Julie Wagner
email: wagner@telecom-paristech.fr

Mathieu Nancel
email: nancel@lri.fr

Sean Gustafson
email: sean.gustafson@hpi.uni-potsdam.de

Stéphane Huot
email: huot@lri.fr

Wendy E Mackay
email: mackay@lri.fr

A Body-centric Design Space for Multi-surface Interaction

Keywords: Multi-surface interaction, Body-centric design space ACM Classification H, 5, 2, Information Interfaces and Presentation: User Interfaces, Theory and methods

We introduce BodyScape, a body-centric design space that allows us to describe, classify and systematically compare multi-surface interaction techniques, both individually and in combination. BodyScape reflects the relationship between users and their environment, specifically how different body parts enhance or restrict movement within particular interaction techniques and can be used to analyze existing techniques or suggest new ones. We illustrate the use of BodyScape by comparing two free-hand techniques, on-body touch and mid-air pointing, first separately, then combined. We found that touching the torso is faster than touching the lower legs, since it affects the user's balance; and touching targets on the dominant arm is slower than targets on the torso because the user must compensate for the applied force.

INTRODUCTION

Multi-surface environments encourage users to interact while standing or walking, using their hands to manipulate objects on multiple displays. Klemmer et al. [START_REF] Klemmer | How bodies matter: five themes for interaction design[END_REF] argue that using the body enhances both learning and reasoning and this interaction paradigm has proven effective for gaming [START_REF] Shotton | Real-time human pose recognition in parts from single depth images[END_REF], in immersive environments [START_REF] Mine | Moving objects in space: exploiting proprioception in virtual-environment interaction[END_REF], when controlling multimedia dance performances [START_REF] Latulipe | Exploring the design space in technology-augmented dance[END_REF] and even for skilled, hands-free tasks such as surgery [START_REF] Wachs | A gesture-based tool for sterile browsing of radiology images[END_REF].

Smartphones and devices such as Nintendo's Wii permit such interaction via a hand-held device, allowing sophisticated control. However, holding a device is tiring [START_REF] Nancel | Mid-air pan-and-zoom on wall-sized displays[END_REF] and limits the range of gestures for communicating with co-located users, with a corresponding negative impact on thought, understanding, and creativity [START_REF] Goldin-Meadow | Action's influence on thought: the case of gesture[END_REF]. Krueger's VIDEOPLACE [START_REF] Krueger | VIDEOPLACE-an artificial reality[END_REF] pioneered a new form of whole-body interaction in which users stand or walk while pointing to a wall-sized display. Today, off-the-shelf devices like Sony's Eyetoy and Microsoft's However, the human body offers a number of potential targets that vary in size, access, physical comfort, and social acceptance. We are interested in exploring these targets to create more sophisticated body-centric techniques, sometimes in conjunction with hand-held devices, to interact with complex data in multi-surface environments. Advances in sensor and actuator technologies have produced a combinatorial explosion of options, yet, with few exceptions [START_REF] Shoemaker | Body-centric interaction techniques for very large wall displays[END_REF][START_REF] Nancel | Mid-air pan-and-zoom on wall-sized displays[END_REF], we lack clear guidelines on how to combine them in a coherent, powerful way. We argue that taking a body-centric approach, with a focus on the sensory and motor capabilities of human beings, will help restrict the range of possibilities in a form manageable for an interaction designer. This paper introduces BodyScape, a design space that classifies body-centric interaction techniques with respect to multiple surfaces according to input and output location relative to the user. We describe an experiment that illustrates how to use the design space to investigate atomic and compound bodycentric interaction techniques, in this case, compound mid-air interaction techniques that involve pointing on large displays to designate the focus or target(s) of a command. Combining on-body touch with the non-dominant hand and mid-air pointing with the dominant hand is appealing for interacting with large displays: both inputs are always available without requiring hand-held devices. However, combining them into a single, compound action may result in unwanted interaction effects. We report the results of our experiment and conclude with a set of design guidelines for placing targets on the human body depending on simultaneous body movements.

BODYSCAPE DESIGN SPACE & RELATED WORK

Multi-surface environments require users to be "physically" engaged in the interaction and afford physical actions like pointing to a distant object with the hand or walking towards a large display to see more details [START_REF] Beaudouin-Lafon | Multi-surface Interaction in the WILD Room[END_REF]. The body-centric paradigm is well-adapted to device-or eyes-free interaction techniques because they account for the role of the body in the interactive environment. However, few studies and designs take this approach, and most of those focus on large displays [START_REF] Li | Virtual shelves: interactions with orientation aware devices[END_REF][START_REF] Shoemaker | Body-centric interaction techniques for very large wall displays[END_REF][START_REF] Nancel | Mid-air pan-and-zoom on wall-sized displays[END_REF].

Today's off-the-shelf technology can track both the human body and its environment [START_REF] Izadi | KinectFusion: real-time 3D reconstruction and interaction using a moving depth camera[END_REF]. Recent research prototypes also permit direct interaction on the user's body [START_REF] Harrison | Skinput: appropriating the body as an input surface[END_REF][START_REF] Lin | PUB -Point Upon Body: exploring eyes-free interactions and methods on an arm[END_REF] or clothes [START_REF] Karrer | Pinstripe: eyes-free continuous input on interactive clothing[END_REF]. These technologies and interaction techniques suggest new types of body-centric interaction, but it remains unclear how they combine with well-studied, established techniques, such as free-hand mid-air pointing, particularly from the user's perspective.

