F Lacour

M Courjal

M.-P Bernal

A Sabac

C Bainier

M Spajer

Nanostructuring Lithium Niobate substrates by focused ion beam milling

Keywords: numbers: 85.40.Ux, 42.70.Qs, 77.84.Dy Nano-structuring, Lithium Niobate, Focused Ion Beam, Reactive Ion Etching

We report on two novel ways for patterning Lithium Niobate (LN) at submicronic scale by means of focused ion beam (FIB) bombardment. The first method consists of direct FIB milling on LiN bO3 and the second one is a combination of FIB milling on a deposited metallic layer and subsequent RIE (Reactive Ion Etching) etching. FIB images show in both cases homogeneous structures with well reproduced periodicity. These methods open the way to the fabrication of photonic crystals on LiN bO3 substrates.

I. INTRODUCTION

The recent development of integrated photonic crystals within planar waveguides can help implementing compact devices with fully integrable functions [1,[START_REF] Zimmermann | Photonic crystal waveguide directional couplers as wavelength selective optical filters[END_REF][START_REF] Wu | Square lattice photonic-crystal collimator[END_REF]. In these devices the light is confined into the crystal by a classical waveguide construction. Lithium Niobate (LN) is a suitable material for 2D photonic crystals because of its high refractive index. Moreover, its high electro-optic coefficient and its low optical losses make this material very adequate for optical communication systems. Therefore, the perspective of fabricating miniature electro-optical and all-optical LN devices is attracting the research on LiN bO 3 nanostructuring [START_REF] Restoin | Fabrication of 1D and 2D structures at submicrometer scale on lithium niobate by electron beam bombardment[END_REF][START_REF] Foglietti | Fabrication of Band gap structures in planar non linear waveguides for second harmonic generation[END_REF]. However, the obtention of good nanometric optical structures in LiN bO 3 continues to be a difficult task due to its wellknown resistivity towards standard machining techniques like wet etching [START_REF]EMIS Data Reviews Series No. 5, Properties of Lithium Niobate[END_REF].

In this paper, we report on two alternative methods based on focused ion beam (FIB) bombardment to produce photonic band gap structures on LiN bO 3 substrates with a spatial resolution of 70nm. The high resolution and the ability to drill holes directly from the sample surface make FIB milling one of the best candidates for designing good optical quality patterns at submicrometer scale [START_REF] Bostan | Line-defect Waveguides in Hexagon-Hole type Photonic Crystal Slabs: Design and Fabrication using Focused Ion Beam Technology[END_REF]. The only constraint is that the sample surface must be metallised and grounded to avoid charge accumulation. Firstly, we describe the method for directly etching the LN substrate by FIB milling through the metal. This method has been already employed to etch sub-micrometric one-dimensional structures in LiN bO 3 [START_REF] Yin | Lithium niobate fibers and waveguides: fabrication and application[END_REF]. The second related method is based on RIE etching after FIB milling of the metal layer which behaves as a mask. The advantage of this alternative solution is a lower exposure time. Another expected advantage whould be a good replication of the mask shape in the whole hole depth. In both cases, the fabricated submicronic patterns are characterized by FIB imaging.

Before describing the two nanostructuring methods, we present the calculated conditions to obtain a photonic band gap for LiN bO 3 substrates.

II. NUMERICAL STUDY

Numerical calculations are carried out using the commercial software BandSolve. For a wavelength λ = 1.55µm, the ordinary and extraordinary indexes of the Z-cut LiN bO 3 substrate are assumed to be n o = 2.2151 and n e = 2.1410, respectively. According to our calculations, a 2D triangular lattice of holes can yield to a total TM photonic bandgap if the diameter D of the holes is larger than 0.4p, where p is the triangular lattice period. The ratio D/p should be as large as possible in order to benefit from a large photonic band gap, but it cannot exceed 0.5 for technological reasons. Indeed, the walls between holes could collapse for diameters larger than 0.5p. Fig. 1 shows the band diagram obtained in the case D/p = 0.5. As it can be seen from the figure, the structure yields to a total TM-photonic band gap for frequencies between 0.321p/(2πc) and 0.349p/(2πc). The difference from the calculation of S. Massy et al. [START_REF] Massy | Structures priodiques bidimensionnelles sur LiN bO3 et HxLi1-xN bO3 ralises par bombardement par faisceau d'lectrons[END_REF] is probably the difference between the refraction index that are chosen in both cases. Having shown the existance of a photonic band gap in lithium niobate, we have investigated the means of nanostructuring it.

III. EXPERIMENTAL

The two fabrication processes are schematically shown in Fig. 2. The first method -Fig. 2(a)-is based on a direct etching of the LiN bO 3 substrate by FIB milling. The second one -Fig. 2(b)-uses the FIB to create the metallic mask and the pattern is then transfered to the LiN bO 3 substrate by RIE. In both cases the sample area is 1cm 2 and the thickness 500µm. A Cr layer is deposited by electron gun evaporation (Balzers B510) and grounded with a conductive paste before introduction into the FIB vacuum chamber (= 2.10 -6 torr). In the case of direct FIB writing the thin Cr metal layer (150nm) does not modify significantly the etching efficiency. In the second case a thicker Cr layer (250nm) is deposited.

The metal-coated substrates are milled using a focused ion beam column Orsay Physics LEO FIB4400 for the case of FIB milling only -Fig. 2(a)-and a FEI Dual Beam Strata 235 for the milling of the metallic mask -Fig. 2(b). This method could be direcly compared with e-beam lithography. The advantage of FIB patterning of the metallic mask is its ability to selectively remove and deposit material without the use of the additional process step of developing a resist layer.

In the first case (Fig. 2 a probe current of 66pA. The pseudo-Gaussian-shaped spot size is estimated to be 70nm on the target. The focused ion beam is scanned on the sample by a computercontrolled deflection field to produce the desired pattern (Elphy Quantum from Raith). A FIB-image cross-section of the cavities is shown in Fig. 3. In order to see the etching depth the sample is tilted 30 o with respect to the FIB axis. As it can be seen from the image, the 4 × 4 array exhibits well defined circular holes. The achieved etching depth is approximately 2µm and the etching time was 12 minutes. At 1µm deep the hole diameter is about 432nm. This conical etching shape is due to material redeposition on the sidewalls while milling. In order to reduce the redeposition there are two possible solutions. If the FIB electronics is fast enough (Elphy Quantum is limited to 300kHz) and the spot size small enough one can scan along the hole sidewalls longer and less on the bottom of the pit. A second possibility that we plan to implement in our system is to use gas assisted milling. In particular, XeF 2 could help to remove N b from the etched substrate decreasing then the redeposition. The etching depth and the real hole diameter can differ from the designed ones and this difference depends on the Ga + dose. This can be clearly seen in Fig. 4 (a) and(b) where this dependence has been measured. It is clear from the graphs that as the milling time is increased, the etching depth increases but with a subsequent increase in the hole diameter due to beam aberrations. In particular, an increase of almost 30% in the hole diameter and of 40% in the etching depth has been measured as the Ga + dose is increased by factor of 4 from the initial value (4.10 4 µAs/cm 2 ). Therefore, for practical uses one should start with a designed hole diameter slightly smaller than the desired one.

The second related process requires lower etching time since the desired photonic structure is fabricated at once. In this case, the FIB bombardment is used to pattern a SiO 2 -Cr mask previously deposited on the LN substrate, as depicted in Fig. 2(b). The first step consists in depositing a 100nm thick layer of SiO 2 by Plasma Enhanced Chemical Vapor Deposition (PECVD). A 250nm thick chrome layer is then deposited on the substrate by sputtering. The metal is used as a mask for the RIE, while the silica layer prevents the diffusion of chrome into the substrate during the RIE plasma processing and the increase of the optical losses. This layer is not needed in the case of direct FIB milling since the etching is done locally and the damaged area is defined by the FIB beam size. The samples are annealed at 250 o C during 2 hours to release stress. The SiO 2 -Cr mask is then nanostructured by FIB patterning, with a current of the sample of 100pA. An exposure time of 3.75s is typically required to etch a 250nm diameter circular hole, which is 11 times lower than the one required in the first process. The pattern (an array of 24 × 20 cylindrical holes) is finally transfered to the substrate by RIE. The relevant parameters of this process are detailed in Table I. It can be noticed that this process requires a very low pressure and a high RF power. In these conditions, the etch rate of the mask is comparable to the etching rate of the substrate. In order to improve the selectivity of etching between the mask and the LiN bO 3 substrate, we start the process with an exposition of the target to a O 2 ionic plasma (pressure =100µBar , power = 60W ). The 250nm thick layer of chrome is then more resistant to the SF 6 RIE. The selectivity of the mask is thus estimated to be 1 : 5 compared to the LiN bO 3 substrate (while the etching selectivity was measured to be of 1 : 2 without the O 2 ionic plasma). The etching rate of the Z-cut substrate is measured to be 50nm/min. This process is applied to fabricate a triangular lattice of holes with D = 250nm and D = 130nm diameters and p = 2D periodicity. Fig. 5 (a) and (b) exhibit the SEM images of the holes after FIB milling and 10min of RIE etching. Fig. 5(a) shows holes with good reproducibility. The etching depth is measured to be 500nm. Fig. 5(b) shows that the 130nm diameter holes were transformed into 130nm diameter rods after RIE etching, while the 250nm diameter holes were well preserved. This is due to a higher etching rate along the sides of the triangular lattice than in the triangle center when the holes are very close to each other. We can infer from these results that the fabrication of small holes (D ≤ 200nm) requires lower RF-power to preserve the initial feature.

IV. CONCLUSION

In this work we have presented two alternative techniques to fabricate submicrometric patterns in LiN bO 3 by means of a FIB milling or a combination of FIB milling and RIE etching. In particular, 500nm-diameter circular holes were etched by FIB milling obtaining an etching depth of 2µm. Material redeposition starts to be a problem for etching depths larger than 1µm A 24 × 20 array of 250nm-diameter circular holes and 500nm periodicity was realized using an alternative method in which the metallic mask is fabricated by FIB milling and the LiN bO 3 etching is obtained by SF 6 RIE. In this case, the etching depth in LiN bO 3 is around 500nm and is limited by the metallic thickness of the mask. Work is in progress to optimize the RIE etching in view of obtaining a higher etching depth. An optical characterization of the patterns is also in progress.

As opposed to the process based on ferro-electric domain inversion, the presented methods are more suitable for implementing nanostructures on both X-cut and Zcut substrates.

