Lorenzo Brandolese
email: brandolese@math.univ-lyon1.fr

Manuel Fernando Cortez

ON PERMANENT AND BREAKING WAVES IN HYPERELASTIC RODS AND RINGS

Keywords: Rod equation, Compressible rod, Camassa-Holm, Shallow water, Wave-breaking, Blowup, Minimization, weighted Poincaré inequality

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point (t0, x0) ∈ R + × S 1 is the identically zero solution. Such conclusion holds provided the physical parameter γ of the model (related to the Finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa-Holm equation, corresponding to γ = 1. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities.

Introduction

Motivations. This paper is devoted to the study of periodic solutions of the rod equation. A first physical motivation comes from the study of the response of hyper-elastic rings under the action of an initial radial stretch. As the nonlinear dispersive waves propagating inside it could eventually lead to cracks, an important problem is the determination of conditions that must be fulfilled in order to prevent their formation. The main issue of the present paper will be a precise description of crack mechanisms inside such rings.

A second reason for studying periodic solutions is that periodic waves spontaneously arise also in hyper-elastic rods: indeed, it has been recently observed that the solitary waves propagating inside an ideally infinite length rod can suddenly feature a transition into waves with finite period as their amplitude increases, see [START_REF] Dai | Solitary shock waves and other travelling waves in a general compressible hyperelastic rod[END_REF].

Our third motivation comes from the study of shallow water waves inside channels. Indeed, the Camassa-Holm equation (at least in the dispersionless case) is a particular case, corresponding to γ = 1, of the rod equation below: if the motion of small amplitude waves is usually modeled by the KdV equation, larger amplitude waves, and in particular breaking waves, are more accurately described by the Camassa-Holm equation. In fact, both the KdV and the Camassa-Holm equation can be rigorously derived as an asymptotic model from the free surface Euler equations for irrotational inviscid flows, in the so-called shallow water regime µ = h 2 /λ 2 < < 1, where h and λ denote respectively the average elevation of the liquid over the bottom and the characteristic wavelength. The Camassa-Holm equation models small, but finite, amplitude waves, i.e. waves such that the dimensionless amplitude parameter ǫ = a/h satisfies ǫ = O( √ µ),

where a is the typical amplitude, whereas the derivation of KdV would require the more stringent scaling ǫ = O(µ). The Camassa-Holm equation thus better captures the genuinely nonlinear behavior of larger amplitude waves and, contrary to KdV, admits both permanent solutions and solutions that blow up in finite time. For a detailed discussion on these issues, see [START_REF] Camassa | A new integrable shallow-water equation[END_REF][START_REF]Wave breaking for nonlinear nonlocal shallow water equations[END_REF][START_REF] Constantin | The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations[END_REF][START_REF] Johnson | Camassa-Holm, Korteweg-de Vries and related models for water waves[END_REF]. Let S = R/Z be the unit circle. The Cauchy problem for the periodic rod equation is written as follows:

(1.1)

   u t + γuu x = -∂ x p * 3 -γ 2 u 2 + γ 2 u 2
x , t ∈ (0, T ), x ∈ S, u(0, x) = u 0 (x).

The real parameter γ is related to the Finger deformation tensor of the material. Both positive and negative values γ are admissible. We refer to [START_REF] Dai | Solitary shock waves and other travelling waves in a general compressible hyperelastic rod[END_REF] for more details on the physical background and the mathematical derivation of the model. The function p in (1.1) is the kernel of the convolution operator (1-∂ 2 x ) -1 . It is the continuous 1-periodic function given by (1.2)

p(x) = cosh(x -[x] -1/2) 2 sinh(1/2) ,
where [•] denotes the integer part. The Camassa-Holm case γ = 1 is somewhat particular, as in this case (1.1) inherits a bi-Hamiltonian structure and the equation is completely integrable, in the sense of infinitedimensional Hamiltonian systems: in suitable variables (action-angle), the flow is equivalent to a linear flow at constant speed on the Jacobi variety associated to a (mostly infinite-dimensional) torus, cf. the discussion in [START_REF] Consantin | A shallow water equation on the circle[END_REF]. Moreover, the Camassa-Holm equation is a re-expression of geodesic flow on the diffeomorphism group of the circle, see the discussion in [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF][START_REF] Kolev | Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations[END_REF]. For these reasons, many important results valid for the Camassa-Holm equation do not go through in the general case. For example, in the case of the Camassa-Holm equation on the real line, a striking necessary and sufficient condition for the global existence of strong solution can be given in terms on the initial potential y 0 = u 0 -u 0,xx , see [START_REF]Breakdown of the Camassa-Holm equation[END_REF]. On the other hand, very little is known on the global existence of strong solutions when γ = 1: it can even happen (when γ = 3) that all nonzero solutions blow up in finite time. Smooth solitary waves that are global strong solutions were constructed at least for some γ, see [START_REF] Dai | Solitary shock waves and other travelling waves in a general compressible hyperelastic rod[END_REF][START_REF] Lenells | Traveling waves in compressible elastic rods[END_REF]. These are essentially the only known examples of global smooth solutions.

Our working assumption will be that u 0 belongs to the Sobolev space H s (S), for some s > 3/2. Then, for any γ ∈ R, the Cauchy problem for the rod equation is locally well-posed, in the sense that there exist a maximal time 0 < T * ≤ ∞ and a unique solution u ∈ C([0, T * ), H s (S)) ∩ C 1 ([0, T * ), H s-1 (S)). Moreover, the solution u depends continuously on the initial data. It is also known that u admits several invariant integrals, among which the energy integral,

E(u) = S (u 2 + u 2 x ) dx.
In particular, because of the conservation of the Sobolev H 1 -norm, the solution u(x, t) remains uniformly bounded up to the time T * . On the other hand, if T * < ∞ then lim sup t→T u(t) H s = ∞ (s > 3/2) and the precise blowup scenario, often named wave breaking mechanism, is the following:

(1.3) T * < ∞ ⇐⇒ lim inf t→T * inf x∈S γu x (t, x) = -∞.
See, e.g. [START_REF] Constantin | Stability of a class of solitary waves in compressible elastic rods[END_REF].

Quick overview of the main results. We will state our two main theorems in the next section after preparing some notations. Loosely, the first theorem asserts that if |γ| is not too small, then there exist a constant β γ > 0 such that if

(1.4) u ′ 0 (x 0 ) > β γ |u 0 (x 0 )| if γ < 0, or u ′ 0 (x 0 ) < -β γ |u 0 (x 0 )| if γ > 0 in at least one point x 0 ∈ S,
then the solution arising from u 0 ∈ H s (S) must blow up in finite time.

Our second theorem quantifies the above result: it precises what "|γ| not too small" means, addressing also the delicate issue of finding sharp estimates for β γ . For example, in the particular case of the periodic Camassa-Holm equation we get that a sufficient condition for the blowup is:

(1.5) ∃ x 0 ∈ S : u ′ 0 (x 0 ) < - 5 2 - 3 2 • cosh 1 2 cosh 3 2 -1 sinh 1 2 sinh 3 2 |u 0 (x 0 )|.
An analogue but weaker result was recently established in our previous paper [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF], where we dealt with non-periodic solutions on the whole real line with vanishing boundary conditions as x → ∞. In the present paper, we take advantage of the specific structure of periodic solutions in order to make improvements on our previous work in two directions.

First of all, the analogue blowup result for the rod equation on R could be established in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF] only in the range 1 ≤ γ ≤ 4 in the non-periodic case. But the relevant estimates on the circle that we will establish turn out to be much stronger. They allow us to cover blowup results of periodic solutions e.g. for arbitrary large γ (or γ negative and arbitrary small). The following corollary of Theorem 2.1 is therefore specific to periodic solutions: Corollary 1.1. There exists an absolute constant β ∞ (it can be checked numerically that β ∞ = 0.295 . . .) with the following property. If u 0 ∈ H s (S), with s > 3/2, is such that for some

x 0 ∈ S, u ′ 0 (x 0 ) > β ∞ |u 0 (x 0 )|, or otherwise u ′ 0 (x 0 ) < -β ∞ |u 0 (x 0 )|,
then the solutions (unique, but depending on γ) of the rod equation (1.1) arising from u 0 blow up in finite time respectively if γ < < -1 or γ > > 1. In both cases, the maximal existence time is

T * = O( 1 |γ| ) as γ → ∞.
There is a second important difference between in the behavior of periodic and non-periodic solutions. It can happen that two initial data u 0 ∈ H s (S) and u 0 ∈ H s (R) agree on an arbitrarily large finite interval, and that periodic solution arising from u 0 blows up, whereas the solution arising from u 0 and vanishing at infinity exists globally. As a comparison, the blowup criterion in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF] for solutions in H s (R) of the Camassa-Holm equation reads inf R (u ′ 0 + |u 0 |) < 0; on the other hand, according to (1.5), in the periodic case the condition inf S (u ′ 0 + 0.515 |u 0 |) < 0 would be enough for the development of a singularity. In general, for γ ∈ [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF][START_REF] Brandolese | Blowup issues for a class of nonlinear dispersive wave equations[END_REF], the coefficient β γ in (1.4) is considerably lower than the corresponding coefficient β γ,R computed in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF] for the blowup criterion of non-periodic solutions.

