ I\T , σ T ) = R (σ I\T , σ ′ ),
where σ ′ is any sequence (possibly empty); If R(σ) has no head, then R (σ I\T , σ T ) = R (σ I\T , σ ′ ), for any σ ′ ∈ S \ S 0 which has the same extremal value as σ T ;

(iv)

a J R σ ∈ Ker(R), hence R (a J R σ ) = 0, for any R of the form T 1 T 2 • • • (finite or infinite); (v) For any R ∈ R, letting R (σ) = b, then |b| max i∈J R σ |a i |, and |b| |a i | for all i ∈ I \ J R
σ (and there is equality for some i, if

I \ J R σ = ∅); Proof. (i) The first equality comes from R (σ) = (R(σ)) = (a I\J R σ )
. By definition of a computation rule, a I\J R σ fulfills associativity, which explains the second equality, while the third one follows from associativity and the definition of .

(ii) From (i) and (C3).

(iii) Suppose R(σ) has a head. Then R (σ) = b is the value in the head, which is strictly greater in magnitude than any value in the tail. Therefore the tail has no effect on the result. Otherwise, b is the extremal value of the tail. Hence the tail can be modified provided that it remains associative and with same extremal value.

(iv) Step 1: We consider first that R = T 1 = ωρ α 1 ρ β 2 ρ 3 and we denote ρ = ρ α 1 ρ β 2 ρ 3 for short. Consider a sequence σ = (a i ) i∈I and let ψ(σ) = ((p 1 , m 1 ), . . . ,

(p k , m k )), ψ(a J R σ ) = ((p ′ 1 , m ′ 1 ), . . . , (p ′ k ′ , m ′ k ′ )). Observe that p ′ 1 > 0, m ′ 1 > 0 since ρ 3 is present in T 1 . Therefore, if ε = ω = ω 1 ω 2 • • • , ω 1 applies
, then the remaining ω 2 , etc., also apply because no

ω i modifies p ′ 1 , m ′ 1 . It then follows that J ω σ = J ω a J R σ
. Now, observe that ρ modifies only

(p 1 , m 1 ), hence J ρ ω(σ) = J ρ ω(a J R σ
) , and thus R(σ J R σ ) = ε.

Step 2: We consider now the case where R is not limited to a single term T 1 and we show the result by induction on the size of the sequence.

Induction hypothesis: for any σ = (a i ) i∈I ∈ S 0 , |σ|

t with t 3, for any R = T 1 T 2 • • • , R(a J R σ ) = ε.
Step 2.1: We show the base case with t = 3. Any sequence in S 0 of three terms is of the form (a, -a, b) with |a| |b|. Then only T 1 is applied in R, and it is easy to check that the result holds in any case.

Step 2.2: We now consider σ = (a i ) i∈I in S 0 with |σ| = t + 1 and representation ψ(σ) = ((p 1 , m 1 ), . . . , (p q , m q )), q 1. Let T 1 = ωρ α 1 ρ β 2 ρ 3 and consider the rule R ′ = T 2 T 3 • • • (which may have only one term, in which case Step 1 of the proof must be used). Observe that since p 1 > 0 and m 1 > 0, ω does not modify p 1 , m 1 and ρ 1 , ρ 2 leave p 1 > 0, m 1 > 0, the action of ρ 3 will decrease p 1 and m 1 by at least one unit each. It then follows that

|T 1 (σ)| |σ| -2 = t -1.
Moreover, by definition we have R(σ) = R ′ (T 1 (σ)) and so J R σ = J T 1 σ ∪ J R ′ T 1 (σ) . Applying induction hypothesis, we find

ε = R ′ (T 1 (σ)) J R ′ T 1 (σ) . We claim that R ′ (T 1 (σ)) J R ′ T 1 (σ) = R(σ J R σ )
, which finishes the proof. Proof of the Claim: We have by definition

R(σ J R σ ) = R ′ (T 1 (σ J R σ )) = R ′ (T 1 (σ J T 1 σ , σ J R ′ T 1 (σ)
))

It remains then to prove

T 1 (σ J T 1 σ , σ J R ′ T 1 (σ) ) = (T 1 (σ)) J R ′ T 1 (σ) . (7)
We distinguish several cases on the structure of R(σ), distinguishing the effect of T 1 and R ′ . In the following figures, the hatched and the grey parts are respectively the effect of T 1 and R ′ . The second hatched block may be empty (if ω = ε), and similarly for the second grey block. We have to show that applying T 1 to the sequence formed by the hatched and grey parts (called σ ′ ) deletes exactly the hatched part. The foregoing analysis shows that only three cases can happen.