We are currently working on the theoretical analysis and the fabrication of active LiN bO 3 2D photonic crystals. In order to operate at 1.55µm our calculations show that we must have a triangular geometry of cylindrical holes of 259nm diameter and 519nm periodicity.

FIG. 1 :

 1 FIG. 1: TE and TM band structure for 2D triangular array.

FIG. 2 :

 2 FIG. 2: TE and TM band structure for 2D triangular array.

  Dose [10 µA/cm ]

FIG. 4 :

 4 FIG. 4: (a) Dependence of the hole depth with respect to the Ga + ion dose with a cross-section at the first hole-line plane. (b) Dependence of the hole diameter with respect to the Ga + dose.

FIG. 5 :

 5 FIG. 5: SEM image of the LiN bO3 substrate covered by 250nm of Chrome after FIB milling and 10min of RIE etching. (a) D = 250nm, (b) D = 130nm.

Acknowledgments

The authors want to thank Jacques Vendeville (FEMTO-ST, Université de Franche-Comté, Besançon) and Eloïse Devaux (Laboratoire des Nanostructures, ISIS, Université Louis Pasteur, Strasbourg) for technical assistance and Jean-Yves Rauch (FEMTO-ST, Université de Franche-Comté, Besançon) for fruitful discussions.


Clément Lévy

As Far as Pynchon "Loves Cameras"

This essay studies photography as a technique and a metaphor in Thomas Pynchon's novels (from V. , 1963, to Against the Day, 2006). As an object, a photograph is very commonplace, but it can also be a fetish and, in some cases, a crucial conveyor of information. Using critical works of Paul Virilio, I show how photography, after it has become a dreadful weapon in Pynchon's works, smashes the ordinary continuity of space and time to pieces. It gives way to contemporary political and ethical problems about which Roland Barthes felt much concern. This could explain why Thomas Pynchon hasn't let any photographer approach him for more than forty years, while he publicly stated his love for cameras.

! !

Photography addresses some of the most famous themes in Thomas Pynchon's novels: media, memory, and surveillance. It is one of the oldest and most present technologies in his works.

Moreover, Pynchon builds many important metaphors around this art, as I will try so show.

And yet, as a person, the author has fought many times to prevent any publication of his own photographic image. This paradox will introduce this reflexion on the part that photography as a technique (and also as an art) and pictures, play in Thomas Pynchon's novels until Against the Day (2006). I will discuss a number of passages where photography is described, in order to show how it is used in Pynchon's novels to convey information and to represent an opposition to the writer's freedom of creation when he describes objects and persons that do not exist in the set of references-space-time-that photography for a long time was thought to best represent.

Thomas Pynchon chose to appear publicly on TV screens as a character in two episodes of season 15 and 16 of The Simpsons, the cartoon series produced by Fox Broadcasting Company. In his first appearance he is asked by Marge's editor to dictate a blurb for her novel, The Harpooned Hearts, a sentimental and shorter version of Moby-Dick. This is ! 1 what he tells her: "Here's your quote: 'Thomas Pynchon loved this book. Almost as much as he loves cameras.'" Shall we take his words seriously?

During this twenty-second speech he is talking to the editor by telephone and is obviously joking about his legendary media shyness. It has been reported that in 1963 Pynchon eluded a photographer sent by Time Magazine to take his picture in Mexico City (Teresi, Ketzan). Since then newspapers and magazines thus can only display pictures which date back from his college and navy years, from sources unknown, or the infamous paparazzo shot that is too indistinct to make a difference.

Yet once cartoon Pynchon has hung up the phone, he faces the street and calls out to indifferent passers-by to take a photo with him under a gigantic billboard that reads "Thomas Pynchon's House. Come on in!" He adds: "Hey, over here, have your picture taken with a reclusive author! Today only, we'll throw in a free autograph! But wait, there's more!" This would mean that Thomas Pynchon really likes cameras and wants his picture to be taken.

The fact that the autograph is offered "today only" seems to show that Pynchon is used to welcoming lots of fans in his neighborhood for these strange photo sessions: this is pure non-sense, as the author is hiding his face under the now famous brown paper bag. And he is very well-known for his secrecy. Pynchon neither likes Marge's novel nor cameras. The billboard and signs are a way to show it very clearly. Nevertheless, two years after this ambiguous declaration of love to cameras, Pynchon's published Against the Day, a novel in which photographers and descriptions of different photographic processes abound to such an extent that there arises a need to reconsider photography in Pynchon's works in general. I want to address this need in the present essay.

!

Pynchon focuses on photography both as an optical and a chemical process, and while it is one among many technologies central to his works, it is also one of the oldest. Against the Day represents both balloon-aeronautics (which began in the 1770s) and photography (invented around 1839) with the Chums of Chance and Merle Rideout respectively, among other characters. Yet one historical person was both a balloonist and a photographer, and he may have inspired the author. Félix Tournachon, known as Nadar, took pictures of the most famous artists of his time, like Victor Hugo, Charles Baudelaire, Gérard de Nerval, Gustave Courbet, Édouard Manet, or Sarah Bernhardt, and as soon as 1855 he even tried to develop aerial reconnaissance using balloons and photography. However, photography in Pynchon's ! 2 work is much more than a historical marker. The very presence of photography at a large scale, and every step of the history of this technique is significant in his novels, as I will show in the following.

Pynchon includes many differents types of camera and processes. Some of them are very famous, as the Kodak Brownie (from 1900 to the 1960s) in Against the Day, used by Wren Provenance to document her research on Ute vestiges, and by Merle Rideout after his apprenticeship with heavy tripods, wooden chambers and glass plates:

!

As years went along, the film got faster, the exposure times shorter, the cameras lighter. Premo came out with a celluloid film pack allowing you to shoot twelve at a time, which sure beat glass plates, and Kodak started selling its "Brownie," a little box camera that weighed practically nothing. Merle could bring it anywhere as long as he held everything steady in the frame, and by then-the old glass plate folding models having weighed in at three pounds plus plates-he had learned to breathe, calm as a sharpshooter, and the images showed it, steady, deep, sometimes, Dally and Merle agreed, more real, though they never got into "real" that far.

(AtD 72)

!

The much smaller and more accurate 24x36 LEICA camera is also mentioned by its brand

!

This German camera is very famous among photographers (notably because its shutter is very quiet, and the lenses very sharp and clear), and Henri Cartier-Bresson did much for its praise, calling it "the extension of [his] eye" in the introduction to The Decisive Moment.

Michael Naumann, who was CEO of Henry Holt publishing company (1996)(1997)(1998) and became the German Secretary of Culture (1998)(1999)(2000)(2001), said in a radio interview that Pynchon once told him that what all he wanted for his birthsday was an old LEICA camera from the 1930s. Not surprisingly then, throughout Gravity's Rainbow German cameras are highly valued and used as a means of exchange:

!

Pirate, driven to despair by the wartime banana shortage, decided to build a glass hothouse on the roof, and persuade a friend who flew the Rio-to-Ascension-to-Fort-Lamy run to pinch him a sapling banana tree or two, in exchange for a German camera, should Pirate happen across one on his next mission by parachute. (GR 6)

! ! 3
For American soldiers, cameras are part of the spoils of war in the Zone, "champagne, furs, cameras, cigarettes" (GR 302), and they were used on black market, as Säure Bummer recalls:

"Remember how the Wilhelmplatz used to be? Watches, wine, jewels, cameras, heroin, fur coats, everything in the world" (GR 376).

Cameras and photographs are so frequently mentioned in Pynchon's novels because they became part of everyday life in Western cultures during the 20th century. Photographic portraits are everywhere, and everyone keeps pictures of their loved ones or of people they admire. We can find "oversize photos of John Dillinger" in the bar called Chicago in Gravity's ! Yet photography's foremost function is to gather and communicate information. Tourists take photographs of their travels to remember them and show them to their friends once they will have returned home, a stereotype Pynchon often exploits. It is parodied when the guide who leads Slothrop down to the Mittelwerke in Nordhausen offers him to rent a camera (GR 300), but the parody is most efficient where it combines with that of national stereotypes. The Japanese in Against the Day, as in Vineland during the strange Chipco incident, cannot seem to help taking snapshots. Each member of the "Japanese trade delegation" in Telluride "carried a pocket Kodak with its shutter ingeniously connected to a small magnesium flashlight, so as to synchronize the two" (AtD 292). And when in the Cosmopolitan Gambling Club Merle Rideout tries to warn them against a nervous Bob Meldrum,

!

[a]ll at once, magnesium flash-lights were exploding everywhere, each producing a column of thick white smoke whose orderly cylindrical ascent was immediatly disarranged by attempts of customers, in some panic, to seek exit, the unexpected combination of brightness and opacity thus quickly spreading to fill every part of the saloon. (AtD 293)

!

This great uproar is all the more funny as it sounds anachronistic. But the tourists taking pictures of a traditional scene of the West are also performing a pyrotechnical display, photography being a technological show in itself.

This use of the photographic medium and its practice as a way to build photo albums and put souvenirs together is quite harmless, even when it drives a whole assembly of ! 4

gamblers and drinkers to such a tumult. But as it is also used for political purposes, a shadow of mistrust is cast on photography. Spying relied on photography in the twentieth century whereas it has had to rely only on traitors for ages before. In Gravity's Rainbow, aerial photo reconnaissance is so widely used by Allied intelligence that "these days, with so much death hidden in the sky, out under the sea, among the blobs and smears of recco photographs, most women's eyes are only functional" (GR 235). Later in the novel there is a direct mention of Constance Babington-Smith, a WAAF officer whose mission was to decipher reconnaissance We thus can read in Pynchon's novels the entire story of photography from its origins and its ! 5 ! Women's Auxiliary Air Force.

use for military purposes until the disappearance of the photographic picture in the very domain where it was supposed to play a major role: war and intelligence.

!