The most important feature of our blowup criteria (1.4)-(1.5) is that it they are local-in-space. This means that these criteria involve a condition only on a small neighborhood of a single point of the datum. Their validity is somewhat surprising, as equation (1.1) is non-local. A huge number of previous papers addressed the blowup issue of solutions to equation (1.1), (see, e.g. [7-11, 19, 22, 23, 27, 33-37]; (the older references only dealt with the Camassa-Holm equation). But the corresponding blowup criteria systematically involved the computation of some global quantities of u 0 : typically, conditions of the form u ′ 0 (x 0 ) < -c γ u 0 H 1 (S 1 ) or some other integral conditions on u 0 , or otherwise antisymmetry conditions, etc.

The main idea (inspired to us combining those of [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF][START_REF] Constantin | Existence of permanent and breaking waves for a shallow water equation: a geometric approach[END_REF][START_REF] Mckean | Breakdown of a shallow water equation[END_REF]) will be to study the evolution of u+βu x and u-βu x along the trajectories of the flow map of γu , for an appropriate parameter β = β(γ) to be determined in order to optimize the blowup result. The main technical issue of the present paper will be the study of a two-parameters family of minimization problems. Such minimization problems arise when computing the best constants in the relevant convolution estimates. Calculus of variation tools have been already successfully used for the study of blowup criteria of the rod equation, see, e.g., [START_REF] Wahlén | On the blow-up of solutions to the periodic Camassa-Holm equation[END_REF][START_REF]On the blowup of solutions to a nonlinear dispersive rod equation[END_REF][START_REF] Zhou | Wave breaking for a shallow water equation[END_REF][START_REF]Local well-posedness and blow-up criteria of solutions for a rod equation[END_REF][START_REF]Blow-up of solutions to a nonlinear dispersive rod equation[END_REF]. The difference with respect to these papers is that our approach requires the minimization of non-coercive functionals. Once such minima are computed, the proof of the blowup can follow the steps of [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF].

We finish this introduction by mentioning a second simple consequence of our main Theorem 2.1, that seems to be of independent interest. Its result applies in particular to the case γ = 1. It is a new result (at best of our knowledge) also for the Camassa-Holm equation.

Corollary 1.2. Let u ∈ C([0, ∞[, H s (S)) ∩ C 1 ([0, ∞[, H s-1 (S)
), be a global solution of the rod equation (1.1) with γ ≤ γ - 1 or γ ≥ γ + 1 (where γ - 1 = -1.036 . . . and γ + 1 = 0.269 . . .). If u vanishes at some point (t 0 , x 0 ) ∈ [0, ∞) × S, then u must be the trivial solution: u(x, t) ≡ 0 for all t ≥ 0 and x ∈ S. Corollary 1.2 improves an earlier result by A. Constantin and J. Escher [START_REF]Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation[END_REF], asserting that the trivial solution is the only global solution of the periodic Camassa-Holm equation such that for all t ≥ 0, ∃ x t ∈ S such that u(t, x t ) = 0. Basically, we manage to replace their condition "∀ t. . . " by the much weaker one "∃ t. . . ". More importantly, unlike [START_REF]Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation[END_REF], our approach will make use of only few properties of the equation, and is more suitable for generalizations.

It might be surprising that our results a priori excludes a small neighborhood of the origin for the parameter γ. In fact, such restriction might be purely technical. However, one should observe that γ = 0 must be excluded. The reason is that for γ = 0 all solutions arising from u 0 ∈ H s (S) are global in time. Indeed, the blowup scenario (1.3) is never fulfilled. It is worth noticing that for γ = 0 the rod equation reduces to the BBM equation introduced by Benjamin, Bona and Mahony in [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF] as a model for surface wave in channels.

We will establish a result similar to that of Corollary 1.2 for non-periodic solutions vanishing for x → ∞ (see Corollary 7.1 at the end of the paper for the precise statement). In this case, we will need that the global solution u is such that u(t 0 , x 1 ) = u(t 0 , x 2 ) = 0 in at least two different points, or otherwise that u(t 0 , •) decays sufficiently fast as x → ∞ to conclude that u(t, x) ≡ 0 for all (t, x) ∈ R + × R.

Organization of the paper. In Section 2 we state our main results, Theorem 2.1 and Theorem 2.2. The first theorem is proved in Section 3. In Section 4 we study a minimization problem that will play an important role for the proof of Theorem 2.2, given in Section 5. Corollary 1.1 will immediately follows from assertion (ii) of this theorem. We will study in more detail the Camassa-Holm equation in Section 6. Corollary 1.2 is established in Section 7 where we also provide some new results for non-periodic solutions. An appendix showing the agreement between our theorems and some numerical results concludes the paper.

The main results

We start preparing some notations. For any real constant α and β, let I(α, β) ≥ -∞ defined by (2.1)

I(α, β) = inf 1 0 p + βp x ) αu 2 + u 2 x dx : u ∈ H 1 (0, 1), u(0) = u(1) = 1 .
Proposition 3.3 below will characterize the set of parameters (α, β) for which the functional appearing in (2.1) is bounded from below as well as the subset for which the infimum is achieved. We also introduce, for γ ∈ R * , the quantity β γ ∈ [0, +∞] defined by (2.2)

β γ = inf β ∈ R + : β 2 + I 3-γ γ , β -3-γ
γ ≥ 0 , with the usual convention that β γ = +∞ if the infimum is taken on the empty set.

Our main result is the following blowup theorem:

Theorem 2.1. Let γ ∈ R * be such that β γ < +∞. Let u 0 ∈ H s (S)
with s > 3/2 and assume that there exists x 0 ∈ S, such that

(2.3) u ′ 0 (x 0 ) > β γ |u 0 (x 0 )| if γ < 0, or u ′ 0 (x 0 ) < -β γ |u 0 (x 0 )| if γ > 0.
Then the corresponding solution u of equation (1.1) in C([0, T * ), H s (S)) ∩ C 1 ([0, T * ), H s-1 (S)) arising from u 0 blows up in finite time. Moreover, the maximal time T * is estimated by

(2.4) T * ≤ 2 γ u ′ 0 (x 0 ) 2 -β 2 γ u 0 (x 0 ) 2
and, for some x(t) ∈ S, the blowup rate is

(2.5) u x (t, x(t)) ∼ - 2 γ(T * -t) as t → T * .
Theorem 2.1 is meaningful only if γ is such that β γ < +∞. The validity of such condition is a priori not clear, as it might happen that the set in equation (2.2) is empty. An important part of the present paper will be devoted to the technical issue of discussing the validity of the condition β γ < +∞ and next estimating β γ . This will be done by establishing some sharp estimates on I(α, β).

In order to state our next theorem, let us introduce the complex number

µ = 1 2 1 + 4(3 -γ)/γ, γ = 0,
where 1 + 4(3 -γ)/γ denotes any of the two complex square roots. We also consider the four constants (that will be constructed in (5.8) below):

γ - 1 = -1.036 . . . γ + 1 = 0.269 . . . γ - 2 = -1.508 . . . γ + 2 = 0.575 . . . Theorem 2.2. (i) For any γ ∈ (-∞, γ - 1 ] ∪ [γ + 1 , +∞), we have β γ < +∞, so that Theorem 2.1 applies in such range. More precisely, if γ ∈ (-∞, γ - 2 ] ∪ [γ + 2 , ∞), then (2.6) β γ ≤ 3 γ - 1 2 -µ • cosh 1 2 cosh µ -1 sinh 1 2 sinh µ . If otherwise γ ∈ [γ - 2 , γ - 1 ] ∪ [γ + 1 , γ + 2 ]
, then β γ can be estimated as in inequality (5.13) below (in (5.13), P ν denotes Legendre function of the first kind of degree ν).