• Case 1:

σ ′ (p 1 , m 1 )
Observe that the action of ω is on the second hatched part, and depends only on (p 1 , m 1 ) and the current (p k , m k ) and not on subsequent values. As these pairs are unchanged, the action of ω on σ ′ or on σ is exactly the same. Now, ρ acts only on (p 1 , m 1 ) and depends only on (p 1 , m 1 ). Hence, the result on σ and σ ′ is again the same.

• Case 2:

(p 1 , m 1 )
Here

(p 1 , m 1 ) is changed into (p ′ 1 , m ′ 1 ) in σ ′ , but both p ′ 1 , m ′ 1
are positive due to the presence of ρ 3 in T 1 . Hence the same reasoning as in case 1 applies.

• Case 3:

(p 1 , m 1 )
This case is treated like case 1.

(v) Clear from the foregoing analysis with head and tail.

Suppose that R(σ) has a head, and that K intersects the head. Then clearly b

′ = b and R (|K ′ | • b, a I\K ′ ) = b, as desired. If on the contrary K ′ is included in the tail, then |b ′ | < |b|. By Lemma 1(iii), R (|K ′ | • b ′ , a I\K ′ ) = b.
Suppose now that R(σ) has no head, which implies that K ′ is included in the tail. Then |b ′ | |b|. If equality occurs, then b ′ = b, a K ′ is associative and the modified tail remains associative. It follows that the desired result holds by Lemma 1(iii). If |b ′ | < |b|, the tail remains also associative because there is no -b. Again, the result holds by Lemma 1(iii).

(ii) We prove now that for each rule, one can find a nonassociative sequence and a subsequence of it on which the symmetric maximum is not decomposable. Consider the sequence σ = (2, -2, 1). Suppose R ∈ R 123 and take the subsequence σ K = (-2, 1). Then R (-2, 1) = -2 and R (σ) = 1. However, R (2, -2, -2) is equal to either 0 or -2, depending whether R uses ρ 2 or not. Hence decomposability fails for K.

Suppose now that R ∈ R 123 and consider the subsequence σ K = (2, -2). Then R (2, -2) = 0 and R (σ) = 0 (supposing that R contains ρ 4 . Otherwise, replace σ by (2, -2, -1)). However, R (0, 0, 1) = 1, and decomposability again fails on K.

Corollary 1. (i) The symmetric maximum is decomposable for no computation rule.

(ii) The symmetric maximum is decomposable for any sequence in S \ S 0 , for any computation rule.

Observe also that • + -is the only rule yielding an extremal operator. Indeed, ( (a i ) i∈I

+ -) = min i∈I a i max i∈I a i .
Any other rule (as shown in [START_REF] Couceiro | On the poset of computation rules for nonassociative calculus[END_REF]) deletes less terms than • + -, hence there are sequences for which other elements than the minimum and maximum are used. This fact motivates the definition of a weaker notion of extremality (see below).

For characterization purposes, we introduce a number of properties.

Example 1 .

 1 Consider the sequence σ = (3, 3, -3, 2, -2, -2, 1, 1, 1, 1). Then θ(σ) = (3, 2, 1); ψ(σ) = ((2, 1), (1, 2)(4, 0)).

Figure 2 :

 2 Figure 2: The sequences σ and R(σ) (white part), with values from left to right in decreasing order of their magnitude. The hatched part is J R σ

Definition 1 .