But as the most powerful states can decline in a few years, we must remember that every photograph is momentary and bears a strong relationship to death. As polaroids appear in the short time of a chemical revelation, older pictures in family albums will tend to fade away. In the process of its making, a silver print reveals itself to the photographer when he plunges the sensitive paper into the developer bath after projection of the negative film. It is not by accident that Pynchon compares the Raketen-Stadt in Gravity's Rainbow to a photograph: "it resembles a Daguerreotype taken of the early Raketen-Stadt by a forgotten photographer in 1856: this is the picture, in fact, that killed him -he died a week later from mercury poisoning after inhaling fumes of the heated metal in his studio" (GR 740) Here as in Against the Day,

Pynchon displays an extended knowledge of the art to his readers. The fluid metal also used in alchemy gives access to a fatal understanding of the city through its photographic representation. Of course, Raketen-Stadt "the ceremonial City, fourfold as expected, an eerie precision to all lines and shadings architectural and human, built in mandalic form like a Herero village" (GR 740) has a perfect architectural and symbolic organization. But its form is always changing: "there seems to be building, or demolition, under way in various parts of the City, for nothing here remains the same, […] engineering changes to the Rocket create new routes of supply, new living arrangements, reflected in traffic densities as viewed from this unusual height" (GR 740). From high above the city (possibly from a balloon?), the viewer can imagine the life of its inhabitants, and the growing of the organism it constitues with the rockets assembly lines. We should add that a daguerreotype, a picture left on a metal plate after an exposure to the natural light, is quite difficult to look at. One has to alternatively lower and raise one's gaze in order to understand plainly what is printed on the plate. This could explain why the image of this city is constantly changing. Another hypothesis could be the uncertain situation in time of this ideal city. It is a strange vision of the future for the main characters of the novel who see the first attempts in rocketry, but if it is portrayed on a Daguerreotype, it could also be an utopian metropolis of the 1850s.

Yet the most striking aspect in this passage is the way an imaginary object, the Raketen-Stadt, is compared to an imaginary photograph the narrator describes to the reader.

This happens many times in Pynchon's works, and it deserves close attention because unlike ! 6

postmodernist writers who includes pictures -drawings, photographs, typographical assemblage -in their fictions (e.g. William T. Vollmann, William Gass, or W. G. Sebald),

Pynchon seems to prefer to compare an object of his fictional world to a fictional photograph which is even absent from his work. In V. , the plastic surgeon Dr. Shoenmaker uses photographs as an evidence of real facts opposed to zany theories:

! "So with Lamarck, who said that if you cut the tail off a mother mouse her children will be tailless also. But this is not true, the weight of scientific evidence is against him, just as every photograph from a rocket over White Sands or Cape Canaveral is against the Flat Earth Society." (V. 47)

!

Of course, Pynchon's novels are full of the phenomena despised by Dr. Shoenmaker, from the Hollow Earth to Bigfoot, UFOs and the underground mail system W.A.S.T.E. But such fantasy is a realm where photography is totally useless.

However, in some occasions, Pynchon, by ways of a metaphor, has his own fantasy meet the photographic picture. In Against the Day, when Miles Blundell goes for a bicycle ride in Flanders with Ryder Thorn, he has a strange feeling that is compared to entering a photography:

! it was like passing through an all-surrounding photographic negative-the lowland nearly silent except for water-thrushes, the harvested fields, the smell of hops being dried in kilns, flax pulled up and piled in sheaves, in local practice not to be retted till the spring, shining canals, sluices, dikes and cart roads, dairy cattle under the trees, the edged and peaceful clouds. Tarnished silver. (AtD 553)

!

Miles has not gone through the looking-glass, but he finds himself in a bright and shiny world, silent, filled with scents, but bearing the anachronistic stain of a future catastrophe that turns out to be World War I. The photographic images bear the morbid aura that Roland Barthes conceives in every photo. In his essay on memory and photography the semiologist writes:

! in Photography I can never deny that the thing has been there. There is a superposition here: of reality and of the past. And since this constraint exists only for Photography, we must consider it, by reduction, as the very essence, the noeme of Photography. (Barthes 76-7)

!

The thing that will have been there (the ça-a-été: "that-has-been" as Barthes puts it) is in this page of Against the Day the peace of a summer that precedes the horrendous massacres and trenches that are described in Brigadier Pudding's story in Gravity's Rainbow (GR 78). Here, ! 7

the comparison with a photo allows a pause that fixes a landscape forever in a form bound to disappear. Pynchon describes photographic images for the fatality they entail. In Vineland, this darkness of death looming in every picture becomes a reason to produces pictures and films, when young activists of the 24fps collective document the repression of junkies and left-wing activists in the 1960s, claiming that "A camera is a gun. An image taken is death performed" (VL 197). This brief anarchist manifesto sentences capitalists to death, and

Frenesi as the camera-operator is the executioner, as her daughter Prairie discovers later. But it is also worth a remark that the People's Republic of Rock 'n' Roll movement in which

Frenesi is involved is also watched closely by her lover Brock Vond (she even gives him her films). Vond is federal prosecutor, and at the College of the Surf, during the revolutionary events,

!

No hour day or night was exempt from helicopter visits, though this was still back in the infancy of overhead surveillance, with a 16mm Arri "M" on a Tyler Mini-Mount being about state of the art as far as Frenesi knew.

!

For Thomas Pynchon, photography is a key element of twentieth-century history because it gives form to our representation of the world, an idea conceived in a similar way by Jean Baudrillard when he makes simulation a crucial concept for his study of our societies and art forms. The fact that photos are everywhere, and do not mean much once they have had their effect on the consciousness and triggered decisions that are always political, helps define the camera as a ubiquitous and panoptic substitute of the eye to which not everybody is given access. But in a fictional world cameras instead become images of an eye that is taken everywhere, and of a gaze that can lead to considerable knowledge but also to dreadful consequences, as in the case of the Mason-Dixon line, which is run across the open space of America after every point of it has been fixed by astronomers gazing at the stars through their optical instruments. The precision of Pynchon's discourse on photography, and the ways he ! 8 uses it as a theme and as a metaphor, makes me think that he loves cameras as much as he fears them.

CELEC, Université Jean Monnet, Saint-Etienne ! Bibliography !

  name in V. (the company has been producing cameras since 1925 to this day), as it is used by Teflon to steal pictures of Paola Maijstral and Benny Profane making love: ! Groan, went the bed. Before either of them knew it: Click, went Teflon's Leica. Profane did what was expected of him: came roaring off the bed, arm terminating in a fist. Teflon dodged it easily. (V. 19)

  Rainbow's Berlin (GR 374), and when Mayva Traverse meets Stray Briggs near the end of Against the Day, they exchange tintypes of their sons Jesse and Reef (AtD 980-1). The sentimental mood of this encounter is made stronger by the contemplation of pictures showing cherished but absent people.

  photographs, and who first identified a V1 on its launching ramp(GR 740). Of course, we also remember how Teddy Bloat used his "midget spy-camera" (GR 17) to take a picture of Slothrop's map, "click zippety click" (GR 19). However, modern spying techniques have rendered the photographer obsolete. Automatic cameras take pictures of areas under surveillance, and these photographs are taken by satellites on orbital flight since the 1960s.The camera thus became ubiquitous, and from then most of the photos taken, after a careful study, are discarded because they do not convey the information wanted. Modern spying techniques thus built a new space of control where the now classical conception of a spacetime continuum does not make sense any more. This is what the French architect Paul Virilio in The Lost Dimension (originally published as L'Espace critique in 1984) calls "speedspace" (96). According to the author, this concept has many applications for modern warfare, geopolitics, and social control. As satellites glide around the Earth, they automatically record and transmit digital pictures that are used to control troops on the ground and missiles in the air. These recent trends in mass destruction, developed during Cold War, seem to concern specifically Pynchon's works. Gravity's Rainbow and Vineland help illustrate the danger of war industries, simply because "The true war is a celebration of markets" (GR 107). But the new ways of finding and transmitting strategic information do not meet the author's interest for photography even if the objects described by Paul Virilio in The Lost Dimension are creations that originate in the desire to control a military operation from above, through optical devices. Air reconnaissance gave birth to satellite surveillance, but digital pictures took the place of photographies, and thus the image as an information medium disappeared: it was replaced by an ever-changing flow of electronic signals, pure speed without any space.

  (VL 209) ! Aerial surveillance is also used much later in the novel, during the War on Drugs led in Vineland County by Kommandant Bopp. The "former Nazi Luftwaffe officer" leads "helicopter and plane crews" of voluntary "antidrug activists" (VL 221). The reconnaissance techniques even involve the use of airborne radar equipments: "AWACS planes in the air round the clock" (VL 222).


J L Vignet

J Pancin

P Gangnant

E Gueroult

GANIL N Renoux

A NEW LOW INTENSITY BEAM PROFILER FOR SPIRAL2

In the framework of SPIRAL 2 ion beams, several beam profile monitors are presently being developed at GANIL. One of them is a low-intensity beam-profile monitor that works as a secondary electron detector. This Emission-Foil Monitor (EFM) will be used in the radioactive beam lines of SPIRAL2 and in the experimental rooms of this new facility. The ions produce secondary electrons when they are stopped in an aluminium emissive foil. The electrons are then accelerated using an electric field and guided using a magnetic field to a double-stage microchannel plate (MCP). A 2D pixellated pad plane placed below the MCP is used to collect the signals. The magnetic field created by permanent magnets in a closed magnetic circuit configuration permits the beam-profile reconstruction to be achieved with a good resolution. The EFM can visualize beam-profile intensities between only a few pps to as much as 10 9 pps and with energies as low as several keV. This profiler has been under development since 2009 and is currently manufactured.

INTRODUCTION

The new Spiral2 facility is under construction at Ganil and the first extracted beams are scheduled for 2014. In order to detect stable and radioactive beams, a new profiler has been developed and will be installed on the Spiral2 transfer lines and in the associated experimental areas in 2014. The EFM (Emissive Foil Monitor) is dedicated to stable and radioactive beams with an energy range in between 5 keV/u and 25 MeV/u with beam intensities from a few pps to 10 9 pps. A prototype has been developed since 2009 [1] [2] and ten EFMs will be constructed in 2013.