Although the literature includes a number of isolated point designs, we lack a higher-level framework that characterizes how users coordinate their movements with, around and among multiple devices in a multi-surface environment. Previous models, such as the user action notation [START_REF] Dievendorf | Extending the user action notation (UAN) for specifying interfaces with multiple input devices and parallel path structure[END_REF], separate interaction into asynchronous tasks and analyze the individual steps according to the user's action, interface feedback, and interface internal state. Loke et al. investigated users' input movements when playing an Eyetoy game [START_REF] Loke | Understanding movement for interaction design: frameworks and approaches[END_REF] and analyzed their observations using four existing frameworks. These models do not, however, account for the body's involvement, including potential interaction effects between two concurrent input movements. Our goal is to define a more general approach to body-centric interaction and we propose a design space that: (i) assesses the adequacy of specific techniques to an environment or use context; and (ii) informs the design of novel body-centric interaction techniques. We are aware of only two design spaces that explicitly account for the body during interaction. One focuses on the interaction space of mobile devices [START_REF] Chen | Extending a mobile device's interaction space through body-centric interaction[END_REF] and the other offers a task-oriented analysis of mixed-reality systems [START_REF] Pederson | Towards a model for egocentric interaction with physical and virtual objects[END_REF]. Both consider proximity to the user's body but neither fully captures the distributed nature of multi-surface environments. We are most influenced by Shoemaker et al.'s [START_REF] Shoemaker | Body-centric interaction techniques for very large wall displays[END_REF] pioneering work, which introduced high-level design principles and guidelines for body-centric interaction on large displays.

BodyScape

BodyScape builds upon Card et al.'s morphological analysis [START_REF] Card | A morphological analysis of the design space of input devices[END_REF], focusing on (i) the relationships between the user's body and the interactive environment; (ii) the involvement of the user's body during the interaction, i.e. which body parts are involved or affected while performing an interaction technique; and (iii) the combination of "atomic" interaction techniques in order to manage the complexity of multisurface environments. These in turn were inspired by early research on how people adjust their bodies during coordinated movements, based on constraints in the physical environment or the body's own kinematic structure [START_REF] Massion | Movement, posture and equilibrium: interaction and coordination[END_REF]. They help identify appropriate or adverse techniques for a given task, as well as the impact they may have on user experience and performance, e.g., body movement conflicts or restrictions.

Relationships Between the Body and the Environment

Multi-surface environments distribute user input and system visual output1 on multiple devices (screens, tactile surfaces, handheld devices, tracking systems, on-body sensors, etc.).

The relative location and body positions of the user thus play a central role in the interactions she can perform. For example, touching a tactile surface while looking at a screen on your back is obviously awkward. This physical separation defines the two first dimensions of BodyScape: User Input (D1) and System Visual Output (D2). Using a body-centric perspective similar to [START_REF] Chen | Extending a mobile device's interaction space through body-centric interaction[END_REF][START_REF] Pederson | Towards a model for egocentric interaction with physical and virtual objects[END_REF], we identify two possible cases for input and output: Relative to the body and Fixed in the world. Such body-environment relationships have been considered in Augmented Reality systems [START_REF] Feiner | Windows on the world: 2D windows for 3D augmented reality[END_REF], but never applied to a body-centric description of interaction techniques. D1: Input -User input may be relative to the body and thus performed at any location in the environment, e.g. onbody touch, or fixed in the world, which restricts the location and body position of the user, e.g. standing next to an interactive table. Different technologies offer users greater or lesser freedom of movement. Some interaction techniques require no devices, such as direct interaction with the user's body [START_REF] Lin | PUB -Point Upon Body: exploring eyes-free interactions and methods on an arm[END_REF] or clothes [START_REF] Karrer | Pinstripe: eyes-free continuous input on interactive clothing[END_REF]. Others require a hand-held device, constraining interaction, but not the orientation of the body. Others further restrict movement, such as mid-air pointing at a wall display, in which the user holds the arm, which is tracked in 3d, in a fixed position relative to an external target.

D2: Visual Output -Multi-surface environments are inevitably affected by the separation of visual output over several devices [START_REF] Su | Put them where? towards guidelines for positioning large displays in interactive workspaces[END_REF][START_REF] Tan | Effects of visual separation and physical discontinuities when distributing information across multiple displays[END_REF]. Users must adjust their gaze, switching their attention to output devices that are relevant to the current task by turning the head and -if that is not sufficient -turning the torso, the entire body, or even walking. Visual output relative to the body is independent of the user's location in the environment, e.g., the screen of a hand-held device. It does not constrain the user's location or body position, except if a limb must hold the device. Conversely, visual output fixed in the world requires users to orient the head towards the target's physical location, e.g., where it is projected on a wall. Users' location and body positions are constrained such that they can see the visual output effectively.

The BodyScape design space differentiates between Touchbased and Mid-air user input, since these can affect performance and restrict the position of the body. Body movements and their coordination depends upon the physical connection with the environment [START_REF] Dickstein | Light touch and center of mass stability during treadmill locomotion[END_REF]. For example, Nancel et al. [START_REF] Nancel | Mid-air pan-and-zoom on wall-sized displays[END_REF] showed that touch-based pan-and-zoom techniques are faster on large displays than mid-air gestures because tactile feedback helps guide input movements. Multi-surface environments may add additional constraints, such forcing users to walk to an interactive tabletop in order to touch it.

Body Restriction in the Environment -We define this as a qualitative measure of how a given interaction technique constrains the user's body position, as determined by a combination of the Input and Visual Output dimensions above, from free to restricted (horizontal axis in Fig. 1). The Input dimension clearly restricts body movement more than Visual Output, and Touch is more restrictive than Mid-air gestures, when the input is fixed in the world. For example, one can watch a fixed display from a distance, at different angles, whereas touch input devices require physical proximity. Together, Input and Visual Output dictate the body's remaining degrees of freedom (translation and rotation) available for other potential interactions or body movements. Note that Body Restriction is not necessarily negative. For example, assigning each user their personal display area in a collaborative multi-surface environment restricts their movement, but can prevent common problems that arise with interactive tables [START_REF] Scott | Territoriality in collaborative tabletop workspaces[END_REF] such as visual occlusions, collisions, conflicts and privacy concerns. Figure 1 shows various atomic interaction techniques in terms of their level of body restriction and the total number of involved and affected body parts, and shows how combining them into a compound technique further restricts body movement.