 1 Let µ : C * → C. (i) µ is extremal w.r.t. R ∈ R if for every σ = (a i ) i∈I in S µ satisfies the replacement property w.r.t. R ∈ R, abbreviated by RP(R), if for every sequence σ = (a i ) i∈I in S, µ(σ) = b ⇒ µ(|J R σ | • b, a I\J R σ ) = b , and µ(a J R σ , |I \ J R σ | • b) = b (iii) µ is retractive w.r.t. R ∈ R, if for every σ = (a i ) i∈I ∈ S we have that µ(a I\J R σ ) = µ(σ);

Documents de travail du Centre d'Economie de la Sorbonne -2016.80

A typical example is C = Z, and in fact any such C is isomorphic to Z or to a subset of it. This is why we may call elements of C "integers" and use the corresponding terminology, like positive and negative numbers, absolute value, etc.

Documents de travail du Centre d'Economie de la Sorbonne -2016.80

This property is called associativity in[START_REF] Bennett | Integer valued means[END_REF], and barycentric associativity in[START_REF] Marichal | Barycentrically associative and preassociative functions[END_REF]. However, the usual meaning of associativity in algebra is different, and the latter is a central property in this paper. To avoid any confusion, we therefore stick to the term "decomposability".

Documents de travail du Centre d'Economie de la Sorbonne -2016.80

Here, we adopt the notation ρ 0 = ε and ρ 1 = ρ. 6 Documents de travail du Centre d'Economie de la Sorbonne -2016.80

3 Weak Z-means based on the symmetric maximum Specifically, we say that µ : C * → C is a weak Z-mean if it satisfies anonymity, internality, monotonicity and restricted decomposability: for every sequence (a i ) i∈I , there exists at least one nonempty K ⊆ I on which decomposability applies:

where k • b, for some nonnegative integer k and some b ∈ C, means b, b, . . . , b (repeated k times). In this case we say that µ is decomposable for K.

We recall that an anonymous function µ : C * → C is associative if µ(a) = a for every a ∈ C, and for every sequence σ = (a i ) i∈I and K ⊂ I, we have µ(σ) = µ(µ(a K ), a I\K ) (see [4, §2.3.1]). We say that an anonymous function µ is weakly associative if associativity holds for associative sequences (i.e., on S \ S 0 ).

We show in this section that the symmetric maximum is the basis for an infinite (uncountable) family of weakly associative weak Z-means, namely ( R ) R∈R .

Lemma 2. Let R ∈ R.

(i) For every sequence σ = (a i ) i∈I , R satisfies decomposability for every K ⊇ J R σ and for every nonempty K ′ ⊆ I \ J R σ ;

(ii) For each rule R, there exists a sequence σ = (a i ) i∈I in S 0 and a set K ⊆ I such that R is not decomposable for K.

Proof. (i) We know by Lemma 1(iv) that R (a J R σ ) = 0. Then, by Lemma 1(i) and the fact that 0 is neutral,

Suppose that R(σ) has a head, and hence b is in the head. Then J R a K = J R σ and |b ′ | |b|. Moreover b ′ = b holds iff K intersects the head of σ, otherwise strict inequality occurs and b ′ is the extremal value of the tail. Then (|K| • b ′ , a I\K ) is associative in any case, because the head does not contain -b and the tail is associative. Its extremal value is b in any case, because either K contains the head, and then b ′ = b, otherwise it does not and b is present in a I\K .

Suppose now that R(σ) has no head. Observe that the sequence restricted to

(iv) µ is weakly retractive w.r.

) is still associative and b is its extremal term. Remark 1. For any R ∈ R, the RP(R) property implies unanimity, i.e., µ(b, . . . , b) = b, for every b ∈ Ran(µ(S \ S 0 )), where Ran(•) indicates the range of a function. Indeed, for σ ∈ S \ S 0 with µ(σ) = b, J R σ = ∅.

Characterizations

We give several characterizations of weak Z-means based on the symmetric maximum. The first one involves decomposability. We define Ker(µ) = {σ ∈ S * | µ(σ) = 0}, where S * is the set of sequences in S not containing 0. Observe that by our convention, Ker(R) = Ker( R ).