EFM DESCRIPTION

Several EFM have already been developed in the past [3,4]. It is however the first time a closed magnetic circuit configuration using permanent magnet is used to achieve a good spatial resolution. The Emissive Foil Monitor (EFM) is fixed on a pneumatic cylinder mounted on a CF150 flange to be installed and inserted easily in the beam line. This detector, oriented at 45° in the beam line, is composed of an aluminum foil (Emission Foil), a magnetic and electrostatic drift area associated with an amplification system with Micro Channel Plates (MCP) (see figure 1) mounted on a pixellated kapton board (see figure 2). Under the impact of the beam nuclei, secondary electrons are emitted from the foil. They are then accelerated by an electrostatic field applied between the emissive foil (EF) (about -2 kV) and a grounded wire grid positioned at 4 mm. Fours NeFeB permanent magnets of 1,4 Tesla persistent magnetic field, inserted in a magnetic circuit made of ARMCO Iron pieces, are used to create a homogeneous magnetic field in between the EF and the MCPs. Once the electrons have been accelerated, they are guided on 55 mm only by the magnetic field and follow a helical trajectory. They are then slowed down by an electric field applied in between a grounded grid and the first MCP surface. This grounded grid is positioned at 4 mm from the MCPs and this gap permits to optimize the electron energy for the MCPs. The electric field is zero in between both grounded grids, which is an advantage to avoid any deformations of the field lines by the detector mechanics or the beam tubes. The electron amplification system is composed of two MCPs (Hamamatsu F1217-01) assembled in chevron structure to provide a high amplification gain of 10 7 . The signal of the electrons exited the MCPs is read using one X-Y matrix of 47 strips with 1 mm pitch. The circular active detection area has a diameter of 42 mm. The matrix board is composed of 500 µm x 500 µm pixels connected alternatively to make X and Y strips (see figure 2). These strips are connected via a Flex Kapton cable to a digital electronic acquisition system developed at GANIL of 2 x 47 channels. Each channel integrates the charge of the associated strip in a capacitor and performs a current-voltage conversion

TRAJECTORY SIMULATION

Electrons trajectories in between the emissive foil and the MCP have been calculated taking into account the electric and magnetic field. Simulations have been performed using Mathematica 7.0 for magnetic field calculation and Simion 8.0 for the electrons transport. Different magnetic geometries were simulated with an adapted electric field ensuring a minimum deviation of the electrons. These shifts depend on the electron initial position and are smaller in the center than on the edges because of the magnetic field non homogeneities.

In the best configuration (figure 3), the systematical deviations are in-between 0.1 and 0.8 mm for an emission area of 30 mm x 30 mm 2 and for a foil and MCP voltage respectively of -2 kV and-1kV.

EXPERIMENTAL RESULTS

The EFM was tested at GANIL with Ar ions of 4.8 MeV/u and 4.8 keV/u. The beam intensity was varied from 10 to 10 10 pps.

Fours Permanent magnets of NdFeB with a magnetic induction of 1.4 Tesla (N50) have been used. Two magnets of 35 mm x 70 mm x 10 mm are assembled on each side to create a homogenate magnetic field of around 300 Gauss in all the drift volume like the figure 3 configuration. The electronic acquisition used was the standard GANIL electronic profiler and obtained profiles were displayed with the GANIL profiler software.

Result with Stable 40 Ar 6+ at 4.8 MeV/u

Test with calibration pattern To estimate the profiler spatial resolution, a mask was installed just before the profiler (see figure 5). The position of each hole was measured with the EFM and reported on figure 5.The beam intensity was 5×10 6 pps and the high voltages applied were -2kV on the foil and -1.2 kV on the microchannel plates. The shift of 3 mm in Y is due to misalignment of the profiler on the beam line. Distances between each wire are 1 mm for the gas profiler and distances between each strip are 1 mm for the horizontal EFM profile and 0.7 mm for the vertical profile (due to the 45° section projection in this plane).

Except the difference between vertical profiles positions (the profiler was not aligned on the beam line), EFM profile is quite similar to the one obtained with the gas profiler.

Results with 40 Ar 6+ at 4.8 KeV/u

The EFM has been mounted on the LIRAT beam line of the GANIL accelerator to test it with very low energy radioactive beam. Foil high voltage was increased to check if the low energy beams can be deflected. The beam intensity was 7x10 9 pps. We can see figure 7 that there are no effect on the horizontal profile and a small effect on the vertical (0.5 mm for a foil high voltage variation between -0.7 and -2 kV). EFM Beam profiles have been compared whit a grid profiler (SEM) at the same position (see figure 8). Beam intensity has been increase to 10 10 pps to be compatible with the range intensity of the grid profiler. EFM profiles are narrowed than SEM profile (respectively -14 and -6 % for the horizontal and vertical).

CONCLUSION

A new beam profile monitor has been developed at Ganil using magnetic and electrostatic field and microchannel plate amplification. Simulations have shown that obtained results are good with a resolution below 1 mm on the whole active area. Experimental tests have also confirmed that the Emissive Foil voltage does not disturb the very low energy beams before hitting the foil. The profiles obtained with the EFM and a wire profiler are quite similar enough to confirm the good working of the device.

This new profiler will be installed on the new Spiral2 facilities in 2014. It might also replace the gas wire profilers installed on the Ganil lines in order to visualize beams profiles at low energy (below 3 MeV/u. Ten of these profiler are built and will be assembled at the end of this year.

Figure 1 :

 1 Figure 1: general view of the EFM head.

Figure 2 :

 2 Figure 2: view of the XY matrix with a zoom on the strip connection.

Figure 3 :

 3 Figure 3: Best magnetic configuration calculated with the simulations The shifts in X and Y directions experienced by the electrons from the EF to the MCP are calculated using the simulations. An example of electrons trajectories projected on the MCP surface is shown on figure 4.

Figure 4 :

 4 Figure 4: SIMION simulation results for secondary electrons emitted from the foil with a high voltage of -2kV and a magnetic field of 280 G.

Figure 5 :

 5 Figure 5: Holes of the mask in blue and measured positions in red.Obtained results are in good agreement with theoretical positions with shifts below 0.5 mm.Profiles comparison EFM beam profiles were compared with profiles from a wire gas profiler installed at the same position on an opposite flange. The wire gas profiler is composed of X and Y tungsten wires enclosed in a volume filled with Ar+CO2 (20%) at 1 bar. Beam energy must be at least 3 MeV/u because of the thickness of the entrance window of the Gas profiler,

Figure 7 :

 7 Figure 7: Centroid profile position versus variation of the foil High Voltage.

Figure 8 :

 8 Figure 8: Profile comparison between grid profiler and EFM.

  [1] JL Vignet et al., GANIL, The new beam profile monitor for SPIRAL2, Proc of DIPAC conf 2009, TUPB07 (2006) 176, http://jacow.org/. [2] C. Jamet et al., GANIL, Beam diagnostic overview of the Spiral2 RIB section, Proc of DIPAC conf 2011, TUPD06 (2011) , http://jacow.org/. [3] O.H. Odland et al., GANIL, a fast positive sensitive microchannel plate detector for ray tracing of charged particules, NIM A,378, (1996) 149-154. [4] D. Shapira et al., A fast accurate position-sensitive timing detector based on secondary electron emission, NIM A454, (2000) 409-420.

Proceedings of IBIC2013, Oxford, UK WEPF14Beam Profile Monitors ISBN 978-3-95450-127-4


Lorenzo Brandolese

Manuel Fernando Cortez

BLOWUP ISSUES FOR A CLASS OF NONLINEAR DISPERSIVE WAVE EQUATIONS

Keywords: Rod equation, Camassa-Holm, shallow water, Wave breaking

In this paper we consider the nonlinear dispersive wave equation on the real line, ut

x x = 0, that for appropriate choices of the functions f and g includes well known models, such as Dai's equation for the study of vibrations inside elastic rods or the Camassa-Holm equation modelling water wave propagation in shallow water. We establish a local-in-space blowup criterion (i.e., a criterion involving only the properties of the data u0 in a neighbourhood of a single point) simplifying and extending earlier blowup criteria for this equation. Our arguments apply both to the finite and infinite energy case, yielding the finite time blowup of strong solutions with possibly different behavior as x → +∞ and x → -∞.

Introduction

The experimental observation by the naval architect Scott Russel of solitary waves propagating in channels at different speeds, and interacting in a nonlinear way before recovering their initial shape, motivated the studies on the mathematical modelling of water wave motion at the end of the XIX century. The first works can be retraced back to Boussinesq, Rayleigh, Korteweg and de Vries. The celebrated KdV equation allows for a first mathematical description of such phenomena. This equation can be derived as an asymptotic model from the free surface Euler equations 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 wavelengt. It models small amplitude waves, i.e. waves such that the dimensionless amplitude parameter ǫ = a/h satisfies ǫ = O(µ), where a is the typical amplitude.

Such small amplitude waves feature both nonlinear and dispersive effects. For larger amplitude waves nonlinear effects become preponderant and wave breaking can eventually occur. As the KdV equation is no longer suitable for the description of breaking mechanisms -its solutions remain smooth for all time -a considerable effort was made toward the modelling of larger amplitude, possibly breaking waves, see, e.g., the monograph [START_REF] Whitham | Linear and nonlinear waves[END_REF]. Such studies culminated with the derivation in 1993, by Camassa and Holm, [START_REF] Camassa | An integrable shallow-water equation with peaked solitons[END_REF][START_REF] Camassa | A new integrable shallow-water equation[END_REF] of an equation obtained from the vertically averaged water wave system, written in Lie-Poisson Hamiltonian form, by an asymptotic expansion preserving the Hamiltonian structure. The scaling of validity of the Camassa-Holm equation is µ < < 1 and ǫ = O( √ µ): such scaling includes that of KdV allowing higher order accuracy. Alternative rigorous derivations of the Camassa-Holm equation are also available, see [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]. Such equation attracted a considerable interest in the past 20 years, not only due its hydrodynamical relevance (it was the first equation capturing both soliton-type solitary waves as well as breaking waves) but also because of its extremely rich mathematical structure. In fact, the Camassa-Holm equation was written for the first time in a different context, as one of the 12 integrable equations classified by Fokas and Fuchsteiner [START_REF] Fuchssteiner | Symplectic structures, their Bäcklund transformations and hereditary symmetries[END_REF] and obtained from a nonlinear operator satisfying suitable defining properties, applying a recursive operator that is an hereditary symmetry. The Camassa-Holm equation is usually written as (1.1)

u t + κu x -u xxt + 3uu x = uu xxx + 2u x u xx , x ∈ R, t > 0.
where u can be interpreted as an horizontal velocity of the water at a certain depth and κ is the dispersion parameter.

The dispertionless case κ = 0 is of mathematical interest as in this case the equation possess soliton solutions (often named peakons) peaked at their crest, of the form u c (t, x) = ce -|x-ct| . Multi-peakon interactions is studied in [START_REF] Camassa | An integrable shallow-water equation with peaked solitons[END_REF][START_REF] Camassa | A new integrable shallow-water equation[END_REF]. For κ > 0 the equation admits smooth solitons.