D3: Body Involvement -BodyScape offers a finer grained assessment of body restriction by considering which parts of the user's body are involved in an interaction technique. Every interaction technique involves the body with varying degrees of freedom, from simple thumb gestures on a handheld device [START_REF] Nancel | Mid-air pan-and-zoom on wall-sized displays[END_REF], to whole-body movements [START_REF] Latulipe | Exploring the design space in technology-augmented dance[END_REF]. We define a group of limbs involved in a technique as the involved body parts. For example, most mid-air pointing techniques involve the dominant arm, which includes the fingers and hand, the wrist, the forearm, the upper arm and the shoulder.

A technique may involve a group of limbs and also affect other limbs. For example, on-body touch interaction involves one hand and the attached arm, and the limb touched by the hand is the affected body part. This implies further restrictions on the body, since affected body parts are unlikely to be involved in the interaction and vice versa, especially when interaction techniques are combined. We define five groups of involved body parts: the dominant arm, the non-dominant arm, the dominant leg, the non-dominant leg and the torso.

We omit the head when considering involved and affected body parts, since the location of the visual output is the primary constraint. Although head orientation has been used to improve access to data on large displays [START_REF] De Almeida | Looking behind bezels: french windows for wall displays[END_REF], this is only a "passive" approach in which the system adapts itself to the user's head orientation. [START_REF] Li | Virtual shelves: interactions with orientation aware devices[END_REF]; b) Skinput [START_REF] Harrison | Skinput: appropriating the body as an input surface[END_REF]; c) Body-centric interaction techniques for large displays [START_REF] Shoemaker | Body-centric interaction techniques for very large wall displays[END_REF]; d) PalmRC [START_REF] Dezfuli | PalmRC: imaginary palm-based remote control for eyes-free television interaction[END_REF]; e) Scanning objects with feedback on a device; f) Pick and Drop [START_REF] Rekimoto | Pick-and-drop: a direct manipulation technique for multiple computer environments[END_REF]; g) Mid-air pointing [START_REF] Nancel | Mid-air pan-and-zoom on wall-sized displays[END_REF]; and h) Multitoe [START_REF] Augsten | Multitoe: high-precision interaction with back-projected floors based on high-resolution multi-touch input[END_REF]. Relative Input / Relative Output -The least restrictive combination lets users move freely in the environment as they interact and obtain visual feedback. VirtualShelf [START_REF] Li | Virtual shelves: interactions with orientation aware devices[END_REF] is a mid-air example in which users use the dominant arm to orient a mobile phone within a spherical area in front of them to enable shortcuts (Fig. 2a). Armura [START_REF] Harrison | On-body interaction: armed and dangerous[END_REF] extends this approach with wearable hardware that detects mid-air gestures from both arms and projects visual feedback onto the user's body. Skinput [START_REF] Harrison | Skinput: appropriating the body as an input surface[END_REF] (Fig. 2b) is a touch example that accepts touch input on the user's forearm and provides body-relative visual output from a projector mounted on the shoulder. The dominant arm is involved and the non-dominant arm is affected by the pointing.

Classification of Body-centric Interaction Techniques

Relative Input / Fixed Output -A more restrictive combination constrains the user's orientation and, if the distance to the display matters, the user's location. Shoemaker's [START_REF] Shoemaker | Body-centric interaction techniques for very large wall displays[END_REF] midair technique involves pointing to a body part and pressing a button on a hand-held device to select a command. Visual output consists of the user's shadow projected on the wall with the available commands associated with body locations. Only the pointing arm is involved and users must remain oriented towards the screen (Fig. 2c). PalmRC [START_REF] Dezfuli | PalmRC: imaginary palm-based remote control for eyes-free television interaction[END_REF] (Fig. 2d) allows free-hand touch operations on a TV set. Users press imaginary buttons on their palm [START_REF] Gustafson | Imaginary Phone: learning imaginary interfaces by transferring spatial memory from a familiar dvice[END_REF] and see visual feedback on the fixed TV screen. One arm is involved in the interaction; the other is affected.

Fixed Input / Relative Output -The next most restrictive approach requires users to stand within a defined perimeter, limiting movement. Here, touch is more constrained than mid-air gestures: standing within range of a Kinect device is less restrictive than having to stand at the edge of an interactive table. A simple mid-air example involves a user who scans a barcode while watching feedback on a separate mobile device (Fig. 2e). Pick and Drop [START_REF] Rekimoto | Pick-and-drop: a direct manipulation technique for multiple computer environments[END_REF] uses touch to transfer an object from a fixed surface to a mobile device (Fig. 2f). Both examples involve the dominant arm and affect the non-dominant arm, which carries the handheld device.

Fixed Input / Fixed Output -The most restrictive combination constrains both the user's location and visual attention. A common mid-air technique uses the metaphor of a laser pointer to point to items on a wall-sized display. Although the interaction is performed at a distance, the user must stand at a specified location in order to accurately point at a target on the wall, making it "fixed-in-the-world" (Fig. 2g). Conventional touch interaction on a tabletop or a large display is highly restrictive, requiring the user to stand in a fixed location with respect to the surface. Multitoe [START_REF] Augsten | Multitoe: high-precision interaction with back-projected floors based on high-resolution multi-touch input[END_REF] is even more constrained, since both touch input and visual output appear on the floor, next to the feet (Fig. 2h).

Body Involvement -Figure 1 shows that most body-centric techniques only involve and affect one or two groups of body parts, usually the arms. We know of only a few "whole-body" techniques that involve or affect the entire body: VIDEOPLACE [START_REF] Krueger | VIDEOPLACE-an artificial reality[END_REF] and its successors for games and entertainment and PinStripe [START_REF] Karrer | Pinstripe: eyes-free continuous input on interactive clothing[END_REF], which enables gestures on the users' clothing.