(ii) The limit β ∞ = lim γ→±∞ β γ does exist and

(2.7) β ∞ ≤ √ 3 1 -cosh 1 2 cos √ 3 2 2 sinh 1 2 sin √ 3 2 - 1 2 = 0.296 . . . .
It is worth observing that we have β γ = 0 if and only if γ = 3. This can be checked directly from the definitions (2.1)-(2.2). Accordingly, the right-hand side in (2.6) vanishes for γ = 3. In particular, we recover the known fact (see [START_REF] Constantin | Stability of a class of solitary waves in compressible elastic rods[END_REF]) that for γ = 3 any nonzero initial datum u 0 ∈ H s (S) (with s > 3/2) gives rise to a solution that blows up in finite time. Moreover, the maximal time

T * satisfies T * ≤ 2 3 -inf x∈S u ′ 0 (x)
. The upper bounds (2.6)-(2.7) are not optimal but, strikingly, they are almost sharp. Indeed we can compute the numerical approximation of β γ with arbitrary high precision. We find that the error between the above bounds and the numerical value is only of order 10 -3 . For example, we find β ∞ = 0.295 . . . that is indeed very close to the bound (2.7). Moreover, in the Camassa-Holm case, we find numerically that β 1 = 0.513 . . .. This is in good agreement with estimate (2.6), that for γ = 1 provided us with the bound β 1 ≤ 0.515 . . . . We will devote the appendix to a more detailed discussion of the numerical results. Such analysis will also show that the effective range of applicability of Theorem 2.1 is slightly larger

than the range γ ∈ (-∞, γ - 1 ] ∪ [γ + 1 , ∞) predicted by Theorem 2.2.
The reader should compare Figure 1 with Figure 9 at the end of the paper: the former plot is obtained analytically and the latter numerically.

This paper does not address the problem of the continuation of the solution after the blowup as this issue is extensively studied in the literature. See, e.g. [START_REF] Bressan | Global conservative solutions of the Camassa-Holm equation[END_REF][START_REF]Global dissipative solutions of the Camassa-Holm equation[END_REF][START_REF] Holden | Global conservative solutions of the generalized hyperelastic-rod wave equation[END_REF][START_REF] Mustafa | Global conservative solutions of the hyperelastic rod equation[END_REF].

First properties of I(α, β) and proof of Theorem 2.1

For any real β, let us consider the 1-periodic function

(3.1) ω(x) = p(x) + βp ′ (x),
where p is the kernel introduced in (1.2) and p ′ denotes the distributional derivative on R, that agrees in this case with the classical a.e. pointwise derivative on R\Z.

We would like to make use of ω as a weight function. The non-negativity condition ω ≥ 0 is equivalent to the inequality cosh(1/2) ≥ ±β sinh(1/2), i.e., to the condition

(3.2) -e+1 e-1 ≤ β ≤ e+1 e-1 .
Figure 1. The upper-bound estimate of β γ given by Theorem 2.2. The estimate is valid outside the interval [γ - 1 , γ + 1 ] (gray region).

Throughout this section, we will work under the above condition on β.

Let us introduce the weighted Sobolev space

(3.3) E β = u ∈ L 1 loc (0, 1) : u 2 E β ≡ 1 0 ω(x)(u 2 + u 2 x )(x) dx < ∞ ,
where the derivative is understood in the distributional sense. Notice that E β agrees with the classical Sobolev space H 1 (0, 1) when |β| < e+1 e-1 , as in this case ω is bounded and bounded away from 0, and the two norms • E β and • H 1 are equivalent: in particular, in this case if u ∈ E β then u = ũ a.e., where ũ ∈ C([0, 1]). The situation is different for β = ± e+1 e-1 as E β is strictly larger than H 1 (0, 1) in this case. Indeed, we have

(3.4) ω(x) = 2e (e -1) 2 sinh(x), x ∈ (0, 1), (if β = e+1 e-1
).

An element u of E (e+1)/(e-1) , after modification on a set of measure zero, agrees with a function ũ that is continuous on (0, 1], but may be unbounded for

x → 0 + (for instance, | log(x/2)| 1/3 ∈ E (e+1)/(e-1)
). In the same way,

(3.5) ω(x) = 2e (e -1) 2 sinh(1 -x),
x ∈ (0, 1), (if β = -e+1 e-1 ).

After modification on a set of measure zero, the elements of E -(e+1)/(e-1) are continuous on [0, 1), but may be unbounded for x → 1 -. Let us consider the closed subspace E β,0 of E β defined as the closure of C ∞ c (0, 1) in E β . Notice that, with slightly abusive notation, consisting in identifying u with its continuous representative ũ we have:

E β,0 = H 1 0 (0, 1) = {u ∈ H 1 (0, 1) : u(0) = u(1) = 0}, if |β| < e+1 e-1 . (3.6)
On the other hand, in the limit cases for β we have the following:

Lemma 3.1. -If u ∈ E (e+1)/(e-1),0 , then u(x) = O( | log x|) as x → 0 + and u(1) = 0. -If u ∈ E -(e+1)/(e-1),0 , then u(x) = O( | log(1 -x)|) as x → 1 -and u(0) = 0.
Proof. We consider only the case β = e+1 e-1 as in the other case the proof is similar. The condition u(1) = 0 follows from the fact that ω is bounded and bounded away from the origin in a left neighborhood of 1. Moreover.

|u(x)| = 1 x u ′ (y) dy ≤ 1 0 ω(y)u ′ (y) 2 dy 1/2 1 x 1 ω(y) dy 1/2 ≤ C u E β | log x|.
The elements of E β,0 satisfy to the weighted Poincaré inequality below:

Lemma 3.2. For all -e+1 e-1 ≤ β ≤ e+1 e-1
, there exists a constant C > 0 such that

(3.7) ∀ v ∈ E β,0 : 1 0 ω(x)v(x) 2 dx ≤ C 1 0 ω(x)v x (x) 2 dx.
Proof. The validity of such inequality is obvious for |β| < e+1 e-1 . Indeed, in this case there exist two constants c 1 and c 2 such that on the interval (0, 1) we have 0 < c 1 ≤ ω(x) ≤ c 2 and the validity of (3.7) is reduced to that of the classical Poincaré inequality without weight.

In the limit case β = e+1 e-1 , we can observe that, from (3.4), the only zero of the function ω(x) = 2e (e-1) 2 sinh(x) in the closure of (0, 1) is of order one. Then the weight ω(x) satisfies the necessary and sufficient condition for the weighted Poincaré inequality to hold, see [START_REF] Stredulinsky | Weighted inequalities and degenerate elliptic partial differential equations[END_REF]. For reader's convenience we prove directly inequality (3.7) exhibiting an explicit constant. Recall that v(x) = O(| log(x)| 1/2 ) as x → 0 + . In particular, for all v ∈ E (e+1)/(e-1),0 , we have ω(x)v(x) 2 1- 0+ = 0. Then integrating by parts and next using the Cauchy-Schwarz inequality we obtain, for v ∈ E (e+1)/(e-1),0 :

1 4 1 0 cosh v 2 2 = 1 0 sinh vv x 2 ≤ 1 0 sinh v 2 1 0 sinh v 2 x .
Next observe that

∀ x ∈ (0, 1), cosh(x) ≥ e 2 + 1 e 2 -1 sinh(x).
Combining these two estimates, we obtain

(3.8) 1 0 sinh(x)v(x) 2 dx ≤ 4(e 2 -1) 2 (e 2 + 1) 2 1 0 sinh(x)v x (x) 2 dx.
Then (3.7) holds, e.g., with C = 4(e 2 -1) 2 (e 2 +1) 2 . In the other limit case β = -e+1 e-1 , ω(x) = 2e (e-1) 2 sinh(1x). Therefore, we can reduce to the previous case with a change of variables.

The constant in (3.8) is far from being optimal. In fact, we will find the best constant in Remark 4.1, as a byproduct of the analysis performed Section 4.

This being observed, let us go back to our minimization problem (3.9)

I(α, β) = inf 1 0 p + βp ′ ) αu 2 + u 2 x dx : u ∈ H 1 (0, 1), u(0) = u(1) = 1 .
Proposition 3.3. We have

I(α, β) > -∞ ⇐⇒ -e+1 e-1 ≤ β ≤ e+1 e-1 , α > -1/C(β),
where C(β) > 0 is the best constant in the weighted Poincaré inequality (3.7). Moreover, if |β| < e+1 e-1 , then I(α, β) is in fact a minimum and there is only one minimizer u ∈ H 1 (0, 1) with u(0) = u(1) = 1.