In the shallow water interpretation, however, κ is proportional to the square root of the water depth and cannot be zero. On the other hand, the same equation, with κ = 0 appears, e.g., in the study of the dynamics of a class of non-Newtonian, second-grade fluids (see [START_REF] Busuioc | On second grade fluids with vanishing viscosity[END_REF]), or when modelling vibrations inside hyper-elastic rods. In the latter case peakons correspond to physical solutions. More in general, the propagation of nonlinear waves inside cylindrical hyper-elastic rods, assuming that the diameter is small when compared to the axial length scale, is described by the one dimensional equation (see [START_REF] Dai | Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod[END_REF]),

v τ + σ 1 vv ξ + σ 2 v ξξτ + σ 3 (2v ξ v ξξ + vv ξξξ ) = 0, ξ ∈ R, τ > 0.
Here v(τ, ξ) represents the radial stretch relative to a pre-stressed state, σ 1 = 0, σ 2 < 0 and σ 3 ≤ 0 are physical constants depending on the material. The scaling transformations

τ = 3 √ -σ 2 σ 1 t, ξ = √ -σ 2 x,
with γ = 3σ 3 /(σ 1 σ 2 ) and u(t, x) = v(τ, ξ), allow us to reduce the above equation to

(1.2) u t -u xxt + 3uu x = γ(2u x u xx + uu xxx ), x ∈ R, t > 0.
Notice that when the physical parameter γ (related to the Finger deformation tensor) is equal to 1, one recovers the dispersionless Camassa-Holm equation. Several positive or negative values of γ correspond to known hyper-elastic materials.

Common important features of the Camassa-Holm and the rod equation include:

-The conservation of the energy integral (u 2 + u 2 x ) for classical and sufficiently decaying solutions.

-The local well-posedness theory: the Cauchy problems for equation (1.1) and (1.2) are well-posed in the Sobolev space H s (R) for s > 3/2 (or in suitable Besov spaces), locally in time. See, e.g., [START_REF]Stability of a class of solitary waves in compressible elastic rods[END_REF][START_REF] Danchin | A few remarks on Camassa-Holm equation[END_REF][START_REF] Li | Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation[END_REF][START_REF] Yin | Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation[END_REF]. -Wave breaking scenario: the maximal existence time T * such that the solution belongs to C([0, T * ),

H s (R)) ∩ C 1 ([0, T * ), H s-1 (R)) is finite if and only if u x is
unbounded from below (or from above when γ < 0) near the blowup time. Up to the time T * , the solution remains uniformly bounded. See [START_REF] Camassa | An integrable shallow-water equation with peaked solitons[END_REF][START_REF]Stability of a class of solitary waves in compressible elastic rods[END_REF][START_REF] Yin | Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation[END_REF]. -Finite time blowup criteria on the initial data and upper bound estimates on T * .

See [START_REF] Constantin | Wave breaking for nonlinear nonlocal shallow water equations[END_REF][START_REF]Stability of a class of solitary waves in compressible elastic rods[END_REF][START_REF] Liu | Global existence and blow-up solutions for a nonlinear shallow water equation[END_REF][START_REF] Mckean | Breakdown of the Camassa-Holm equation[END_REF][START_REF] Yin | Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave 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]. -Exponentially decaying solitary wave solutions. See, e.g. [START_REF] Camassa | A new integrable shallow-water equation[END_REF][START_REF] Dai | Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod[END_REF][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][START_REF] Yin | Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation[END_REF].

-Infinite propagation speed and persistence results in weighted spaces. See [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].

-The existence of global conservative or dissipative weak solutions. See [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]Global conservative solutions of the Camassa-Holm equation. A Lagrangian point of view[END_REF].

Beside such common features, the Camassa-Holm equation is considerably better understood than the rod equation. Indeed, contrary to the rod equation with γ = 1, equation (1.1) possess a bi-hamiltonian structure that makes the equation formally integrable via the inverse scattering method. Elegant geometric interpretations (see [START_REF] Constantin | Geodesic flow on the diffeomorphism group of the circle[END_REF][START_REF] Constantin | A shallow water equation on 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][START_REF] Misio | A shallow water equation as a geodesic flow on the Bott-Virasoro group[END_REF]) are available: for example, equation (1.1) gives rise to a geodesic flow of a right invariant metric on the Bott-Virasoro group. This equation also admits infinitely many conservation laws. Moreover, it has solitary waves interacting like solitons that are orbitally stable, see [START_REF] Constantin | Stability of peakons[END_REF]. The global existence of strong solutions of the Camassa-Holm equation can be obtained putting suitable sign conditions on the associated initial potential y 0 = u 0 -(u 0 ) xx , see [START_REF] Constantin | Existence of permanent and breaking waves for a shallow water equation: A geometric approach[END_REF][START_REF] Mckean | Breakdown of the Camassa-Holm equation[END_REF]. When γ = 1, as the sign of the potential is no longer conserved by the flow, such global existence criterion is no longer valid. In fact, we know of no general condition on the initial datum guaranteeing that the corresponding solution of the rod equation remains in H s (s > 3/2) for all time. The value of the parameter γ plays a crucial role: two limit situations occur when γ = 0 and γ = 3. In the latter case any non-zero solution eventually develops a singularity (see [START_REF]Stability of a class of solitary waves in compressible elastic rods[END_REF]). On the other hand, when γ = 0, no blowup can occur. (When γ = 0, the rod equation boils down to the BBM equation, a model for the unidirectional evolution of long waves, [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]). Physically, the formation of a singularity corresponds to a formation of a crack inside the rod. Both behaviors in the two limit cases γ = 0 and γ = 3 are not physically realistic for real rods, but it is interesting to observe that there are known materials such that γ is indeed close to 3 (e.g., γ = 2.668) and others such that γ is close to zero (e.g. γ = -0.539). We refer to [START_REF] Dai | Solitary shock waves and other travelling waves in a general compressible hyperelastic rod[END_REF] for a list of physically acceptable values of γ (ranging from -29.476 to 3.417).

Blowup for the generalized hyper-elastic rod equation

In this paper we will consider the Cauchy problem for a nonlinear dispersive wave equation including both (1.1) and (1.2) as a particular case:

(2.1)

u t -u txx + [f (u)] x -[f (u)] xxx + g(u) + f ′′ (u) 2 u 2 x x = 0.
Equation (2.1) is often referred as the generalized hyper-elastic rod wave equation, see [START_REF] Holden | Global conservative solutions of the generalized hyperelastic-rod wave equation[END_REF]. The Camassa-Holm equation corresponds to the choice f (u) = u 2 /2, g(u) = κu + u 2 and the rod equation to the choice f (u) = γ 2 u 2 and g(u

) = 3-γ 2 u 2 . When f (u) = u Q+1 Q+1 and g(u) = κu+ Q 2 +3Q
2(Q+1) u Q+1 one recovers from (2.1) another class of equations with interesting mathematical properties, studied in [START_REF] Hakkaev | Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation[END_REF].

From now on, we will study the Cauchy problem for the generalized rod equation, written in the non-local form, formally equivalent to (2.1):

(2.2)    u t + f ′ (u)u x + ∂ x p * g(u) + f ′′ (u) 2 u 2 x = 0, x ∈ R, t > 0, u(x, 0) = u 0 (x),
x ∈ R.

Here p(x) = 1 2 e -|x| is the fundamental solution of the operator 1 -∂ 2

x . The problem (2.2) is thus written in the abstract form

du dt + A(u) = H(u), u(x, 0) = u 0 (x), with A(u) = f ′ (u)∂ x and H(u) = -∂ x (1-∂ 2 x ) -1 g(u)+ f ′′ (u) 2 u 2
x . The local existence theory can be developped applying classical Kato's approach [START_REF] Kato | Quasi-linear equation of evolution, with applications to partial differential equations[END_REF]. For reader's convenience we collect in a single theorem the main results of the recent paper of Tian, Yan and Zhang [START_REF] Tian | The Cauchy problem for the generalized hyperelastic rod equation[END_REF] on the problem (2.2).

Theorem 2.1 (See [START_REF] Tian | The Cauchy problem for the generalized hyperelastic rod equation[END_REF]).

(1) Assume that f, g ∈ C ∞ (R). Let u 0 ∈ H s (R), s > 3/2. Then there exists T > 0, with T = T (u 0 , f, g) and a unique solution u to the Cauchy problem

(2.2) such that u ∈ C([0, T ), H s (R)) ∩ C 1 ([0, T ), H s-1 (R)). The solution has constant energy integral R (u 2 + (u x ) 2 ) = R (u 2 0 + (u ′ 0 ) 2 ) = u 0 2 H 1 .
Moreover, the solution depends continuously on the initial data: the mapping

u 0 → u is continuous from H s (R) to C([0, T ), H s (R)) ∩ C 1 ([0, T ), H s-1 (R)). (2) Assume in addition that f ′′ ≥ γ > 0: i) (Blowup scenario and rate) Let 0 < T * ≤ ∞ be the maximal time of the solution in C([0, T * ), H s (R)) ∩ C 1 ([0, T * ), H s-1 (R)). Then T * < ∞ if and only if lim t→T * inf x∈R u x (t, x) = -∞.
In this case, the blowup rate of

inf x∈R u x (t, x) as t → T * is O( 1 T * -t ). ii) (Blowup criterion) Assume that there exists a point x 0 ∈ R such that (2.3) u ′ 0 (x 0 ) < - 4 sup |v|≤ u 0 H 1 |g(v)| + u 0 2 H 1 sup |v|≤ u 0 H 1 f ′′ (v) γ .
Then u blows up in finite time and

T * ≤ 1 √ 2C 0 γ log √ γ/2 u ′ 0 (x 0 )- √ C 0 √ γ/2 u ′ 0 (x 0 )+ √ C 0
, where

C 0 = C 0 ( u 0 H 1 , f, g) is given by C 0 ≡ 2 sup |v|≤ u 0 H 1 |g(v)| + u 0 2 H 1 2 sup |v|≤ u 0 H 1 f ′′ (v).
In the first item, the existence time T can be taken independent on the parameter s in the following sense: if u 0 also belongs to H s 1 (R) with s 1 > 3/2, then we have also [START_REF] Tian | The Cauchy problem for the generalized hyperelastic rod equation[END_REF] include lower bound estimates for the existence time T * and the lower semi-continuity of the existence time. Let us also mention the construction of global conservative weak solutions for such equation [START_REF] Holden | Global conservative solutions of the generalized hyperelastic-rod wave equation[END_REF] (see also [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] Coclite | Global weak solutions to a generalized hyperelastic-rod wave equation[END_REF] for earlier results on weak solutions for more specific choices of the functions f and g).