Compound Techniques in Multi-surface Environments

Complex tasks in multi-surface environments combine several interaction techniques: (i) in series, e.g., selecting an object on one touch surface and then another; or (ii) in parallel, e.g., simultaneously touching one object on a fixed surface and another on a handheld device.

Serial Combinationa temporal sequence of interaction techniques. The combined techniques can be interdependent (sharing the same object, or the output of one as the input of the other), but the first action should end before the second starts. For example, the user can select an object on a tactile surface (touch and release) and then apply a function to this object with a menu on a mobile device. Serial compound techniques do not increase the restrictions imposed by each atomic technique in the sequence, nor the involved or affected body parts. However, one must still design serial combinations to avoid awkward movements, such as having to constantly walk back and forth, move a device from one hand to another or repeatedly switch attention between fixed and relative displays.

Parallel Combinationperforming two techniques at the same time. The techniques may be independent or dependent. For example, the user might touch two tactile surfaces simultaneously in order to transfer an object from one to the other [START_REF] Wilson | Combining multiple depth cameras and projectors for interactions on, above and between surfaces[END_REF]. Unlike serial combinations, these compound techniques may significantly restrict the body's movement and raise conflicts between involved and affected body parts.

The constraint on body movement is determined by the more restrictive of the combined techniques. Thus, combining a "fixed-in-the-world" with a "relative-to-the-body" technique will be as restrictive as "fixed-in-the-world". Touchprojector [START_REF] Boring | Touch projector: mobile interaction through video[END_REF] illustrates this well (see Fig. 1). The user uses one device as a lens to select objects on a distant display, orienting it towards the target (mid-air fixed input and fixed output) while simultaneously touching the device's tactile screen to select the target (touch relative input + relative output). Touchprojector is thus considered a "touch fixed input and fixed output" technique in BodyScape. The advantage of minimizing body restrictions with relative-to-thebody technique is overridden by requiring a fixed input. Even so, Touchprojector offers other advantages, since users can interact directly with a remote display without having to move to the display or use another interaction device.

BODYSCAPE EXPERIMENT: COMBINING ON-BODY TOUCH AND MID-AIR POINTING

Our work with users in complex multi-surface environments highlighted the need for interaction techniques that go beyond simple pointing and navigation [START_REF] Beaudouin-Lafon | Multi-surface Interaction in the WILD Room[END_REF]. Users need to combine techniques as they interact with complex data spread across multiple surfaces. The BodyScape design space suggests a number of possibilities for both atomic and compound interaction techniques that we can now compare and contrast.

This section illustrates how we can use the BodyScape design space to look systematically at different types of body-centric interaction techniques, both in their atomic form and when combined into compound interaction techniques. We chose two techniques, illustrated in Figure 2d, ON-BODY TOUCH input, and 2g, MID-AIR POINTING input, both with visual output on a wall display, which is where our users typically need to interact with their data. Although the latter has been well-studied in the literature [START_REF] Nancel | Mid-air pan-and-zoom on wall-sized displays[END_REF], we know little of the performance and acceptability trade-offs involved in touching one's own body to control a multi-surface environment. Because it is indirect, we are particularly interested in on-body touch for secondary tasks such as confirming a selection, triggering an action on a specified object, or changing the scope or interpretation of a gesture. We are also interested in how they compare with each other, since MID-AIR POINTING restricts movement more than ON-BODY TOUCH (Fig. 2g vs. 2d), while ON-BODY TOUCH affects more body parts than MID-AIR POINTING (Fig. 1).

Finally, we want to create compound interaction techniques, so as to increase the size of the command vocabulary and offer users more nuanced control. However, because this involves coordinating two controlled movements, we need to understand any potential interaction effects. The following experiment investigates the two atomic techniques above, which also act as baselines for comparison with a compound technique that combines them. The two research questions we are addressing are thus:

1. Which on-body targets are most efficient and acceptable?

Users can take advantage of proprioception when touching their own bodies, which enables eyes-free interaction and suggests higher performance. However, body targets differ both in the level of motor control required to reach them, e.g., touching a foot requires more balance than touching a shoulder, and in their social acceptability, e.g., touching below the waist [START_REF] Karrer | Pinstripe: eyes-free continuous input on interactive clothing[END_REF].

2. What performance trade-offs obtain with compound bodycentric interaction techniques? Users must position themselves relative to a target displayed on the wall and stabilize the body to point effectively. Simultaneously selecting on-body targets that force shifts in balance or awkward movements may degrade pointing performance. In addition, smaller targets will decrease pointing performance, but may also decrease ON-BODY TOUCH performance.

Method Participants

We recruited sixteen unpaid right-handed volunteers (13 men, average age 28); five had previous experience using a wallsized display. All had good to excellent balance (median 4 on a 5-high Likert scale) and practiced at least one activity that requires balance and body control. All wore comfortable, non-restrictive clothing.

Apparatus

Participants stood in front of a wall-sized display consisting of 32 high-resolution 30" LCD displays laid out in an 8×4 matrix (5.5m ×1.8m) with a total of 20480×6400 pixels (100.63 ppi). Participants wore passive infra-red reflective markers that were tracked in three dimensions by ten VICON cameras with sub-millimeter accuracy at a rate of up to 200 Hz. Markers were mounted on a wireless mouse held in the user's dominant hand to track pointing at a target on the wall, on the index finger of the non-dominant hand to track on-body touches, and on protective sports gear -belt, forearms, shoulders and legs -to track on-body targets. The latter were adjustable to fit over the participants' clothing. VICON data was filtered through the 1Euro filter [START_REF] Casiez | 1e filter: a simple speed-based low-pass filter for noisy input in interactive systems[END_REF].