Proof. Putting u = v + 1 and observing that

1 0 ω(x) dx = 1, we see that (3.10) I(α, β) = α + inf J(v) : v ∈ H 1 0 (0, 1)
, where

(3.11) J(v) = 1 0 ω(x)(αv 2 + v 2 x + 2αv)(x) dx.
Assume that I(α, β) > -∞. Then |β| ≤ e+1 e-1 , otherwise we would get a contradiction by taking a sequence of the form χ(x) cos(nx) with χ smooth and such that supp(χ) ⊂ supp(ω -) ∩ (0, 1), where ω -denotes the negative part of ω. To prove the second inequality α > -1/C(β), we only have to treat the case α < 0. Applying the inequality

1 0 ω(αn 2 v 2 + n 2 v 2 x + 2αnv) ≥ I(α, β) -α
valid for all v ∈ H 1 0 (0, 1) and all n ∈ N and letting n → ∞, we get

1 0 ωv 2 ≤ - 1 α 1 0 ωv 2 x .
Then we get α ≥ -1/C(β). But in fact the inequality is strict, as otherwise we could take a sequence (v n ) such that (-α x defines on E β,0 an equivalent norm. As α < -1/C(β), the symmetric bilinear form

1 0 ωv 2 n )/( ωv 2 x,n ) →
B(v 1 , v 2 ) = 1 0 ω(αv 1 v 2 + v ′ 1 v ′ 2
) is coercive on the Hilbert space E β,0 . Applying the Lax-Milgram theorem yields the existence and the uniqueness of a minimizer v ∈ E β,0 for the functional J. But H 1 0 (0, 1) ⊂ E β,0 , so in particular, we get

I(α, β) > -∞. Moreover, if |β| < e+1 e-1 , then recalling E β,0 = H 1 0 we see that I(α, β) is in fact a minimum, achieved at ū = 1 + v ∈ H 1 .
The next proposition provides some useful information on I(α, β).

Proposition 3.4. The function (α, β) → I(α, β) ∈ R ∪ {-∞}, defined for all (α, β) ∈ R 2 ,
is concave with respect to each one of its variables and is even with respect to the variable β. Moreover,

(3.12) ∀ α ∈ R, ∀ |β| ≤ e+1 e-1 , -∞ ≤ I(α, e+1 e-1 ) ≤ I(α, β) ≤ I(α, 0) ≤ α.
Proof. The concavity property follows from the fact that I(α, β) is defined as an infimum of affine functions of the variables α and β. To prove that I(α, β) = I(α, -β), we can observe that and conclude making the change of variable y = 1 -x inside the integral in (3.9).

(p + βp ′ )(x) = (p -βp ′ )(1 -x)
To prove the last inequality in (3.12), consider the constant function u ≡ 1 and observe that

I(α, 0) ≤ α 1 0 (p + βp ′ )(x) dx = α.
The other inequalities follow from the concavity and parity properties of the map β → I(α, β). The result of Proposition 3.3 is also needed for the second inequality.

Next lemma motivates the introduction of the quantity I(α, β) in relation with the rod equation.

Lemma 3.5. For any α, β ∈ R and all u ∈ H 1 (S) the following convolution estimate holds:

(3.13) ∀ x ∈ S, (p + βp ′ ) * αu 2 + u 2 x )(x) ≥ I(α, β) u(x) 2
and I(α, β) is the best possible constant.

Remark 3.6. This lemma is interesting when I(α, β) > -∞, but its statement is true also when I(α, β) = -∞. In this case the sentence "-∞ is the best possible constant" should be understood as a negative result: the convolution estimate (3.13) breaks down whenever (α, β) does not fulfill the conditions of Proposition 3.3.

Proof. Let δ = δ(α, β) be some (possibly negative) constant. Because of the invariance under translations, we have that

(3.14) (p + βp ′ ) * αu 2 + u 2 x ) ≥ δ u 2
holds true for all u ∈ H 1 (S) if and only if

(p + βp ′ ) * αu 2 + u 2 x )(1) ≥ δ u 2 (1)
holds true for all u ∈ H 1 (S). But on the interval (0, 1), (p

+ βp ′ )(1 -x) = (p -βp ′ )(x). Hence, (p + βp ′ ) * αu 2 + u 2 x )(1) = 1 0 (p -βp ′ )(x)(αu 2 + u 2 x )(x) dx.
Normalizing to obtain u(1) = 1 (and hence u(0) = 1 by the periodicity) we get that the best constant δ in inequality (3.14) satisfies δ = I(α, -β) = I(α, β).

Proof of Theorem 2.1. Let s > 3/2. By the result of [START_REF] Constantin | Stability of a class of solitary waves in compressible elastic rods[END_REF] we know that there exists a unique solution u of the rod equation (1.1), defined in some nontrivial interval [0, T ), and such that

u ∈ C([0, T ), H s (S)) ∩ C 1 ([0, T ), H s-1 (S)). Moreover, the map u 0 → u is continuous from H s (S) to C([0, T ), H s (S)) ∩ C 1 ([0, T ), H s-1 (S)
). Owing to this well-posedness result, we can reduce to the case s ≥ 3. Indeed, if u 0 ∈ H s (S) with 3/2 < s < 3, we can approximate u 0 in the H s (S)-norm using a sequence of data (u 0 ) n belonging to H 3 (S) and satisfying condition (2.3). The relevant estimates, including that of T * , will pass to the limit as n → ∞.

If, by contradiction, T * = +∞, then T > 0 can be taken arbitrarily large. As in [START_REF] Constantin | Existence of permanent and breaking waves for a shallow water equation: a geometric approach[END_REF][START_REF] Mckean | Breakdown of a shallow water equation[END_REF], the starting point is the analysis of the flow map q(t, x), defined by (3.15) q t (t, x) = γu(t, q(t, x)), t ∈ (0, T ), x ∈ R, q(0, x) = x.

For any x ∈ S, the map t → q(t, x) is well defined and continuously differentiable in the whole time interval [0, T ). It is worth pointing out that for γ = 1, (3.15) is the equation defining the geodesic curve of diffeomorphisms, issuing from the identity in the direction of u 0 , cf. the discussion in [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF][START_REF] Kolev | Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations[END_REF]. However, no such geometric interpretation is available for γ = 1.

From the rod equation

(3.16) u t + γuu x = -∂ x p * 3-γ 2 u 2 + γ 2 u 2
x , differentiating with respect to the x variable and applying the identity ∂ 2

x p * f = p * f -f , we get

u tx + γuu xx = 3 -γ 2 u 2 - γ 2 u 2 x -p * 3 -γ 2 u 2 + γ 2 u 2 x = 3 2(α + 1) αu 2 -u 2 x -p * (αu 2 + u 2 x ) .
(3.17)

Here we set

(3.18) α = 3 -γ γ .
Let us introduce the two C 1 -functions of the time variable, depending on β, f (t) = (-u x + βu)(t, q(t, x 0 )), and g(t) = -(u x + βu)(t, q(t, x 0 )).

Computing the time derivative using the definition of the flow q, next using equations (3.16)-(3.17), we get

df dt (t) = (-u tx -γuu xx ) + β(u t + γuu x ) (t, q(t, x 0 )) = 3 2(α + 1) -αu 2 + u 2 x + (p -βp ′ ) * (αu 2 + u 2 x ) (t, q(t, x 0 )), (3.19)
and

(3.20) dg dt (t) = 3 2(α + 1) -αu 2 + u 2 x + (p + βp ′ ) * (αu 2 + u 2 x ) (t, q(t, x 0 )).
Let us first consider the case γ > 0. Then α > -1. From the definition of β γ (2.2) and the condition β γ < ∞, we deduce that there exist β ≥ 0 such that (3.21)

β 2 ≥ α -I(α, β).
Applying the convolution estimate (3.13) and recalling I(α, -β) = I(α, β), we get, for all β ≥ 0 satisfying (3.21),

df dt (t) ≥ 3 2(α + 1) u 2 x -α -I(α, -β) u 2 (t, q(t, x 0 )) ≥ 3 2(α + 1) u 2 x -β 2 u 2 (t, q(t, x 0 )) = 3 2(α + 1) f (t)g(t).
In the same way,

dg dt (t) ≥ 3 2(α + 1) u 2 x -α -I(α, β) u 2 (t, q(t, x 0 )) ≥ 3 2(α + 1) u 2 x -β 2 u 2 (t, q(t, x 0 )) = 3 2(α + 1) f (t)g(t).
The assumption u ′ 0 (x 0 ) < -β γ |u(x 0 )| guarantees that we may choose β satisfying (3.21), with β -β γ > 0 is small enough in a such way that u ′ 0 (x 0 ) < -β|u 0 (x 0 )| For such a choice of β we have f (0) > 0 and g(0) > 0.

The blowup of u will rely on the following basic property:

Lemma 3.7. Let 0 < T * ≤ ∞ and f, g ∈ C 1 ([0, T * ), R) be such that, for some constant c > 0 and all t ∈ [0, T * ), df dt (t) ≥ cf (t)g(t), dg dt (t) ≥ cf (t)g(t). (3.22)
If f (0) > 0 and g(0) > 0, then

T * ≤ 1 c f (0)g(0) < ∞. Proof. Let τ = inf t ∈ [0, T * ) : f (t) = 0 or g(t) = 0 .
The positivity of f (0) and g(0) implies that τ > 0. Observe that we cannot have τ < T * . Indeed, otherwise (exchanging if necessary f with g) we would get f (τ ) = 0 and f, g ≥ 0 on [0, τ ]. The differential inequality on f implies that f is monotone increasing on [0, τ ] leading to the contradiction 0 = f (τ ) ≥ f (0) > 0. Hence, f and g are both monotone strictly increasing and positive in the whole interval [0, T * ). Now take h(t) = f (t)g(t). Using again assumption (3.22), next the arithmetic-geometric mean inequality we find, on [0, T * ),

dh dt ≥ c 2 √ f g f g(f + g), ≥ cf g = ch 2 , with h(0) = f (0)g(0). Substituting v = 1/f , we find that T * ≤ 1 c √ h(0)
. Estimate (2.4) immediately follows in the case γ > 0. When γ < 0 the proof is the same, excepted for the fact that, as α < -1, the inequalities must be reversed.