u ∈ C([0, T ), H s 1 (R)) ∩ C 1 ([0, T ), H s 1 -1 (R)). Additional results in
Main results. The purpose of this paper is to establish a new blowup criterion for equation (2.2), considerably simplifying (2.3), and extending our previous result established in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF] in the special case of the classical rod equation. Our second goal is to handle more general boundary conditions in order to encompass the case of solutions not necessarily vanishing at infinity. In particular, in the present paper we will be able to cover the case f (u) = u 2 and g(u) = κu + u 2 corresponding to the Camassa-Holm equation with dispersion (κ > 0), a case that was not covered in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF]. Contrary to previously known blowup criteria, like those in [START_REF] Camassa | A new integrable shallow-water equation[END_REF][START_REF] Constantin | Existence of permanent and breaking waves for a shallow water equation: A geometric approach[END_REF][START_REF] Constantin | Wave breaking for nonlinear nonlocal shallow water equations[END_REF][START_REF]Stability of a class of solitary waves in compressible elastic rods[END_REF][START_REF] Li | Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation[END_REF][START_REF] Tian | The Cauchy problem for the generalized hyperelastic rod equation[END_REF][START_REF] Yin | Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation[END_REF][START_REF] Zhou | Wave breaking for a shallow water equation[END_REF], our criterion has the specific feature of being purely local in the space variable: indeed our blowup condition only involves the values of u 0 (x 0 ) and u ′ 0 (x 0 ) in a single point x 0 of the real line. On the other hand, for earlier criteria, checking the blowup conditions involved the computation of global quantities (typically, the u 0 H 1 norm, as in criterion (2.3) above) or other global conditions like antisymmetry assumptions or sign conditions on the associate potential.

As we shall see, in order to establish such blowup result we will need to restrict the choice of the admissible functions f and g. On the other hand, when available, our criterion is applicable to a wider class of initial data. We are now in the position of establishing our theorem. Roughly speaking, under appropriate conditions on f and g, we get the finite time blowup as soon as

(2.4) ∃ x 0 ∈ R such that u ′ 0 (x 0 ) < -β u 0 (x 0 ) -c ,
where β and c are two real constants depending on the shape of the functions f and g. We obtain two slightly different versions when g is bounded from below, or when g is bounded from above. Both cases turn out to be physically interesting.

Theorem 2.2. Let u 0 ∈ H s (R), with s > 3/2. Let f, g ∈ C ∞ (R) with f ′′ ≥ γ > 0. The maximal time T * of the solution u to problem (2.2) in C([0, T * ), H s (R))∩C 1 ([0, T * ), H s-1 (R))
must be finite, if at least one of the two following conditions (1) or (2) is fulfilled:

(1) -∃ c ∈ R such that m = g(c) = min R g.
-The map φ : R → R given by φ

= 1 γ (g -m) is K-Lipschitz with 0 ≤ K ≤ 1, -∃ x 0 ∈ R such that u ′ 0 (x 0 ) < -1 2 √ 1 + 8K 2 -1 |u 0 (x 0 ) -c|. (2) Or, otherwise, -∃ c ∈ R such that M = g(c) = max R g.
-The map ψ : R → R given by ψ

= 1 γ (M -g) is K-Lipschitz with 0 ≤ K ≤ 1 √ 8 -∃ x 0 ∈ R such that u ′ 0 (x 0 ) < -1 2 1 - √ 1 -8K 2 |u 0 (x 0 ) -c|.
More precisely, the following upper bound estimate for T * holds:

(2.5)

T * ≤ 4 γ 4u ′ 0 (x 0 ) 2 - √ 1 ± 8K 2 -1 2 u 0 (x 0 ) -c 2 ,
where in the term ±8K 2 one has to take the positive sign under the conditions of Part (1) and the negative sign under the conditions of Part (2).

We will also establish a variant of this theorem for solutions that are not in H s (R). See Theorem 4.1 below. This variant will apply to a large class of infinite energy solutions, as well as to solutions with non-vanishing and possibly different asymptotics as x → +∞ and x → -∞. Another possible variant is obtained assuming that f is strictly concave, rather than strictly convex: in this case the blowup condition on u 0 should be of the form:

u ′ 0 (x 0 ) > β|u 0 (x 0 ) -c|. Remark 2.3 (Application to the Camassa-Holm equation). The case f (u) = 1
2 u 2 and g(u) = κu + u 2 corresponds to the Camassa-Holm equation with dispersion (1.1). Situation (1) of Theorem 2.2 applies (with c = -κ/2, φ(u) = u 2 + κu + κ 2 /4 and K = 1). We then immediately get the following corollary:

Corollary 2.4. Let u 0 ∈ H s (R), with s > 3/2 be such that at some point x 0 ∈ R we have u ′ 0 (x 0 ) < -u 0 (x 0 ) + κ 2 .
Then the corresponding solution of the Camassa-Holm equation breaks down in finite time.

This corollary could be also obtained from the special case κ = 0 (established in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF]) via the change of unknown v(t, x) = u(t, x -κ 2 t) + κ 2 . Indeed, if u solves equation (1.1), then v solves the Camassa-Holm equation without dispersion. However, one should check that the proof of [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF] does indeed go through when applied to v, which requires slight changes (the point is that the solution v does not vanish as x → ∞ as it was required in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF]).

Remark 2.5 (Application to the classical rod equation). In the case f (u) = γ 2 u 2 and g(u) = 3-γ 2 u 2 , corresponding to the classical rod equation, the conditions of our theorem are satisfied if and only if 1 ≤ γ ≤ 4. Namely, situation (1) applies for 1 ≤ γ ≤ 3 and situation (2) applies for 3 ≤ γ ≤ 4. Making explicit our blowup condition in this case, we see that solutions of the classical rod equation break down in finite time as soon as, at some point x 0 ∈ R, we have

(2.6) u ′ 0 (x 0 ) < - 1 2 √ γ 12 -3γ - √ γ |u 0 (x 0 )| (1 ≤ γ ≤ 4).
This conclusion allow us to recover the result in [START_REF]Local-in-space criteria for blowup in shallow water and dispersive rod equations[END_REF]. Outside the range 1 ≤ γ ≤ 4 it seems difficult to get purely "local-in-space" blowup criteria in the same spirit as in Theorem 2.2. But non local-in-space blowup conditions involving the computation of the u 0 H 1 as in (2.3) still apply outside the above range for the parameter γ, see, . e.g., [START_REF]Stability of a class of solitary waves in compressible elastic rods[END_REF], [START_REF] Guo | Wave breaking and persistence properties for the dispersive rod equation[END_REF].

Next section will be devoted to the proof of Theorem 2.2. The most immediate application of this theorem is that one can recover earlier blowup criteria, like that in (2.3), by a simple application of Sobolev imbedding theorems. In the last section we discuss some further consequences of of Theorem 2.2 by establishing three corollaries. Such theorem and its variant, Theorem 4.1, imply that global solutions must satisfy quite stringent properties, including a not too fast decay of u(t, x)-c as |x| → ∞, and sign restrictions for u(t, x)-c. Namely, we prove that the finite time blowup must occur if u 0 (x) = c+o(e -β|x| ) as |x| → ∞. The finite time blowup must occur also if there exist x 1 < x 2 such that u(t, x 1 ) > c > u(t, x 2 ). Here, the constants constants β > 0 and c ∈ R are as in Theorem 2.2.

The proof of Theorem 2.2

Proof of Theorem 2.2. We can assume, without restriction, that s ≥ 3. Indeed, if 3/2 < s < 3, and u 0 ∈ H s (R) satisfies a condition of the form (2.4), then we can approximate u 0 with a sequence of data belonging to H 3 (R) and satisfying the same condition (2.4), and next use the well-posedness result recalled in Theorem 2. [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]. In what follows we then consider a solution

u ∈ C([0, T * ) ∩ H 3 ) ∩ C 1 ([0, T * ), H 2 ).
We start with an useful lemma:

Lemma 3.1. Let 1 R ± denote one of the two indicator functions 1 R + or 1 R -.
(1) If f and g satisfy the condition as in Theorem 2.2 -(1) then the following estimate holds:

(p1 R ± ) * g(u) + f ′′ (u) 2 u 2 x ≥ α 2 g(u) -m + m 2 (3.1) with (3.2) α = 1 4K 2 1 + 8K 2 -1 .
(2) If f and g satisfy the condition as in Theorem 2.2 -(2), then we have:

(p1 R ± ) * g(u) + f ′′ (u) 2 u 2 x ≥ α 2 g(u) -M + M 2 (3.3) with (3.4) α = 1 4K 2 1 -1 -8K 2 .
In the case g = m = M be a constant function (this corresponds to K = 0), the right-hand side of the above convolution estimates reads

(p1 R ± ) * g + f ′′ (u) 2 u 2 x ≥ g/2.
Proof. Let us consider the case of the indicator function 1 R + . Under the conditions of Theorem 2.2 -(1) for f and g, we see that g -m ≥ 0 and that the zeros of g -m are of order at least two. Then φ = 1 γ (g -m) is differentiable, and by the Lipschitz condition |φ ′ | ≤ K. This in turn implies

(g ′ ) 2 ≤ 4γK 2 (g -m).
Consider the quadratic polynomial in λ,

P (λ) = γ 2 λ 2 -αg ′ (u)λ + b g(u) -m ,
where α and b are constants to be determined below. We have P ≥ 0 on the real line if and only if α 2 (g ′ (u)) 2 ≤ 2γb g(u) -m . We then choose b = 2α 2 K 2 , thus ensuring that indeed P ≥ 0. Recalling f ′′ (u) ≥ γ and using the fact that P (u x (ξ)) ≥ 0 for all real ξ, we now deduce that

(p1 R + ) * b g(u) -m + f ′′ (u) 2 u 2 x (x) = e -x 2 x -∞ e ξ b g(u) -m + f ′′ (u) 2 u 2 x (ξ) dξ ≥ e -x 2 x -∞ e ξ αg ′ (u)u x (ξ) dξ = α e -x 2 x -∞ e ξ d dξ g(u) -m (ξ) dξ = α 2 g(u) -m (x) -α(p1 R + ) * g(u) -m (x).
Hence,

(3.5) (p1 R + ) * (b + α) g(u) -m + f ′′ (u) 2 u 2 x (x) ≥ α 2 g(u) -m (x).
In order to get the first estimate of the lemma we need

b + α = 1.
This boils down to the equation 2K 2 α 2 + α -1 = 0. Taking the largest real root we obtain expression (3.2) for α. Now observing that (p1 R + ) * m = m R p1 R + = m/2, leads to estimate (3.1). The reasoning when f and g satisfy the conditions of Theorem 2.2 -( 2) is entirely analogous: one has (g ′ ) 2 ≤ 4γK 2 (M -g). We can thus consider the same polynomial as before, but choosing now b = -2α 2 K 2 . Reproducing the same computation as above (with M instead of m) we find as before the relation α + b = 1. Eliminating b gives the equation 2K 2 α 2 -α + 1 = 0. We thus need 0 ≤ K ≤ 1/ √ 8 for this equation to admits real roots. Choosing now the smallest root gives (3.4) and estimate (3.3).