Based on pilot studies, we defined 18 body target locations distributed across the body (Fig. 3), ranging in size from 9cm on the forearm to 16cm on the lower limbs, depending upon location and density of nearby targets, grouped as follows: Dominant Arm: 4 targets on dominant arm (D ARM = upper arm, elbow, forearm, wrist) Dominant Upper Body: 4 targets on dominant side of upper body (D UPPER = thigh, hip, torso, shoulder) Non-dominant Upper Body: 4 targets on non-dominant side of upper body (ND UPPER = thigh, hip, torso, shoulder) Dominant Lower Leg: 3 targets on dominant side of lower leg (D LOWER = knee, tibia, foot) Non-dominant Lower Leg: 3 targets on non-dominant side of lower leg (ND LOWER = knee, tibia, foot)

In ON-BODY TOUCH conditions, participants wore an IR tracked glove on the non-dominant hand with a pressure sensor in the index finger. The system made an orthogonal projection from the index finger to the touched limb segment using a skeletonbased model to calculate the closest body target.

Wall pointing tasks varied in difficulty from easy (diameter of the circular target was 1200px or 30cm) to medium (850px or 21.25cm) to hard (500px or 12.5cm). Wall targets were randomly placed 4700px (117.5cm) from the starting target.

Data Collection

We collected timing and error data for each trial, as follows: TRIAL TIME: from trial start to completion. POINTING REACTION TIME: from appearance of on-screen target to cursor displacement of more than 1000px. POINTING MOVEMENT TIME: from initial cursor movement to entry into goal target. CURSOR READJUSTMENT TIME: from leaving goal target to relocating cursor onto goal target. BODY REACTION TIME: from appearance of trial stimulus to leaving starting position. BODY POINTING TIME: from leaving start position to touching on-body target. BODY ERRORS: number of incorrect touches detected on body target2 ; includes list of incorrect targets per error.

We debriefed participants at the end of the experiment and asked them to rank on a Likert scale: (i) perceived comfort of each body target according to each MID-AIR POINTING condition ('1=very uncomfortable' to '5=very comfortable'); and (ii) social acceptability of each on-body target:"Would you agree to touch this body target in a work environment with colleagues in the same room?" ('1=never' to '5=certainly').

Procedure

Each session lasted about 60 minutes, starting with a training session, followed by blocks of trials of the following conditions, counter-balanced across subjects using a Latin square. TASK: Participants were asked to perform trials as quickly and accurately as possible. They were asked to point and select on-body targets using their non-dominant hand's index finger in the BODY ONLY condition, and to point and select walltargets using a mouse device held in the dominant hand in the POINTING ONLY condition. The compound POINTING+BODY condition asked users to point to the wall-target and keep the cursor inside before selecting an on-body target.

BODY ONLY (Fig. 4b): The starting position involves standing comfortably facing the wall display, with the non-dominant hand at the thigh (Fig. 4a). The trial begins when an image of a body silhouette appears on the wall, with a red circle indicating the location of the on-body target to acquire. The participant touches that target with the index finger of the non-dominant hand as quickly and accurately as possible.

Participants were asked to avoid crouching or bending their bodies, which forced them to lift their legs to reach lower-leg targets. The trial ends only when the participant selects the correct target; all intermediate incorrect selections are logged.

Figure 5 shows how different body parts interact for different on-body targets. The non-dominant arm is always involved, since it is responsible for pointing at the target. However, some on-body targets also affect other body parts, which may have adverse effects, such as shifting one's balance to touch the foot (Fig. 5c).

POINTING ONLY (Fig. 4c): The starting position involves standing comfortably facing the wall display and using the dominant hand to locate a cursor within a circular target displayed in the center of the wall. The trial begins when the starting target disappears and the goal target appears between 0.5s and 1s later, to reduce anticipatory movements and learning effects. The participant moves the dominant hand to move the cursor to the goal target and selects by pressing the left button of the mouse bearing the optical marker used for pointing. The trial ends only when the participant successfully clicks the mouse button while the cursor is inside the goal target. POINTING+BODY (Fig. 4d): The starting position combines the above, with the non-dominant hand at the thigh and the dominant hand pointing to the starting target on the wall. The trial begins with the appearance of a body-target illustration and the goal target on the wall display. The participant first points the cursor at the goal target, then completes the trial by touching the designated on-body target. The trial ends only when the on-body touch occurs while the cursor is inside the goal target on the wall. As in the BODY ONLY condition, multiple body parts may be involved, sometimes with adverse effects. Fig. 5e shows the interaction between the dominant arm, which is trying to point to a target on the wall and the non-dominant arm, which is pointing at the dominant arm.

Training

Participants began by calibrating the system to their bodies, visually locating, touching and verifying each of the 18 body targets. They were then exposed to three blocks of six BODY ONLY trials, with the requirement that they performed two onbody touches in less than five seconds. They continued with three additional blocks to ensure they could accurately touch each of the targets. Next, they were exposed to all three levels of difficulty for the POINTING ONLY condition: easy, medium and hard, in a single block. Finally, they performed three additional blocks of the compound POINTING+BODY technique.

Results

Q1: Efficiency & acceptability of on-body targets

Our first research question asks which on-body targets are most efficient and which are socially acceptable. We conducted a full factorial ANOVA on the BODY ONLY condition, with PARTICIPANT as a random variable based on the standard repeated measures (REML) technique from the JMP 9 statistical package. We found no fatigue or learning effects.

Figure 6 shows the times for touching all 18 on-body targets, grouped into the five body areas. We found significant effects of BODY TARGET on BODY POINTING TIME: touching lower body targets is slower. Since BODY POINTING TIME is consistent for targets within a given target group, we report results according to target group, unless otherwise stated.

Overall, we found a main effect of BODY TARGET GROUP on TRIAL TIME (F 4,60 = 21.2, p < 0.0001). A post-hoc Tukey test revealed two significantly different groups: body targets located on the upper torso required less than 1400ms to be touched whereas targets on the dominant arm and on the lower body parts required more than 1600ms. Results are similar for BODY POINTING TIME with a significant effect of BODY TARGET GROUP only for the D UPPER group (F 3,45 = 5.07, p = 0.004), specifically, targets on the dominant thigh are touched more slowly than those on the shoulder or torso. For BODY REACTION TIME, despite a significant effect, values are very close for each BODY TARGET GROUP (530ms ± 20ms).