Let us establish the blowup rate (2.5). More precisely, we prove that lim

t→(T * ) - (T * -t) u x t, q(t, x 0 ) = - 2 γ .
We will obtain such blowup rate adapting the arguments of [START_REF] Constantin | Existence of permanent and breaking waves for a shallow water equation: a geometric approach[END_REF]. Namely, by Lemma 3.7 and the fact that, for some constant c > 0, we have the estimate u(t) ∞ ≤ c u 0 H 1 , we see that m(t) ≡ -γu x (t, q(t, x 0 )) → +∞ as t → T * . But m satisfies the equation

dm dt (t) = 3 2(α + 1) -αu 2 + u 2 x -p * (αu 2 + u 2 x ) (t, q(t, x 0 )) = 3 m(t) 2 2(α + 1) + R(t),
where R(t) = 3 2(α+1) (-αu 2 -p * (αu 2 + u 2 x ))(t, q(t, x 0 )). Using again u(t) ∞ ≤ c u 0 H 1 and observing that p * (αu 2 + u 2

x ) ∞ ≤ c(α) u 0 2 H 1 by Young inequality, we get that |R(t)| is uniformly bounded on (0, T * ) by a constant depending only on α and u 0 H 1 .

Let ǫ > 0. Taking 0 < t 0 < T * such that t 0 is close enough to T * in a such way that -ǫ ≤ R(t)/m(t) 2 ≤ ǫ on (t 0 , T * ), we deduce that, on such interval,

3 2(α + 1) -ǫ ≤ d dt -1 m(t) ≤ 3 2(α + 1) + ǫ.
Now integrating these inequalities on (t, T * ) we get the blowup rate (2.5) with x(t) = q(t, x 0 ).

4. The minimization problem in the limit case β = e+1 e-1 and in the case β = 1 4.1. The limit case β = e+1 e-1 . We will start considering the limit case

β = e + 1 e -1 .
In this case, according to formula (3.1), the weight function becomes (4.1) ω(x) = 2e (e -1) 2 sinh(x),

x ∈ (0, 1).

Because of Proposition 3.4, we are led to assume also (4.2) α > -1/C( e+1 e-1 ).

We start observing that I(α, e+1 e-1 ) ≥ I(α, e+1 e-1 ), where

I(α, e+1 e-1 ) ≡ inf 1 0 p + βp ′ ) αu 2 + u 2
x dx : u ∈ E (e+1)/(e-1) and u(1

) = 1 = α + inf J(v) : v ∈ E (e+1)/(e-1),0
and the functional J is given by (3.11). Indeed, the inequality I(α, e+1 e-1 ) ≥ I(α, e+1 e-1 ) follows from the inclusion H 1 0 (0, 1) ⊂ E (e+1)/(e-1),0 . The unique minimizer v ∈ E (e+1)/(e-1),0 of the functional J (whose existence was obtained in the proof of Proposition 3.3) satisfies the Euler-Lagrange equation complemented with the right-boundary condition

(4.3) (ωv x ) x -αωv = αω in x ∈ (0, 1), v(1) = 0.
Problem (4.3) is undetermined, but we will see that among its solutions only one belongs to E (e+1)/(e-1),0 .

4.2.

The case β = e+1 e-1 and α > -1/C( e+1 e-1 ). The Euler-Lagrange equation (4.3) reads

(4.4) sinh(x)v xx (x) + cosh(x)v x (x) -α sinh v(x) = α sinh(x), for x ∈ (0, 1).
To find the general solution, consider the change of unknown v(x) = f (y(x)), with y = cosh(x). Then equation (4.4) can be rewritten in the y variable as (4.5) (1 -y 2 )f yy -2yf y (y) + αf (y) = -α, with y ∈ (1, cosh 1).

The constant function y → -1 is a particular solution. Substituting α = ν(ν + 1), we recognize the usual form of a non-homogeneous second order Legendre ODE. We thus set

(4.6) ν(α) = - 1 2 + 1 2 √ 1 + 4α ∈ {z ∈ C : Im(z) ≥ 0},
where the complex square root is taken in {z ∈ C : Im(z) ≥ 0}. The general solution of equation (4.5) is thus

(4.7) f λ,µ = -1 + λP ν(α) (y) + µQ ν(α) (y), y ∈ (1, cosh 1),
where P ν and Q ν are the two associate Legendre functions respectively of the first and of the second kind, of degree ν. We recall that Q ν has a logarithmic singularity at 1 + , whereas P ν is bounded as y → 1 + . Moreover, P ν is a polynomial when ν is an integer. Notice that the function

P ν(α) (cosh) does belong to E (e+1)/(e-1) , but Q ν(α) (cosh) does not, because Q ν(α) (cosh x) = O( | log(x)|) as x → 0 + .
Hence, the only solution v = v λ,µ of (4.4), such that v ∈ E (e+1)/(e-1),0 is obtained taking µ = 0 and λ = 1/P ν(α) [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]. Thus, (4.8) v(x) = -1 + P ν(α) (cosh x) P ν(α) (cosh 1) , x ∈ (0, 1).

Expression (4.8) makes sense provided the denominator is nonzero. Therefore, we can recast condition (4.2) on α, guaranteeing that I(α, e+1 e-1 ) > -∞, in the following equivalent form: α > α 0 ≃ -6.113, where α 0 is the largest zero of the function α → P ν(α) (cosh 1) (see Figure 4.2). The representation of P ν (z) as a hyper-geometric series, convergent for

| 1 2 -z 2 | < 1, is P ν (z) = 2 F 1 -ν, ν + 1, 1 2 - z 2 = +∞ k=0 Γ(-ν + k)Γ(ν + 1 + k) (k!) 2 Γ(-ν)Γ(ν + 1) 1 2 - z 2 k .
Such representation holds true for z = cosh 1, but computing α 0 analytically seems to be difficult. On the other hand, α 0 can be estimated via Newton's method. For α > α 0 = -1/C( e+1 e-1 ), using that v solves (4.3), vx (0 + ) = 0, and

1 0 ω = 1, we have I(α, e+1 e-1 ) ≥ α + J(v) = α + 1 0 ω(αv 2 + v2 x + 2αv) = α + 1 0 (ωv x ) x v + ωv 2 x + αωv = α + ωv x v 1 - 0 + + α 1 0 ωv = 1 0 (ωv x ) x = (ωv x )(1 -) = (e + 1) 2 2e P ′ ν(α)
P ν(α) (cosh 1).

(4.9)

Remark 4.1. Incidentally, we proved that c = -(1/α 0 ) ≃ 0.164 is the best constant in the weighted Poincaré inequality below (valid for all v ∈ C 1 ([0, 1]) such that v(1) = 0,

1 0 sinh(x)v(x) 2 dx ≤ c 1 0 sinh(x)v x (x) 2 dx. 4.3. The case β = 1.
The computation of I(α, β) in the case 0 ≤ β < e+1 e-1 is different than that of the previous section, as standard variational methods apply in the usual Sobolev space H 1 0 (0, 1). Moreover, in the case β = 1, the associated Euler-Lagrange boundary value problem can be explicitly solved. According to Proposition 3.3, the computation below will be valid for α > -1/C(1). As a byproduct of our calculations, we will find the explicit expression

C(1) = 4/(1 + 4π 2 )
for the weighted Poincaré inequality.

Recall that I(α, β) = α + inf{J(v) : v ∈ H 1 0 (0, 1)}, where J(v) was introduced in (3.11) and the weight function is positive and bounded away from 0 on the interval (0, 1). The unique minimizer v ∈ H 1 0 is the solution of (4.11)

ωv xx + ω x v x -αωv = αω x ∈ (0, 1) v(0) = v(1) = 0.
For β = 1, the weight ω reduces to ω(x) = e x /(e -1). Thus, the general solution of equation (4.11) for such choice of ω (at least for α = -1/4) is

(4.12) v(x) = -1 + λe ν(α)x + µe ν(α)x , λ, µ ∈ R,
where the complex number ν(α) is understood as in (4.6). To simplify further the result let us set

(4.13) µ = 1 2 √ 1 + 4α ∈ {z ∈ C : Im(z) ≥ 0}.
Imposing the boundary conditions v(0) = v(1) = 0 we finally get the expression of minimizer of I(α, 1):

v(x) = -1 + √ e sinh(µ(α)x) + sinh(µ(α)(1 -x))
e x/2 sinh(µ(α)) .