The convolution estimate involving p1 R -can be proved in a similar way. Indeed, under, e.g. the assumptions of Theorem 2.2 -(1) we have, for any λ ∈ R,

P (λ) = γ 2 λ 2 + αg ′ (u)λ + b(g(u) -m) ≥ 0.
Thus,

(p1 R -) * b g(u) -m + f ′′ (u) 2 u 2 x (x) = e x 2 ∞ x e -ξ b g(u) -m + f ′′ (u) 2 u 2 x (ξ) dξ ≥ e x 2 +∞ x e -ξ -αg ′ (u)u x (ξ) dξ = α 2 g(u) -m (x) -α(p1 R -) * g(u) -m (x),
and we can proceed as in the previous case.

Let us go back to the proof of Theorem 2.2. Taking the space derivative in equation (2.2), and recalling that (1 -∂ 2

x )p equals the Dirac mass at the origin, we get

(3.6) u tx + f ′ (u)u xx = - f ′′ (u) 2 u 2 x + g(u) -p * g(u) + f ′′ (u) 2 u 2
x . We now consider the flow map, defined by

(3.7) q t (t, x) = f ′ (u(t, q(t, x)), t > 0, x ∈ R, q(0, x) = x x ∈ R,
where u is the solution of the problem (2.2) given by Theorem 2.1. Notice that the assumptions made on f and u imply that q ∈ C 1 ([0, T * ) × R, R) is well defined on the whole time interval [0, T * ). We now proceed putting the conditions of Theorem 2.2 (1). From (3.6), the uniform convexity condition f ′′ ≥ γ > 0, and summing up the two convolution estimates for 1 R + and 1 R -in (3.1), we get

d dt u x (t, q(t, x))] = u tx + f ′ (u)u xx (t, q(t, x)) = - f ′′ (u) 2 u 2 x + g(u) -p * g(u) + f ′′ (u) 2 u 2 x ≤ - γ 2 u 2 x + (1 -α) g(u) -m (t, q(t, x)). (3.8)
By the definition of α (3.2) we see that 0 < α ≤ 1. We can express K in terms of α as

(3.9) 2K 2 = 1 -α α 2 . The Lipschitz condition on φ = 1 γ (g -m)
provides the estimate φ(u) ≤ K|u -c|. We thus obtain

d dt u x (t, q(t, x))] ≤ - γ 2 u 2 x + (1 -α)γφ(u) 2 ≤ - γ 2 u 2 x + (1 -α)K 2 γ(u -c) 2 = γ 2 (1 -α) 2 α 2 (u -c) 2 -u 2 x .
(3.10)

Let us set

(3.11) β := 1 -α α = 2K 2 α = 1 2 √ 1 + 8K 2 -1 .
Next introduce (3.12)

A(t, x) = β(u -c) -u x (t, q(t, x)) and (3.13) B(t, x) = β(u -c) + u x (t, q(t, x))
We then obtain from (3.10)

(3.14) d dt u x (t, q(t, x))] ≤ γ 2 (AB)(t, x).
On the other hand, the kernel p satisfies the identity (both in the distributional and a.e. pointwise sense)

∂ x p = p1 R --p1 R + .
Then we get, recalling the inequality

f ′′ ≥ γ, A t (t, x) = β(u t + f ′ (u)u x ) -(u tx + f ′ (u)u xx ) = f ′′ (u) 2 u 2 x -g(u) + (p -β∂ x p) * g(u) + f ′′ (u) 2 u 2 x ≥ γ 2 u 2 x -g(u) + (1 + β)p1 R + * g(u) + f ′′ (u) 2 + (1 -β)p1 R - * g(u) + f ′′ (u) 2 .
(3.15)

We now would like to apply the convolution estimates (3.1). This can be done, provided we have -1 ≤ β ≤ 1. Such additional condition is equivalent to α ≥ 1/2 and this last condition is ensured by the restriction 0 ≤ K ≤ 1 made in the assumptions of Theorem 2.2 -( 1). Now applying estimates (3.1) and reproducing the same calculations as in (3.10) gives

A t (t, x) ≥ γ 2 u 2 x + (α -1) g(u) -m = γ 2 u 2 x -(1 -α)γφ(u) 2 ≥ γ 2 u 2 x -β 2 (u -c) 2 = - γ 2 (AB)(t, x). (3.16)
Similar computations yield the estimate

B t (t, x) ≤ - γ 2 u 2 x + (1 -α) g(u) -m ≤ - γ 2 u 2 x + (1 -α)γφ(u) 2 ≤ γ 2 β 2 (u -c) 2 -u 2 x = γ 2 (AB)(t, x).
(3.17)

By our assumption on the initial datum made in Part (1) of Theorem 2.2,

u ′ 0 (x 0 ) < - 1 2 1 + 8K 2 -1 |u 0 (x 0 ) -c|.
According to the definition of β (3.11), this can be re-expressed as

u ′ 0 (x 0 ) < -β u 0 (x 0 ) -c , or, equivalently, as A(0, x 0 ) > 0 and B(0, x 0 ) < 0. Let τ = sup t ∈ [0, T * ) : A(•, x 0 ) > 0 and B(•, x 0 ) < 0 on [0, t] .
By continuity, τ > 0. If τ < T * , then at least one of the inequalities A(τ, x 0 ) ≤ 0 and B(τ, x 0 ) ≥ 0 must hold true. This contradicts the fact that on the interval [0, τ [, we have (AB)(•, x 0 ) < 0, hence A(τ, x 0 ) ≥ A(0, x 0 ) > 0 by (3.16) and B(τ, x 0 ) ≤ B(0, x 0 ) < 0 by (3.17). Thus τ = T * . Summarizing, we can say that during the whole existence time [0, T * ):

(3.18)        A(•, x 0
) is positive and increasing, B(•, x 0 ) is negative decreasing, AB(•, x 0 ) negative and decreasing.

We are now in the position of proving that T * < ∞: let us consider

h(t) = -(AB)(t, x 0 ).
Computing the time derivative of h, next applying the differential inequalities (3.16) and (3.17) and the geometric-arithmetic mean inequality

A-B 2 (t, x 0 ) ≥ h(t), we get dh dt (t) = - A t B + AB t 2 √ -AB (t, x 0 ) ≥ γ -AB (A -B) 4 √ -AB (t, x 0 ) ≥ γ 2 h 2 (t).
But h(0) = -AB(0, x 0 ) > 0. Hence the solution blows up in finite time and T * < 2 γh(0) . Recalling the definitions of A, B we thus get the estimate for

T * T * ≤ 2 γ u ′ 0 (x 0 ) 2 -β 2 u 0 (x 0 ) 2 -c 2
that agrees with that given in (2.5)

The necessary changes to deal with the conditions of Part (2) of the theorem are slight. First of all, the relation between K and α is now

2K 2 = α -1 α 2 .
instead of (3.9). On the other hand, owing to (3.4), we now have α ≥ 1. Then we can replace estimates (3.8)-(3.10) with

d dt u x (t, q(t, x))] = u tx + f ′ (u)u xx (t, q(t, x)) ≤ - γ 2 u 2 x + (α -1) M -g(u) (t, q(t, x)) ≤ - γ 2 u 2 x + (α -1)γψ(u) 2 ≤ γ 2 (α -1) 2 α 2 (u -c) 2 -u 2 x .
As the coefficient β now given by β := 2K 2 α = 1 2 1 -√ 1 -8K 2 , the required condition -1 ≤ β ≤ 1 does not bring any additional restriction on the Lipschitz constant K. The last part of the proof proceeds in the same manner. Remark 3.2. In some cases, the equality holds in (3.1) and (3.3) for some specific choices of the function u. This happens (e.g. in the case (1)) when the functions g and f are such that

(3.19) 2α 2 (φ ′ ) 2 = (1 -α)f ′′ in all points where φ = 1 γ (g -m) does not vanish.
Indeed, such condition tells us that the discriminants of P and P vanish. In this case, take u such that, at some point x 0 , (3.20)

u x (t, ξ) = αg ′ (u(t,ξ)) f ′′ (u(t,ξ)) , if ξ < x 0 , u x (t, ξ) = -αg ′ (u(t,ξ)) f ′′ (u(t,ξ)) , if ξ > x 0 .
Then we have P (u x (t, x 0 )) = P (u x (t, x 0 )) = 0. This in turn implies that the equality holds at the point x 0 in (3.1). In the particular case of the Camassa-Holm equation,

f (u) = u 2 /2, g(u) = u 2 and α = 1/2. Thus condition (3.19) is satisfied.
Next imposing the continuity of u, the system (3.20) boils down to the differential equations defining the peaked solitons u c (x, t) = ce |x-ct| .

Futher consequences and conclusions

We established the finite time blowup for solutions of the generalized rod equation

u t -u txx + [f (u)] x -[f (u)] xxx + g(u) + f ′′ (u) 2 u 2 x x = 0,
under appropriate conditions on the functions f and g, provided the initial datum u 0 (s > 3/2), satisfies

∃ x 0 ∈ R such that u ′ 0 (x 0 ) < -β u 0 (x 0 ) -c
, where β and c are two real constants depending on the shape of the functions f and g.

In stating Theorem 2.2, we considered for simplicity solutions C([0, T * ), H s (R)) ∩ C 1 ([0, T * ), H s-1 (R)). These are known to uniquely exist provided the initial datum is such that u 0 ∈ H s (R), with s > 3/2. In particular, such solutions are of finite energy and vanish as |x| → ∞.