Characterization by decomposability

Theorem 3. Let µ : C * → C and let R ∈ R. Then µ = R if and only if µ satisfies (C1), (C2), (C3) on C2 , is anonymous, weakly associative, decomposable on J R σ for every sequence σ, and satisfies µ(σ) = 0 for every σ ∈ Ker(R) (i.e., Ker(R) ⊆ Ker(µ)).

The following observation is useful in many subsequent proofs.

Observation 1. Let µ : C * → C. If µ satisfies (C1), (C2), (C3), and weak associativity, then µ = on S \ S 0 . Indeed, by Theorem 2 (C1), (C2), (C3) and weak associativity imply that µ coincides with on C2 . It thus follows by weak associativity that µ(σ) = (σ) for every associative sequence σ.

Proof. The "only if" part is obtained from previous results (in particular Lemma 2).

For the "if" part, by Observation 1, µ = on S \ S 0 . Consider now sequences in S 0 and, for the sake of a contradiction, suppose that µ = R on S 0 . Take σ = (a i ) i∈I in S 0 such that µ(σ) = R (σ) and |Ran(σ)| is minimum, where Ran(σ) is the range of σ (i.e., the set of all integers in σ).

Suppose first that |Ran(σ

By definition of σ and by applying decomposability on J R σ , it follows that Now, by Lemma 1(iv), a J R σ ∈ Ker(R) ⊆ Ker(µ), and since 0 is neutral for R , we get

where the last equality comes from Lemma 1(i). Combining the latter result with (8) yields µ(σ) = R (σ), a contradiction.

Suppose now that |Ran(σ)| = 2 (note that a lower value is impossible). Then necessarily σ is such that ψ(σ) = (p 1 , m 1 ) with p 1 > 0, m 1 > 0. Two cases arise. Suppose first that a J R σ = σ. Then σ ∈ Ker(R), hence µ(σ) = 0 = R (σ), a contradiction. Suppose on the contrary that a

Observe that by the assumption |Ran(σ)| = 2, we have necessarily |Ran(a

-, then decomposability on J R σ is trivially satisfied and therefore is not needed in the axiomatization. Indeed, J

Characterization by the replacement property and others

We introduce further characterizations based on the properties in Definition 1. First, we observe that the kernels of µ and R are somehow related. Proof. Note that by (C2) Ker(µ) = ∅. For the sake of a contradiction, suppose that there exists σ = (a i ) i∈I ∈ Ker(µ) \ Ker(R), i.e., µ(σ) = 0 and R (σ) = 0. As before, take such a σ with minimum |Ran(σ)|. As µ satisfies (C1) and (C2), we may assume that |Ran(σ)| ≥ 2.

The case |Ran(σ)| = 2 is dealt with as above. So suppose that |Ran(σ)| ≥ 3. Observe that J R σ = ∅ is impossible. Indeed, this would mean that σ ∈ S * \ S 0 , and by Observation 1, 0 = µ(σ) = (σ), a contradiction. Hence, J R σ = ∅. Then

. By the minimality assumption on σ, it then follows from (10) that

13 Documents de travail du Centre d'Economie de la Sorbonne -2016.80

By Lemma 1 (i) and the fact that 0 is neutral in R , we then have that

, which contradicts the choice of σ. Hence there is no σ ∈ Ker(µ) \ Ker(R), and thus Ker(µ) ⊆ Ker(R).

The last assertion is obvious. Then the following assertions are equivalent:

(iii) µ is weakly associative and it is retractive w.r.t. R;

(iv) µ is weakly associative, weakly retractive, and | R | ≥ |µ|.

Proof. By Lemma 1(i), (i) is equivalent to (ii) and (i) implies (iii). Also, if µ satisfies (C1), (C2), (C3) on C2 and it is weakly associative and retractive w.r.t. R, then

that is, µ is extremal w.r.t. R, and hence (iii) implies (ii). Also, by Lemma 3 we have that (i) implies (iv). To show that the converse implication also holds, note that by Remark 3 and | R | ≥ |µ| we get | R | = |µ| and, by weak retractiveness and Observation 1, it follows that R = µ. Lemma 5. Suppose that µ : C * → C satisfies (C1), (C2), (C3) on C2 , and that µ is weakly associative, anonymous, and satisfies RP(R) and | R | ≥ |µ| with both of the same sign (i.e., R (σ) ≥ 0 if and only if µ(σ) ≥ 0), for some R ∈ R. Then µ = R .