Participants were able to quickly touch on-body targets with an accuracy of 92.4% on the first try. A post-hoc Tukey test showed that targets on the dominant arm were more prone to errors than other body target areas (14.8% vs. 6% for dominant and non-dominant upper body and 2.9% for nondominant lower body targets). Most errors obtained when targets were close to each other, i.e. when the participant's hand touched the boundary between the goal and a nearby target or when the dominant arm was held close to the torso, making it difficult to distinguish between the torso and arm targets. Touching lower body parts is, not surprisingly, slower, since these targets are further from the starting position and require more complex movements. However, the difference is small, about 200ms or 12% of global trial time.

Qualitative measures of Preference and Social Acceptance

Figure 7a shows that participants' preferences (median values of Likert-scale) for and within each BODY TARGET GROUP were consistent with performance measures: targets on the upper body parts were preferred over lower body parts (consistent with [START_REF] Karrer | Pinstripe: eyes-free continuous input on interactive clothing[END_REF]) and the torso were slightly more preferred than on the dominant arm.

Interestingly, preferences for non-dominant foot and the dominant arm decrease when on-body touch interaction is combined with mid-air pointing (Fig. 7b). The latter is surprising, given that the most popular location for on-body targets in the literature is on the dominant arm. This suggests that interaction designers should explore alternative on-body targets as well. Social acceptability varies from highly acceptable (upper body) to unacceptable (lower body) (Fig. 7c).

Q2: Performance Trade-offs for compound techniques

The second research question examines the effect of combining two atomic interaction techniques, in this case BODY ONLY and POINTING ONLY, into a single compound technique. We treat these atomic techniques as baseline values to help us better evaluate the compound task.

Pointing Only task

Not surprisingly, hard pointing tasks are significantly slower (TRIAL TIME of 1545ms avg., F 2,30 = 40.23, p < 0.0001) than medium (1216ms) or easy (1170ms) tasks, which are not significantly different from each other (Fig. 8a). POINT-ING REACTION TIME is also significantly slower for difficult (498ms) as opposed to medium (443ms) or easy (456ms) tasks. POINTING MOVEMENT TIME is significantly different for all three levels of difficulty: hard (708ms), medium (511ms) and easy (435ms).

Participants made few errors but occasionally had to relocate the cursor inside the goal target before validating the selection with the mouse. This occurred rarely (1.8% of all trials), but was significantly more likely for difficult pointing tasks (15%) (F 2,30 = 8.02, p = 0.0016) and accounts for the differences in TRIAL TIME and POINTING MOVEMENT TIME.

Compound Pointing plus Body task

Figure 8b shows that the combined MID-AIR POINTING and ON-BODY TOUCH task is significantly slower than MID-AIR POINTING alone for all levels of difficulty. TRIAL TIME is significantly slower for difficult MID-AIR POINTING (2545ms) than both medium (1997ms) and easy (1905ms) tasks. In fact, the easiest compound task is significantly slower that the hardest POINTING ONLY task.

BODY TARGET GROUP also has an effect on TRIAL TIME (F groups). The main effect of MID-AIR POINTING is also similar to those observed before, showing that difficult pointing tasks make simultaneous body touching slower than medium or easy pointing task. Obviously, these are all significantly slower than the BODY ONLY baseline.

More interesting, the MID-AIR POINTING×BODY TARGET GROUP interaction effect reveals the actual impact of MID-AIR POINT-ING on ON-BODY TOUCH. As shown in Figure 11: (i) the increasing difficulty of the pointing task increases BODY POINTING TIME. In fact, despite the fact that our task required body target selection as the final action, the reaction times indicate that both tasks start almost simultaneously (ON-BODY TOUCH even before MID-AIR POINTING). (ii) The increase in difficulty does not change the difference between the groups of targets, but rather amplifies them. ND UPPER and D UPPER remain the groups of targets that require less time to be touched.

In summary, the compound POINTING+BODY task involves interaction effects between the two atomic techniques, which not only incur a time penalty when the tasks are performed simultaneously, but also degrades pointing performance for MID-AIR POINTING (fixed in the world) when combined with a body-relative technique that involves and affects multiple limbs. However, our results also reveal that ON-BODY TOUCH on the lower parts of the body significantly impair the movement phase of pointing, and that the overall negative impact increases with the difficulty of the pointing task, especially when targeting on the pointing arm.

CONCLUSION

The BodyScape design space uses a body-centric approach to classify both existing and novel interaction techniques. The distributed nature of multi-surface environments highlights the need for combining interaction techniques, in series or in parallel, to accomplish more complex tasks. A bodycentric approach can help predict possible interaction effects of body movements by (i) analyzing the spatial body-device relationship and (ii) proposing ways to decompose individual techniques into groups of body parts that are either involved in or affected by the interaction. We argue that studying compound interaction techniques from a body-centric perspective will lead to powerful guidelines for interaction design, both with and without physical devices.

We illustrate BodyScape by examining the effects of combining two multi-surface interaction techniques: mid-air pointing and on-body touch. This novel combination enables an eyes-free interaction with on-body targets while offering a rich set of mid-air pointing commands to access a remote virtual target on a large display. We ran a controlled experiment to study both techniques individually and in combination, investigating performance and acceptability of 18 on-body targets, as well as any interaction effects that obtain when the two techniques are combined. Participants were most efficient with targets on the torso and least efficient with targets on the lower body and on the dominant arm, especially in the combined condition: Reaching targets on the lower legs requires additional balance and touching the dominant arm impairs the precision of mid-air pointing because of the force applied on the pointing arm. Users consistently preferred targets located on the upper body.