However, a closer look to the proof of Theorem 2.2 reveals that our arguments go through also in the case of infinite energy solutions, possibly non-vanishing at infinity. Of course, the problem arises of finding a suitable functional setting in order to get the local existence and the uniqueness of this type of solutions. This problem has been successfully addressed in [START_REF] Grunert | Global conservative solutions of the Camassa-Holm for initial data with nonvanishing asymptotics[END_REF], at least in the Camassa-Holm case and for solutions admitting possibly distinct limits as x → ±∞.

Rather than restating Theorem 2.2 relying on well-posedness results more general than those in the H s -setting, let us assume that we are given a priori a solution u ∈ C 1 ([0, T * ), C 2 (R)) of the generalized rod equation, written in its non-local form (2.2), for some 0 < T * ≤ +∞. We need to put also two a priori growth conditions on u:

(i) We first assume that u is such that the integrals in Lemma 3.1 converge (in particular the convolution term in (2.2) makes sense): this is a very mild condition, that would allow us to take into account also unbounded solutions (e.g. solutions with polynomial growth, when g and f are polynomials).

(ii) We assume that the flow map in (3.7) is well defined on the whole time interval [0, T * ): this leads us to restrict ourselves to solutions u such that |f ′ • u| is bounded, uniformly on compact time intervals, by an affine function of the x variable.

The above conditions, in particular, encompass the case of smooth solutions u to (2.2), such that both u(t, •) and u x (t, •) are bounded on R (uniformly with respect to t on compact time intervals). Hence, we obtain the following variant of Theorem 2.2: In particular, if g is constant and u 0 is non-constant, then u blows up in finite time.

Theorem 4.1. Let f, g ∈ C ∞ (R) with f ′′ ≥ γ > 0. Let u ∈ C([0, T * ), W 1,∞ (R)) ∩ C 1 ([0, T * ), C 2 (R)), with 0 < T * ≤ ∞,
The very last statement of Theorem 4.1 generalizes the known fact, [START_REF]Stability of a class of solitary waves in compressible elastic rods[END_REF], that for the classical rod equation in the limit case γ = 3 (that corresponds to g ≡ 0), any nonzero initial data gives rise to a solution that blows up.

The above discussion applies in particular to periodic solutions: hence, the statement of Theorem 2.2 remains valid for solutions u ∈ C([0, T * ), H s (S)) ∩ C 1 ([0, T * ), H s-1 (S)), with s > 3/2, where S denotes the one-dimensional torus. It should be pointed, however, that the estimates of Lemma 3.1 (that are optimal for u ∈ H s (R), at least for a few specific choices of f and g) are no longer optimal when u ∈ H s (S). Improving such estimates would require the application of variational methods. Therefore, we expect that in the case of the torus, the restriction on the Lipschitz constant K appearing in Theorem 2.2 could be relaxed and the estimate on the coefficient β appearing in (2.4) improved. See [START_REF] Brandolese | On permanent and breaking waves in hyperelastic rods and rings[END_REF] for results in this direction in the case of the classical period rod equation.

We finish this paper establishing three simple corollaries of our blowup results. The first corollary establishes a relation between the behavior at the spatial infinity and the blow up. Corollary 4.2. Let f, g ∈ C ∞ (R) with f ′′ ≥ γ > 0 be such that at least one of the two following conditions is satisfied (the maps φ and ψ are as in Theorem 2.2):

(1) min R g = g(c) and φ is K-Lipschitz with 0 ≤ K ≤ 1, or otherwise (2) max R g = g(c) and ψ is K-Lipschitz with 0 ≤ K ≤ 1 √ 8 If u ∈ C([0, T * ), W 1,∞ (R)
) is a smooth solution of the generalized rod equation (2.1) arising from an initial datum u 0 ≡ c such that

where β = 1 2 √ 1 + 8K 2 -1 (in the case (1)), or β = 1 2 1 - √ 1 -8K 2 (in the case (2)
), then u must blow up in finite time.

In particular, a blow up occurs if u 0 ≡ c is such that u 0 (x) = c + o e -β|x| for |x| → ∞.

A result in the same direction as in Corollary 4.2 appeared in [START_REF] Brandolese | Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces[END_REF] in the case of the Camassa-Holm equation (see also [START_REF] Himonas | Persistence properties and unique continuation of solutions of the Camassa-Holm equation[END_REF] for earlier results in the same spirit), where wellposedness issues in weighted spaces were thoroughly discussed. However, the proof given in [START_REF] Brandolese | Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces[END_REF] does not go through for equation (2.2), as it deeply relies on Mc Kean's necessary and sufficient condition for the global existence of solutions of the Camassa-Holm equations.

The second corollary establishes a relation between blow up and sign changes for u 0 -c. Then, for all t ≥ 0, i) Either u(t, x) > c for all x ∈ R, ii) or u(t, x) < c for all x ∈ R, iii) or ∃ x t ∈ R such that u(t, •) ≤ c in (-∞, x t ] and u(t, x) ≥ c in [x t , +∞). In this case, if x → u(t, x) is equal to c at two distinct points of the real line, then x → u(t, x) must be constant = c in the whole interval between them.

Our last corollary is a unique continuation result valid for periodic solutions. Let S = R/Z denote the one-dimensional torus. The local-in-time well posedness result of periodic solutions in C([0, T ), H s (S)) ∩ C 1 ([0, T ), H s-1 (S)) for s > 3/2, together with the mean S u(t) dx = S u 0 dx and the energy S (u 2 + u 2

x )(t) dx = S (u 2 0 + (u 0 ) 2 x ) dx conservation laws, can be established following the steps of [START_REF] Tian | The Cauchy problem for the generalized hyperelastic rod equation[END_REF]. We then conclude that e -βa u(t, a) -c ≤ e -βb u(t, b) -c .

Summarizing, we proved that x → e βx u(x, t)-c is monotone increasing in any interval where u(•, t) ≥ c and x → e -βx u(x, t) -c is monotone increasing in any interval where u(•, t) ≤ c. Corollary 4.2, 4.3 and 4.4 now easily follow from these monotonicity properties. Indeed, if u 0 (x 0 ) > c at some point x 0 ∈ R, then for all x > x 0 , we have u 0 (x) -c > e -β(x-x 0 ) (u 0 (x 0 ) -c), hence lim inf x→+∞ e βx u 0 (x) -c > 0. In the same way, we see that if u 0 (x 0 ) < c, at some point x 0 , then lim sup x→-∞ e -βx u 0 (x) -c < 0. The claim of Corollary 4.2 immediately follows. On the other hand, if we exclude the two situations i) and ii) in Corollary 4.3, then u(t, x t ) = c at some point x t ∈ R; in the interval (-∞, x t ] we must have u(t, x) ≤ c: otherwise, if u -c is strictly positive at some point x ′ < x t , then we would get, for all x ≥ x ′ , u(t, x) -c > e -β(x-x ′ ) (u(t, x ′ ) -c) and in particular u(t, x t ) could not be equal to c. In the same way, we see that we must have u(t, x) ≥ c in the interval [x t , ∞). Such argument proves also that as soon u(t, •) vanishes at two points x t < y t , then u(t, •) must vanish in the whole interval [x t , y t ]. The claim of Corollary 4.3 follows. In the periodic case, if u is a global smooth solution such that u 0 (x 0 ) = c at some point x 0 , then by Corollary 4.3 u 0 ≡ c. Inside the class of energy-preserving solutions, this in turn implies that u ≡ c. Corollary 4.4 is thus established.

In the special case of the Camassa-Holm equation without dispertion (κ = 0), Corollary 4.3 ensures that a sufficient condition for the breakdown is that the initial data satisfy the sign condition ∃ x 1 < x 2 such that u 0 (x 1 ) > 0 > u 0 (x 2 ). This should be compared with the blowup condition of McKean's theorem, that reads: ∃ x 1 < x 2 such that y 0 (x 1 ) > 0 > y 0 (x 2 ), where y 0 = u 0 -(u 0 ) xx is the associated potential. McKean's theorem is more precise, as it provides a necessary and sufficient condition for the wave-breaking of the dispersionless Camassa-Holm equation, see [START_REF] Mckean | Breakdown of the Camassa-Holm equation[END_REF]. However, as already observed, McKean's arguments deeply rely on the persistence properties of the sign of the potential y during the evolution -a byproduct of the bi-hamiltonian structure -that are no longer valid for the general equation (2.2).

e

  βx u 0 (x) -c ≤ 0 and lim sup x→-∞ e -βx u 0 (x) -c ≥ 0,

Corollary 4 . 3 .

 43 Let f, g ∈ C ∞ (R) as in Corollary 4.2 and let u ∈ C([0, ∞), W 1,∞ (R)) be a global smooth solution of the rod equation (2.1).

Corollary 4 . 4 . 2 √ 1 + 1 - √ 1 -

 442111 Under the conditions of Corollary 4.2 for f and g, the identically constant solution u ≡ c (where c = arg min g, or c = arg max g) is the only global, smooth and spatially periodic solution of the generalized rod equation (2.1) with time-independent energy, such that u 0 (x 0 ) = c for some some x 0 ∈ R. Proof of Corollary 4.2, 4.3 and 4.4. If u is a global smooth solution in C([0, ∞), W 1,∞ (R)) of the generalized rod equation (2.1) then, by Theorem 4.1, for all t ≥ 0 and for all x ∈ R we must have u x (t, x) ≥ β|u(t, x) -c|, with β = 1 8K 2 -1 (in the case (1)), or β = 1 2 8K 2 (in the case (2)). If [a, b] is an interval where u(t, x) ≥ c for all x ∈ [a, b], then we have b a βe βx u(t, x) -c dx ≤ b a e βx u x (t, x) dx = e βb u(t, b) -c -e βa u(t, a) -c -β b a e βx ( u(t, x) -c dx. This implies that e βa u(t, a) -c ≤ e βb u(t, b) -c . If u(t, •) ≤ c on [a, b], then applying again Theorem 4.1 we have u x (t, x) ≥ β γ u(t, x)-c . This in turn implies b a βe -βx u(t, x) -c dx ≤ b a e -βx u x (t, x) dx = e -βb u(t, b) -c -e -βa u(t, a) -c + β b a e -βx u(t, x) -c dx.

  be a solution of the Cauchy problem for the generalized rod equation (2.2). Assume that at least one of the conditions (1) or (2) of Theorem 2.2 is fulfilled. T * < ∞ and T * is bounded from above by (2.5).

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. 1