The minimum I(α, 1) is thus given by

I(α, 1) = α + J(v) = ω(1)v x (1) -ω(0)v x (0) = - 1 2 + µ • cosh 1 2 cosh µ -1 sinh 1 2 sinh µ . (4.14)
The above expression makes sense provided sinh(µ(α)) = 0, i.e. for α = -1 4 -k 2 π 2 , with k ∈ Z. But α = -1/4 is in fact a removable singularity in (4.14). The restriction to be imposed on α is thus α > - 1 4 -π 2 .

Proof of Theorem 2.2

We are now in the position of proving Theorem 2.2.

Proof of Theorem 2.2. Let us recall the definition of β γ , (5.1)

β γ = inf β ∈ R + : β 2 + I(α, β) -α ≥ 0 ,
where the one-to-one relation between α and γ is

(5.2) α = 3 -γ γ , or γ = 3 1 + α .
Using the results of the previous section we can now give explicit bounds from below for I(α, β) that can be used for the estimate of β γ .

Using the concavity properties of the function β → I(α, β) we see that, for any fixed α > α 0 , the infimum I(α, β) is bounded from below by piecewise affine function of the β variable. Namely,

(5.3)    I(α, β) ≥ I(α, 1), if 0 ≤ β ≤ 1 I(α, β) ≥ e-1 2 I(α, e+1 e-1 ) -I(α, 1) β + e+1 2 I(α, 1) -e-1 2 I(α, e+1 e-1 ), if 1 ≤ β ≤ e+1 e-1 .
We denote by R(α, β), the function defined by the right-hand in (5.3), for α > α 0 and 0 ≤ β ≤ e+1 e-1 . The condition α > α 0 ensures that I(α, 1) ≥ I(α, e+1 e-1 ) > -∞, so that R(α, β) is finite. The first issue is to find the condition on α guaranteeing β γ < +∞. Owing to the lower bound (5.3), a sufficient condition for this is that α is chosen in a such way that ∃ β such that 0 ≤ β ≤ e+1 e-1 : β 2 + R(α, β) -α ≥ 0 (5.4) But condition (5.4) is equivalent to the following one:

(5.5)

1 + I(α, 1) -α ≥ 0 or e + 1 e -1 2 + I α, e+1 e-1 -α ≥ 0. Indeed, one implication is obvious and the converse one easily follows applying to the β-variable the elementary properties of quadratic polynomials. Let us make more explicit the last condition: because of the definition of I(α, β) and Proposition (3.4), the two functions α → I(α, e+1 e-1 ) and α → I(α, 1) are both increasing, concave and vanishing at α = 0. Therefore, there exist

α - 1 < 0 < α + 1 such that (5.6) e + 1 e -1 2 + I α, e+1 e-1 -α ≥ 0 ⇐⇒ α - 1 ≤ α ≤ α + 1 .
For the same reason, there exist α -

2 < 0 < α + 2 such that (5.7) 1 + I(α, 1) -α ≥ 0 ⇐⇒ α - 2 ≤ α ≤ α + 2 .
The above zeros can be easily estimated via Newton's method. We find in this way

α - 1 < α - 2 < 0 < α + 2 < α + 1
, see also Figures 5-5.

According to (5.2), let us introduce the four constants In particular, for α - 1 ≤ α ≤ α + 1 , we find that the set in (5.1) is nonempty: this means that in the range γ ∈ (-∞, γ - 1 ] ∪ [γ + 1 , +∞) we get β γ < +∞, and Theorem 2.1 and Corollary 1.2 apply. On the other hand, under the more restrictive condition α - 2 ≤ α ≤ α + 2 , we can make use of inequality (5.7) to get the estimate (5.9)

(5.8) γ - 1 = 3 1+α - 1 = -1.036 . . . γ + 1 = 3 1+α + 1 = 0.269 . . . γ - 2 = 3 1+α - 2 = -1.508 . . . γ + 2 = 3 1+α + 2 = 0.575 . . .
β γ ≤ α -I(α, 1) ≤ 1.
Now recalling the expression of I(α, 1) computed in (4.14) we obtain the explicit estimate, valid

for γ ∈ (-∞, γ - 2 ] ∪ [γ + 2 , +∞):
(5.10)

β γ ≤ 3 γ - 1 2 -µ • cosh 1 2 cosh µ -1 sinh 1 2 sinh µ ,
where µ is the complex number

µ = 1 2 1 + 4(3 -γ)/γ.
The choice between the two complex square roots of 1 + 4(3 -γ)/γ does not affect the result and the radical in (5.10) (or in (5.13) below) is nonnegative because of the last inequality in (3.12).

Observe that taking here γ = 1 gives the blowup criterion (1.5) for the Camassa-Holm equation.

If otherwise α ∈ [α - 1 , α + 1 ]\[α - 2 , α + 2 ]
then, from the inequality (5.3) I(α, β) ≥ R(α, β), we obtain the bound β γ ≤ βγ , where βγ is the only zero of the quadratic polynomial

β → β 2 + R(α, β) -α inside the interval [1, e+1 e-1 ]. Thus, letting b ≡ e -1 2 I(α, e+1 e-1 ) -I(α, 1) = - e -1 2 (e + 1) 2 2e P ′ -1/2+µ P -1/2+µ (cosh 1) + 1 2 -µ • cosh 1 2 cosh µ -1 sinh 1 2 sinh µ . (5.11)
and c ≡ e+1 2 I(α, 1) -e-1 2 I(α, e+1 e-1 ) -α

= e + 1 2 - 1 2 + µ • cosh 1 2 cosh µ -1 sinh 1 2 sinh µ - e -1 2 (e + 1) 2 2e • P ′ -1/2+µ P -1/2+µ
(cosh 1) , (5.12) we get the estimate, valid for γ

∈ [γ - 2 , γ - 1 ] ∪ γ ∈ [γ + 1 , γ + 2 ]:
(5.13)

β γ ≤ - b 2 + 1 2 b 2 -4c.
By our construction, for γ = γ - 2 or γ = γ + 2 , the equality holds in (5.10) and in (5.13) and we have in this case β γ -

2 = β γ + 2 = 1.
The limit β ∞ = lim γ→∞ β γ is obtained taking α = -1 in (5.1). Going back to the estimate (5.10), letting γ → ∞ we get

(5.14) β ∞ ≤ √ 3 1 -cosh 1 2 cos √ 3 2 2 sinh 1 2 sin √ 3 2 - 1 2 .
The three claims of Theorem 2.2 are now established.

The case of the Camassa-Holm equation

In the case of the Camassa-Holm equation (α = 2), it is remarkable that the Euler-Lagrange equation (6.1) (ω x v x ) x -2ωv = 2ω, x ∈ (0, 1), associated with the minimization of I(2, β) can be explicitly solved for any β. Indeed, observing that from (4.10) we have ω = ω xx , on (0, 1), the associate homogeneous equation (ωv x ) x -2ωv = 0 clearly possess v(x) = ω x (x) as a solution. We can compute a second independent solution of this homogeneous equation of the form v(x) = ω x f (x). Then f must satisfy

f xx + ( 2ω ωx + ωx ω )f x = 0, provided ωω x = 0. Thus, f x (x) = 1 ωω 2
x in intervals where ω x = 0. Observe that if β > e -1 e + 1 , then ω x > 0 on the interval (0, 1) if 0 < β < e -1 e + 1 , then ω x = 0 for x ∈ (0, 1) and

x = x β ≡ 1 2 log( (1-β)e 1+β ).
On the other hand, using again ω = ω xx and integration by parts in the indefinite integrals below we see that, for x = x β and an arbitrary constant

C ∈ R ω x (x) 1 ωω 2 x = -ω x (x) 1 ω 2 • 1 ω x x = - 1 ω(x) 2 + Cω x (x) -2ω x (x) 1 ω 3
The expression on the right-hand side is well defined on the whole interval (0, 1), for all 0 ≤ β < e-1 e+1 . Therefore, the general solution of (6.1) is

(6.2) v(x) = -1 + ω x (x) λ + 2µ 1 ω 3 + µ ω(x) 2 ,
x ∈ (0, 1).