Our results suggest three guidelines for designing on-body interaction: G1 Task difficulty: On-body targets should be placed on stable body parts, such as the upper torso, when tasks require precise or highly coordinated movements. G2 Body balance: Anticipatory movements, such as shifts in balance, can be detected to accommodate corresponding perturbations in a primary task, e.g. freezing an on-screen cursor. The precision of a pointing task can be adversely affected if users must also touch on-body targets that require a shift in balance or coordination, in particular, touching the dominant arm while it is performing a separate task. G3 Interaction effects: Designers should consider which body parts negatively affect users' comfort while touching on-body targets as well as side effects of each task, such as reduced attention or fatigue that may lead to unexpected body positions or increases in errors.

Future work will develop more detailed descriptions of each limb's involvement in the interaction. We also plan to increase the predictability of BodyScape, following Card et al. [START_REF] Card | A morphological analysis of the design space of input devices[END_REF], such as developing a Fitts-style pointing model for on-body touch.

Figure 1 .

 1 Figure 1. Atomic body-centric interaction techniques in BodyScape, according to the level of Body Restriction in the Environment and number of Involved and Affected limbs. Compound techniques (colored background) are linked to their component atomic techniques.

Figure 2 .

 2 Figure 2. BodyScape presents a taxonomy of atomic body-centric interaction techniques, organized according to Input and Visual Output. a) Virtual Shelves[START_REF] Li | Virtual shelves: interactions with orientation aware devices[END_REF]; b) Skinput[START_REF] Harrison | Skinput: appropriating the body as an input surface[END_REF]; c) Body-centric interaction techniques for large displays[START_REF] Shoemaker | Body-centric interaction techniques for very large wall displays[END_REF]; d) PalmRC[START_REF] Dezfuli | PalmRC: imaginary palm-based remote control for eyes-free television interaction[END_REF]; e) Scanning objects with feedback on a device; f) Pick and Drop[START_REF] Rekimoto | Pick-and-drop: a direct manipulation technique for multiple computer environments[END_REF]; g) Mid-air pointing[START_REF] Nancel | Mid-air pan-and-zoom on wall-sized displays[END_REF]; and h) Multitoe[START_REF] Augsten | Multitoe: high-precision interaction with back-projected floors based on high-resolution multi-touch input[END_REF].

Figure 2

 2 Figure 2 lays out atomic body-centric interaction techniques from the literature along the Input and Visual Output dimensions, illustrating their impact on body restrictions in the environment. Each technique involves performing an elementary action, e.g. moving a cursor or selecting a target.

Figure 3 .

 3 Figure 3. 18 body targets are grouped into five categories.

Figure 4

 4 Figure 4. a) Starting position b) BODY ONLY c) POINTING ONLY d) POINTING+BODY Starting position: non-dominant hand at the hip and/or dominant hand points to a starting target on the wall display. BODY ONLY and POINTING ONLY are atomic conditions; POINTING+BODY is compound: a body touch triggers the selected wall target.the location of body targets randomized and no two successive trials involved the same body target group. POINTING ONLY trials were organized into blocks of five and all wall pointing trials were counterbalanced across difficulty. The two atomic interaction techniques, BODY ONLY and POINTING ONLY serve as baseline comparisons for performance with the compound interaction technique, POINTING+BODY.

Figure 5 .

 5 Figure 5. Body parts involved when touching the (a) torso, (b) arm, (c) leg; (d) mid-air pointing; and (e) in parallel, when the dominant hand points in mid-air and non-dominant hand touches the dominant arm.

Figure 6 .

 6 Figure 6. Mean BODY POINTING TIME is faster for both upper body targets (D UPPER and ND UPPER) compared to other targets. Horizontal lines indicate group means; performance within groups is consistent.

Figure 7 .

 7 Figure 7. Median preference and acceptability rankings of on-body targets (from green = acceptable to red = unacceptable).

Figure 8 .Figure 9 .

 89 Figure 8. TRIAL TIME for (a) Pointing Only and (b) Pointing + Body, by pointing difficulty.

Figure 11 .

 11 Figure 11. Interaction Pointing×Body on BODY POINTING TIME.

  4,60 = 34.1, p < 0.0001) with the same significant groups as for BODY ONLY: TRIAL TIME is significantly faster when touching upper body targets (ND UPPER = 1794ms, D UPPER = 1914ms) than lower body targets (ND LOWER = 2267ms, D LOWER = 2368ms) or the dominant arm (D ARM = 2401ms). BODY REACTION TIME is faster than POINTING REACTION TIME, regardless of pointing difficulty.Although we can see that the individual techniques are both more efficient than the compound technique, the question is why? Just how does ON-BODY TOUCH affect MID-AIR POINTING? Figure9shows interaction effects between the two elements of the compound tasks, by both BODY TARGET GROUP and pointing difficulty. While POINTING MOVEMENT TIME is close to the pointing baseline for all difficulties when MID-AIR POINTING is Figure 10. Effect of Pointing difficulty and BODY TARGET GROUP on CURSOR READJUSTMENT TIME.combined with ON-BODY TOUCH on the upper body parts, we observe a stronger negative effect for the lower body parts and the dominant arm, especially for difficult pointing tasks. This impact of ON-BODY TOUCH on the MID-AIR POINTING task does not only relate to the movement phase but also cursor readjustments. For the combined POINTING+BODY task, 31% of the trials required the participants to relocate the cursor inside of the target before validating the selection with a body touch, compared to only 6% for POINTING ONLY. Thus, we found significant effects of MID-AIR POINTING (F 2,30 = 59.64, p < 0.0001), BODY TARGET GROUP (F 5,75 = 23.03, p < 0.0001) and MID-AIR POINTING×BODY TARGET GROUP (F 10,150 = 8.45, p < 0.0001) on CURSOR READJUSTMENT TIME. As shown in Figure10, CURSOR READJUSTMENT TIME increases significantly for each level of difficulty of MID-AIR POINTING but selecting body targets on some BODY TARGET GROUP, especially in D LOWER and D ARM, affects the body configuration and requires even more time to relocate the cursor inside of the on-screen target. The main effect of BODY TARGET GROUP is similar to the baseline (with ND UPPER and D UPPER significantly faster than all other