Proof. First we show that | R | = |µ|. To this end, note that from (C1), (C2), (C3) and weak associativity it follows by Observation 1 that for any sequence σ = (a i ) i∈I ∈ S 0 ,

Let b = µ(σ). Since µ and R have the same sign and µ is odd, we may assume that b ≥ 0. By assumption and (12), we have that

By RP(R), we have that µ

However, R is extremal w.r.t. R and from the fact that 0

, and thus by ( 14) We can now state further characterizations as mentioned earlier.

Theorem 4. Let µ : C * → C be an anonymous and weakly associative function that satisfies (C1), (C2), (C3) on C2 . For every R ∈ R, the following assertions are equivalent.

(ii) µ decomposable on J R σ for every sequence σ, and Ker(R) ⊆ Ker(µ). (iii) µ is extremal w.r.t. R.

(iv) µ is retractive w.r.t. R. Proof. By Theorem 3 and Proposition 1 we have that (i) to (v) are all equivalent. By Lemma 5 we also have that (vi) implies (i), and by Lemma 3 it follows that (i) implies (vi).

Independence of the axioms

As a preliminary remark, we must distinguish the roles of the different axioms introduced. Axioms (C1), (C2), (C3) pertain only to the case of a binary operator and are clearly independent. Weak associativity permits to extend to associative sequences, and by Observation 1, the operator is obtained. Hence, the four axioms are independent.

The remaining axioms only serve to determine what happens for nonassociative sequences. Anonymity permits to work on ordered sequences and is clearly independent of the other remaining axioms.

Therefore, our task consists in proving that, for any rule R (except for R = • + -, see Remarks We begin with Theorem 3 and suppose that µ satisfies (C1), (C2), (C3), anonymity, and weak associativity. Consider any rule R = • + -in R. Clearly, we only have to consider nonassociative sequences σ = (a i ) i∈I . Let C = Z.

(i) Define µ(σ) as min(a I\J R σ ) for any nonassociative σ s.t. I \J R σ = ∅, and 0 otherwise. Then µ satisfies Ker(R) ⊆ Ker(µ) but not decomposability. Indeed, by Lemma 1(iv)

, in which case µ(a J R σ ) = 0, so that (C2) is satisfied. Then, Ker(R) ⊆ Ker(µ) is not satisfied. We claim that, assuming that µ is decomposable on J R σ , µ is well defined on S 0 (note that by (C2), we already have µ(a, -a) = 0 for every a.). Indeed, by decomposability, µ

) is associative, and therefore µ(σ) = (σ ′ ). Otherwise, σ I\J R σ contains -α as extremal term and possibly other terms smaller in magnitude. By repeatedly applying decomposability, say k ≥ 1 times, we can delete the occurrence of -α in the subsequence that remains undeleted, and thus we arrive at a sequence σ

) that is associative. To illustrate this fact, suppose that σ = (2 • 3, 7 • (-3)), R = • 0 = ρ * 3 , and that µ(n • 3, n • (-3)) = α = 3, for n ≥ 2. Then the first application of decomposability produces σ

By reapplying decomposability, we obtain σ

and in this case we have σ

I\J σ (2) = (7 • 3). By applying decomposability for a third time, we get σ

), which is now associative, and from which it follows that µ(σ) = µ(σ ′ ) = µ(σ (2) ) = µ(σ (3) ) = (σ (3) ) = 3.