We compute The last integral can be easily computed distinguishing the cases B > 0, B = 0 and B < 0. For example, in the case B > 0, that corresponds to 0 < β < 1, we have

1 ω(x) 3 dx = (e -1) 3 (1 + β) 3 y(y 2 -B) B(y 2 + B) 2 + B -3/2 arctan(t/ √ B) = 2 sinh(1/2) 2 1 -β 2 • ω x (x) ω(x) 2 + 8 sinh(1/2) 3 (1 -β 2 ) 3/2 arctan 1+β e(1-β) e x .
The minimizer v of I(α, β), with 0 ≤ β < e+1 e-1 is obtained choosing in the above expression the coefficients λ and μ solving the linear system

     ω x (0) λ + 1 ω(0) μ = 1 ω x (1) λ + 2ω x (1) 1 ω(1) 2 + 1 0 1 ω 3 μ = 1.
Once v is computed as indicated, the minimum is given, for 0 ≤ β < e+1 e-1 , by

I(2, β) = 2 + J(v) = ω(1)v x (1) -ω(0)v x (0) = ω(1) 2 λ + 2μ 1 0 1 ω 3 -λω(0). (6.3)
The explicit expression of I(2, β) is thus easily written in terms of elementary functions. We do not reproduce the complete formula here as it is too long to be really useful. We just provide here a few particular values and its plot in the interval [0, e+1 e-1 ] (see Figure 7).

I(2, 0) = 1 + arctan(sinh 1 2 ) sinh 1 2 + arctan(sinh 1 2 )(sinh 1 2 ) 2 = 1.737 . . . The value of I(2, 0) was computed also in [START_REF] Wahlén | On the blow-up of solutions to the periodic Camassa-Holm equation[END_REF] or [START_REF]Blow-up of solutions to a nonlinear dispersive rod equation[END_REF].

I(2, 1) = - 1 2 + 3 2 • cosh 1 2 cosh 3 2 -1 sinh
Remark 6.1. The last formula in (6.4), in fact, is obtained in a slight different way, as in our previous computations we excluded the limit case β = e+1 e-1 . Of course, this formula agrees with that obtained from the more general one (4.9) for α = 2, as ν(2) = 1 and P 1 (y) = y. But we can easily prove it without relying on the more complicated approach used to get (4.9). Indeed, when β = e+1 e-1 , we can write the general solution of (6.1) as (6.5) v λ,µ (x) = -1 + λ cosh(x) + µ 1 + 1 2 cosh(x) log cosh(x) -1 cosh(x) + 1 , x ∈ (0, 1). The appropriate boundary condition for β = e+1 e-1 is v λ,µ ∈ E (e+1)/(e-1),0 . The only possibility to achieve this is to choose µ = 0 (recall that the elements of E (e+1)/(e-1),0 are O( | log(x)|) as x → 0+). The remaining condition v(1) = 0 requires λ = 1/ cosh [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]. The minimizer of the functional J in the case α = 2 and β = e+1 e-1 , is then (6.6) v(x) = -1 + cosh(x) cosh [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF] . This indeed leads to the last formula in (6.4).

In view of the application of Theorem 2.1. it is interesting to make the numerical study of the zeros of the function β → β 2 + I(2, β) -2, on the interval [0, e+1 e-1 ]. The only zero is β 1 = 0.513 . . . According to Theorem 2.1, solutions of Camassa-Holm blowup provided u ′ 0 (x 0 ) < -β 1 |u 0 (x 0 )|. On the other hand, Theorem 2.2 predicted β 1 ≤ 0.515 . . . (see criterion (1.5)). This is another confirmation that the estimate provided by Theorem 2.2 is almost sharp.

To go one step further, one might ask what is the best constant β * 1 with the property that if

inf x∈S u ′ 0 (x) + β * 1 |u 0 (x)| < 0,
then the solution of the Camassa-Holm equation arising from u 0 blows up in finite time. We do not know how to exactly compute β * 1 in the periodic case. However, the following estimates hold: 0.462 • • • = e-1 e+1 ≤ β * 1 ≤ β 1 = 0.513 . . . To establish the lower bound, we can make use of a property specific of the case γ = 1. Namely, the fact that if the initial potential y 0 = u 0 -u 0,xx has a constant sign then the corresponding solution exist globally. See [START_REF] Constantin | Existence of permanent and breaking waves for a shallow water equation: a geometric approach[END_REF][START_REF] Danchin | A few remarks on the Camassa-Holm[END_REF]. Let us take a sequence y n 0 of smooth, periodic, non-negative functions converging in the distributional sense to a Dirac comb. Then the corresponding initial data u n 0 = p * y n 0 give rise to global smooth periodic solutions. But

u n 0,x (0) u n 0 (0) → -sinh(1/2) cosh(1/2) = -e-1 e+1
. Therefore, a condition of the form u ′ 0 (x 0 ) < -c|u 0 (x 0 )|, in general, does not imply the blowup, unless c ≥ e-1 e+1 .

7. Proof of Corollary 1.2 and the case of non-periodic solutions.

Proof of Corollary 1.2. We assume that γ ≥ γ + 1 , so in particular that γ > 0. In the case γ ≤ γ - 1 the proof is similar. From the assumption of the corollary, T * = +∞. Then Theorems 2.1-2.2 apply, implying that for all t ≥ 0, and all x ∈ R we have u We then conclude that e -βγ a u(t, a) ≤ e -βγ b u(t, b).

x (t, x) ≥ -β γ |u(t, x)|. If [
Summarizing, we proved that x → e βγ x u(x, t) is monotone increasing in any interval where u(•, t) ≥ 0 and x → e -βγ x u(x, t) is monotone increasing in any interval where u(•, t) ≤ 0. The condition u(x 0 , t 0 ) = 0 together with the periodicity of u imply that u(•, t 0 ) ≡ 0 and so u ≡ 0 because of the conservation of the H 1 -norm.

As a consequence of this Corollary, we can deduce that if u 0 is not identically zero, but S u 0 = 0, then the corresponding solution must blow up in finite time. Indeed, the zero-mean condition implies of course that u 0 must vanish in some point x 0 ∈ S. We recover in this way a known conclusion for the Camassa-Holm equation (see [START_REF]Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation[END_REF]), extended for the rod equation (at least for γ outside a neighborhood of the origin) in [START_REF] Hu | Blowup phenomena for a new periodic nonlinearly dispersive wave equation[END_REF].

A digression on non-periodic solutions. The above proof applies also global solutions u

∈ C([0, ∞), H s (R)) ∩ C([0, ∞), H s-1 (R)) of the rod equation (1.1). Notice that in this case the kernel p of (1 -∂ 2 x ) -1 is given by p(x) = 1 2 e -|x| , x ∈ R.
Indeed, let us recall that by the result in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF], if γ ∈ [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF][START_REF] Brandolese | Blowup issues for a class of nonlinear dispersive wave equations[END_REF] and u 0 ∈ H s (R), with s > 3/2, is such that

(7.1) ∃ x 0 ∈ R such that u ′ 0 (x 0 ) < -- 1 2 + 3 γ - √ 12 -3γ 2 √ γ |u(x 0 )| then the unique local-in-time solution u ∈ C([0, T ], H s (R)) ∩ C 1 ([0, T ], H s-1 (R)
) must blow up in finite time. In the Camassa-Holm case γ = 1 the above coefficient equals -1 and is known to be optimal, see [START_REF] Danchin | A few remarks on the Camassa-Holm[END_REF]. Reproducing the proof of Corollary 1.2 in the case of non-periodic solution, taking (7.2)

β γ = - 1 2 + 3 γ - √ 12 -3γ 2 √ γ
we get the following result.

Corollary 7.1. Let γ ∈ [1, 4] and s > 3/2. Let u ∈ C([0, +∞), H s (R)) ∩ C 1 ([0, +∞), H s-1 (R)),
be a global solution of the rod equation (1.1) (with p(x) = 1 2 e -|x| and x ∈ R). i) For all t ≥ 0, either u(t, x) > 0 for all x ∈ R, or u(t, x) < 0 for all x ∈ R, or ∃ x t ∈ R such that u(t, •) ≤ 0 in (-∞, x t ] and u(t, x) ≥ 0 in [x t , +∞). In the latter case, if x → u(t, x) vanishes at two distinct points of the real line, then x → u(t, x) must vanish in the whole interval between them. ii) If, at some time t 0 ≥ 0, lim inf x→+∞ e βγ x u(t 0 , x) ≤ 0 and lim sup x→-∞ e -βγ x u 0 (t 0 , x) ≥ 0, where β γ is given by equation (7.2), then u is the identically zero solution.

In particular, if 0 ≡ u 0 ∈ H s (R) is such that u 0 (x) = o e -βγ |x| for |x| → ∞, then the corresponding solution of the rod equation must blow up in finite time.

Indeed, all these claims follow from the fact that if u is a global solution than x → e βγ x u(x, t) is monotone increasing in any interval where u(•, t) ≥ 0 and x → e -βγ x u(x, t) is monotone increasing in any interval where u(•, t) ≤ 0, as seen before.