In general, if k ≥ 1 applications of decomposability suffice to delete the occurrences of -α in σ I\J R σ (i.e., σ (k) is associative), then


Damien Graux
email: damien.graux@inria.fr

Louis Jachiet
email: louis.jachiet@inria.fr

Pierre Genevès
email: pierre.geneves@cnrs.fr

Nabil Layaïda
email: nabil.layaida@inria.fr

SPARQLGX : Une Solution Distribuée pour RDF Traduisant SPARQL vers Spark

sparql est un langage de requête standardisé par le w3c permettant d'interroger des données exprimées au format rdf (Resource Description Framework ). Avec l'augmentation des volumes de données rdf disponibles, de nombreux efforts de recherche ont été faits pour permettre l'évaluation distribuée et efficace de requêtes sparql. Dans ce contexte, nous proposons et partageons sparqlgx : notre solution de stockage rdf distribuée utilisant Apache Spark pour évaluer des requêtes sparql et stockant les données via des infrastructures Hadoop (hdfs). sparqlgx repose sur un traducteur de requêtes sparql vers une séquence d'instructions exécutables par Spark en adoptant des stratégies d'évaluation selon (1) le schéma de stockage des données utilisé et (2) des statistiques sur les données. Nous montrons que sparqlgx permet l'évaluation de requêtes sparql sur plusieurs milliards de triplets rdf répartis sur plusieurs noeuds. Nous comparons aussi sparqlgx à d'autres solutions issues de l'état-de-l'art. Nous démontrons ainsi les performances obtenues en permettant aux participants de reproduire par eux-mêmes les résultats présentés grâce à différents scénarios mettant directement en compétition plusieurs solutions de l'état-de-l'art. Nous montrons dans ce travail que tout en ayant une architecture relativement simple, sparqlgx représente une alternative intéressante dans de nombreux cas d'utilisation.

INTRODUCTION

sparql is the standard query language for retrieving and manipulating data represented in Resource Description Framework (rdf) [START_REF] Hayes | RDF semantics[END_REF]. sparql constitutes one key technology of the semantic web and has become very popular since it became an official w3c recommendation, first in 2008 [START_REF]SPARQL query language for RDF[END_REF] and then in 2013 as an improved version [START_REF]SPARQL 1.1 overview[END_REF]. In this study, we investigate the problem of evaluating the conjunctive sparql queries i.e. the Basic Graph Pattern fragment. The construction of efficient sparql query evaluators faces several challenges. First, rdf datasets are increasingly large, with some already containing more than a billion triples. To handle efficiently this growing amount of data, we need systems to be distributed and to scale. Furthermore, semantic data often have the characteristic of being dynamic (frequently updated). Thus being able to answer quickly after a change in the input data constitutes a very desirable property for a sparql evaluator.

In this context, we propose sparqlgx: an engine designed to evaluate sparql queries based on Spark [START_REF] Zaharia | Resilient distributed datasets: A fault-tolerant abstraction for in-memory cluster computing[END_REF]: a MapReduce-like data-parallel engine. sparqlgx relies on a compiler of sparql conjunctive queries which generates Scala code that is executed by the Spark infrastructure.

SPARQLGX: ARCHITECTURE

Data Storage Model. In order to process rdf datasets, we adopt the vertical partitioning approach introduced by Abadi et al. in [START_REF] Abadi | Scalable semantic web data management using vertical partitioning[END_REF] which stores a triple (s p o) in a file named p whose contents keeps only s and o entries. Converting rdf data into a vertically partitioned dataset is straightforward while (1) tending to minimize the memory footprint and the datasets size on disks and (2) reducing response time when queries have bounded predicates since searches are limited to the relevant files.

Compilation of SPARQL BGPs. Conjunctions of triple patterns (tps) are translated in terms of primitives of the Apache Spark framework expressed in Scala code. Each bgp is first translated in terms of a list of filters, which are then joined based on common variables. SPARQL Fragment Extension. Moreover, we support the selection of distinguished variables in addition of the "SE-LECT *" form. We also support sparql solution modifiers e.g. removing duplicates with DISTINCT, sorting with OR-DER BY, returning only few answers with LIMIT.

Furthermore, we also easily translate two additional sparql keywords: UNION and OPTIONAL, provided they are located at top-level in the WHERE clauses. Indeed, Spark allows to aggregate sets having similar structures with union and is also able to add data if possible with leftOuterJoin. Thus sparqlgx natively supports a slight extension (unions and optionals both at top level) of the extensively studied sparql fragment made of conjunctions of tps. Optimized Joins With Statistics. The evaluation process (using Spark) first evaluates tps and then joins these subsets according to their common variables; thus, minimizing the intermediate set sizes involved in the join process reduces evaluation time (since communication between workers is then faster). Thereby, statistics on data and information on intermediate results sizes provide useful information that we exploit for optimisation purposes. Thereby, to rank each tp, we compute statistics on datasets counting all the distinct subjects, predicates and objects. This is implemented in a compile-time module that sorts tps in ascending order of their selectivities before they are translated.