Remark 7.2. In the last conclusion of the corollary we recover the known fact that solutions of the Camassa-Holm equation arising from compactly supported data (see [START_REF] Himonas | Persistence properties and unique continuation of solutions of the Camassa-Holm equation[END_REF]), or more in general (see [START_REF] Brandolese | Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces[END_REF]) from data decaying faster than peakons -solitons with profile ce -|x| -feature a wave breaking phenomenon after some time. The proofs given in [START_REF] Brandolese | Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces[END_REF][START_REF] Himonas | Persistence properties and unique continuation of solutions of the Camassa-Holm equation[END_REF] relied on McKean's necessary and sufficient condition [START_REF]Breakdown of the Camassa-Holm equation[END_REF] for wave breaking and were not suitable for the generalization to γ = 1. Moreover, the sign condition on global solutions u contained in the first item of our corollary is in the same spirit as the sign condition on the associated potential u -u xx provided by McKean's theorem.

The method that we used in our previous paper [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF] in the non-periodic case looks simpler than that of the present paper. Indeed, in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF] we relied on elementary estimates for bounding from below the convolution term p * (αu 2 + u 2

x ) without making use of variational methods. On the other hand, the result obtained therein is much weaker as the range of applicability for the parameter γ is considerably narrower. We point out that applying the variational method of the present paper in the non-periodic case we can recover, but not improve, the results in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF]. Indeed, the main issue is the study of the minimization problem (the analogue of (2.1) but now with p(x) = 1 2 e -|x| ):

(7. Moreover, the function β → I R (α, β) is constant on the interval [-1, 1] and under the above restrictions on α and β we have

(7.5) I R (α, β) = - 1 2 + 1 2 √ 1 + 4α.
The proof of this proposition, that we only sketch, relies on the identity p(x) + βp ′ (x) = For α > 0, one easily find solving the associate Euler-Lagrange equation the minimizer ū ∈ H 1 (R + ) given by ū(x) = exp x( 1 2 -1 2 √ 1 + 4α) and the minimum by formula (7.5). For -1 4 ≤ α < 0, ū does not belong to H 1 (R + ). However, the inequality I R (α, β) ≥ -1 2 +

1 2 √ 1 + 4α does hold, as proved in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF]. To see that equality (7.5) remains true in this case, we can construct a minimizing sequence taking u ℓ = ū on [0, ℓ] and u ℓ (x) = C ℓ exp(-√ -αx) for x ≥ ℓ, with C ℓ chosen in a such way that u ℓ is continuous. In this way, +∞ ℓ e -x αu 2 + u 2

x dx = 0 and letting ℓ → ∞ we find ℓ 0 e -x αu 2 + u 2

x dx → -1 2 + 1 2 √ 1 + 4α. For α < -1/4, we choose b > 1/2 such that α + b 2 < 0. Considering now continuous functions on R + of the form u ℓ (x) = e bx on [0, ℓ] and C ℓ exp(-√ -αx) for x ≥ ℓ and letting ℓ → ∞ we easily see that I(α, β) = -∞ in this case.

Notice that the condition α ≥ -1/4 gives the restriction γ ≤ 4. On the other hand, the formula β γ = inf{β ≥ 0 : β 2 + I R (α, β) -α ≥ 0} immediately gives β γ = +∞ if γ < 1. We recover in this way that we must restrict ourselves to γ ∈ [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF][START_REF] Brandolese | Blowup issues for a class of nonlinear dispersive wave equations[END_REF] and that β γ is given by formula (7.2) in the non-periodic case.

Appendix: numerical analysis

In this appendix we briefly revisit Theorem 2.1 using a numerical approach. Rather than obtaining estimates for β γ by making use of some exact formulas, as we did in Theorem 2.2, we can evaluate β γ numerically. In order to achieve this, we need first to approximate I(α, β). This can be done by approaching the solutions of the boundary value problem (4.11), for any fixed α and β, with 0 ≤ β < e+1 e-1 . This can be done with arbitrary high precision applying e.g. a finite difference technique with Richardson extrapolation. In order to get an error smaller than 0.001 we needed a large number of grid points (a few thousands), especially when β is getting close to the critical value e+1 e-1 . The following table provides the values of β γ computed numerically, corresponding to the values of γ listed in [START_REF] Dai | Solitary shock waves and other travelling waves in a general compressible hyperelastic rod[END_REF] and associated with known hyper-elastic materials. Only in one case (γ = -0.539) our theorems are not applicable. 2), we see that Theorem 2.1 can be applied for γ ∈ (-∞, -0.817 . . .] ∪ [0.262 . . . , +∞), which is a range slightly larger than the range γ ∈ (-∞, γ - 1 ]∪[γ + 1 , ∞), predicted by Theorem 2.2. Relying in this numerical analysis, we claim that Corollary 1.2 remains true in such a larger range.

Figure 9 provides the plot of the the function γ → β γ as computed numerically. The curve is very close to that in Figure 1 (reproduced here through a dotted line) and the two curves almost overlap, in particular for γ > γ + 1 or γ < γ + 2 (only when γ ∈ [γ - 2 , γ - 1 ] the difference between the two curves becomes visible).

  1 and αωv n → -∞ to get a contradiction. Conversely, assume that |β| ≤ e+1 e-1 . By the weighted Poincaré inequality (3.7), the map v → 1 0 ωv 2

Figure 2 .

 2 Figure 2. The set of points (α, β) such that I(α, β) > -∞ (gray region). The infimum is achieved in H 1 0 on its interior and in the larger space E β,0 on the upper and lower boundaries β = ± e+1 e-1 . The abscissa α 0 = -1/C(∓ e+1 e-1 ) of the vertical dashed segment is the largest zero of a suitable Legendre function (see Subsection 4.2).

( 4 Figure 3 .

 43 Figure 3. The function α → P ν(α) (cosh 1) and its largest zero α 0 .

Figure 4 .

 4 Figure 4. A few minimizers as given by formula (4.9), for α = 2, 1, -1, -2 (from bottom to top).

Figure 5 .

 5 Figure 5. The plot of α → I(α, e+1 e-1 ) (see equation (4.9)) and of the straight line α → α-e+1 e-1 2 , intersecting the curve at α - 1 and α + 1 .

Figure 6 .

 6 Figure 6. Plot of α → I(α, 1) (see equation (4.14)) and of the straight line α → α -1, intersecting the curve at α - 2 and α + 2 .

1 ω 3 3 (

 133 making the change of variables y = e x : 1 ω(x) 3 dx = (e -1)

1 2 sinh 3 2 = 1

 21

Figure 7 .

 7 Figure 7. The plot of I(2, β) and the construction of β 1 = 0.513 . . .

  a, b] is an interval where u(t, x) ≥ 0 for all x ∈ [a, b], then we have b a β γ e βγ x u(t, x) dx ≤ b a e βγ x u x (t, x) dx = e βγ b u(t, b) -e βγ a u(t, a) -β γ b a e βγ x u(t, x) dx. This implies that e βγ a u(t, a) ≤ e βγ b u(t, b). If u(t, •) ≤ 0 on [a, b], then applying again Theorem 2.1 we have u x (t, x) ≥ β γ u(t, x). This in turn implies b a β γ e -βγ x u(t, x) dx ≤ b a e -βγ x u x (t, x) dx = e -βγ b u(t, b) -e -βγ a u(t, a) + β γ b a e -βγ x u(t, x) dx.

  3) I R (α, β) = inf R p + βp x ) αu 2 + u 2 x dx : u ∈ H 1 (R), u(0) = 1 .The analogue of Proposition (3.3) and Proposition 3.4 in our present setting is provided by the following one:Proposition 7.3. We have(7.4) I R (α, β) > -∞ ⇐⇒ α ≥ -1 4and -1 ≤ β ≤ 1.

1+β 2 e x 1 R

 1 -+ 1-β 2 e -x 1 R + , implying that I R (α, β) = inf ∞ 0 e -x αu 2 + u 2 x dx : u ∈ H 1 (R + ), u(0) = 1 .

γ - 29 .Figure 8 .

 298 Figure 8. Set of points (α, β) such that β ≥ 0 and β 2 + I(α, β) -α ≥ 0 (dark gray region). Right: two zooms on the upper-left and upper-right corners.

Figure 9 .

 9 Figure 9. The plot of β γ in the interval γ ∈ (-∞, -0.817] ∪ [0.262, +∞) (continuous line) and the upper estimate of β γ provided by Theorem 2.2 (dotted line).

Acknowledgements

The authors are grateful to the referee for his useful suggestions and for pointing out a few relevant geometrical interpretations of the model studied in the present paper.

The authors are supported by the French ANR Project DYFICOLTI. The second author is also supported by the Secretaría Nacional de Educación Superior, Ciencia, Tecnología e Innovación.