C1 C2 C3 F1 F2 F3 F4 F5 L1 L2 L3 L4 L5 S1 S2 S3 S4 S5 S6
Finally, we also want to avoid cartesian products. Given an ordered list l of tps we compute a new list l by repeating the following procedure: remove from l and append to l the first tp that shares a variable with a tp of l . If no such tp exists, we take the first.

S-DE: SPARQLGX as a Direct Evaluator.

In certain situations, queried data might be subject to updates; in others users might only need to evaluate a single query once (for data cleaning purposes for instance). In such cases, it is interesting to limit as much as possible both the preprocessing time and the query evaluation time. For this purpose, sparqlgx provides a specific tool, called sde, capable of directly evaluating sparql queries without preprocessing.

DEMONSTRATION DETAILS

We report on our experimental comparisons of sparqlgx against other open source hdfs-based distributed rdf systems such as PigSPARQL [START_REF] Schätzle | PigSPARQL: Mapping SPARQL to pig latin[END_REF], RYA [START_REF] Punnoose | Rya: a scalable RDF triple store for the clouds[END_REF], CliqueSquare [START_REF] Goasdoué | Cliquesquare: Flat plans for massively parallel RDF queries[END_REF], S2RDF [START_REF] Schätzle | S2RDF: RDF querying with SPARQL on spark[END_REF] and RDFHive. Furthermore, we consider several datasets coming from two popular bgp benchmarks: LUBM [START_REF] Guo | LUBM: A benchmark for OWL knowledge base systems[END_REF] and WatDiv [START_REF] Aluç | Diversified stress testing of RDF data management systems[END_REF].

We present in Figure 1 the response times obtained with WatDiv1k (about 100 million triples). This illustrates that, for this dataset: (1) SDE always outperforms other tested "direct evaluators" (i.e. PigSPARQL and RDFHive); [START_REF]SPARQL 1.1 overview[END_REF] sparqlgx is able to answer all the queries unlike RYA and CliqueSquare; (3) it shares with CliqueSquare the same order of magnitude for linear queries; (4) it outperforms its competitors on complex queries. To further illustrate the performance of sparqlgx, we provide an interactive GUI. Attendees can interact directly with our engines (i.e. sparqlgx and sde) and test them against state-of-the-art solutions.

CONCLUSION

  (c) 2016, Copyright is with the authors. Published in the Proceedings of the BDA 2016 Conference (15-18 November, 2016, Poitiers, France). Distribution of this paper is permitted under the terms of the Creative Commons license CC-by-nc-nd 4.0. (c) 2016, Droits restant aux auteurs. Publié dans les actes de la conférence BDA 2016 (15 au 18 Novembre 2016, Poitiers, France). Redistribution de cet article autorisée selon les termes de la licence Creative Commons CCby-nc-nd 4.0. .

Figure 1 :

 1 Figure 1: Query Response Times with WatDiv1k.

  sparqlgx outperforms several related implementations in many cases, while implementing a simple architecture exclusively built on top of open source and publicly available technologies. The sparqlgx implementation is available from: http://github.com/tyrex-team/sparqlgx Related Publications • SPARQLGX: Efficient Distributed Evaluation of SPARQL with Apache Spark. D. Graux, L. Jachiet, P. Genevès, N. Layaïda. The 15th International Semantic Web Conference, Kobe, Japan, October 17-21, 2016, Proceedings, Part II, pages 80,87. • SPARQLGX in action: Efficient Distributed Evaluation of SPARQL with Apache Spark. D. Graux, L. Jachiet, P. Genevès, N. Layaïda. Proceedings of the ISWC 2016 Posters & Demonstrations Track co-located with 15th International Semantic Web Conference, Kobe, Japan, October 19, 2016.