	Mean of Body Pointing Time (ms)	500 1000 1500 2000 2500	Pointing Task	none	easy	medium difficult	100 200 300 400 500 600 700 800 900 1000 1100 Mean of Cursor Readjustment Time (ms) 1200 1300	Pointing task	easy	medium difficult
		0	NDupper Dupper NDlower Dlower	Darm	0	none NDupper Dupper NDlower Dlower	Darm
				Body Target Group				Body Target Group
								This result reveals two important things: (1) touching the
								dominant arm while pointing affects the precision of pointing
								and requires "force-balance" (targets on D ARM); (2) touching
								targets on the lower body parts affects the precision of point-
								ing and requires "movement-balance" (targets on ND LOWER
								and D LOWER). Overall, since the impact of both D LOWER and
								D ARM is similar, we observe that maintaining force-balance
								is as difficult as maintaining movement-balance during the
								pointing task, and that the difficulty in movement-balance is
								not only caused by standing on one leg, but by simultane-
								ously crossing the body's sagittal plane (difference between
								D LOWER and ND LOWER).
								Similarly, we studied the effect of MID-AIR POINTING on ON-
								BODY TOUCH by performing an ANOVA with the model MID-
								AIR POINTING[none /easy/medium/difficult]×BODY TARGET GROUP. We
								did not find any effect on BODY REACTION TIME. On BODY
								POINTING TIME, we did find a significant effect of BODY TARGET

GROUP (F 4,60 = 38.69, p < 0.0001), of MID-AIR POINTING (F 3,45 = 78.15, p < 0.0001) and a significant MID-AIR POINTING×BODY TARGET GROUP interaction (F 12,180 = 2.28, p = 0.01).

We do not consider auditory feedback since sound perception does not depend upon body position, except in environments featuring finely tuned spatial audio.

Includes both system detection and user errors.

ACKNOWLEDGEMENTS

We wish to thank the participants for their time and effort, as well as the anonymous reviewers for their helpful comments.


Stéphane Gerbi
email: stephane.gerbi@univ-savoie.fr

Belkacem Said-Houari
email: saidhouarib@yahoo.fr

Global existence and exponential growth for a viscoelastic wave equation with dynamic boundary conditions

Keywords: Damped viscoelastic wave equations, global solutions, exponential growth, blow up in finite time, dynamic boundary conditions

The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time.

Introduction

We consider the following problem

                 u tt -∆u -α∆u t + t 0 g(t -s)∆u(s)ds = |u| p-2 u,
x ∈ Ω, t > 0 , u(x, t) = 0, x ∈ Γ 0 , t > 0 ,

u tt (x, t) = - ∂u ∂ν (x, t) - t 0 g(t -s) ∂u ∂ν (x, s)ds + α∂u t ∂ν (x, t) + h (u t ) x ∈ Γ 1 , t > 0 , u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) x ∈ Ω , (1)
where u = u(x, t) , t ≥ 0 , x ∈ Ω , ∆ denotes the Laplacian operator with respect to the x variable, Ω is a regular and bounded domain of R N , (N ≥ 1), ∂Ω = Γ 0 ∪ Γ 1 , mes(Γ 0 ) > 0, Γ 0 ∩ Γ 1 = ∅ and ∂/∂ν denotes the unit outer normal derivative, α is a positive constant, p > 2, h and g are functions whose properties will be discussed in the next section, u 0 , u 1 are given functions. Nowadays the wave equation with dynamic boundary conditions are used in a wide field of applications. See [START_REF] Meurer | Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator[END_REF] for some applications. Problems similar to (1) arise (for example) in the modeling of longitudinal vibrations in a homogeneous bar in which there are viscous effects. The term ∆u t , indicates that the stress is proportional not only to the strain, but also to the strain rate, see [START_REF] Caroll | Singular and Degenerate Cauchy Problems[END_REF] fore more details.

From the mathematical point of view, these problems do not neglect acceleration terms on the boundary. Such type of boundary conditions are usually called dynamic boundary conditions. They are not only important from the theoretical point of view but also arise in several physical applications. For instance in one space dimension and for g = 0, problem (1) can modelize the dynamic evolution of a viscoelastic rod that is fixed at one end and has a tip mass attached to its free end. The dynamic boundary conditions represents Newton's law for the attached mass, (see [START_REF] Budak | A collection of problems on mathematical physics[END_REF][START_REF] Andrews | Second order evolution equations with dynamic boundary conditions[END_REF][START_REF] Conrad | Stabilization of a flexible beam with a tip mass[END_REF] for more details). In the two dimension space, as showed in [START_REF] Goldstein | Derivation and physical interpretation of general boundary conditions[END_REF] and in the references therein, these boundary conditions arise when we consider the transverse motion of a flexible membrane Ω whose boundary may be affected by the vibrations only in a region. Also some dynamic boundary conditions as in problem (1) appear when we assume that Ω is an exterior domain of R 3 in which homogeneous fluid is at rest except for sound waves. Each point of the boundary is subjected to small normal displacements into the obstacle: this type of dynamic boundary conditions are known as acoustic boundary conditions, see [START_REF] Beale | Spectral properties of an acoustic boundary condition[END_REF] for more details.

Littman and Markus [START_REF] Littman | Stabilization of a hybrid system of elasticity by feedback boundary damping[END_REF] considered a system which describe an elastic beam, linked at its free end to a rigid body. The whole system is governed by the Euler-Bernoulli Partial Differential Equations with dynamic boundary conditions. They used the classical semigroup methods to establish existence and uniqueness results while the asymptotic stabilization of the structure is achieved by the use of feedback boundary damping.

In [START_REF] Grobbelaar-Van Dalsen | On fractional powers of a closed pair of operators and a damped wave equation with dynamic boundary conditions[END_REF] the author introduced the model

u tt -u xx -u txx = 0, x ∈ (0, L), t > 0, (2)
which describes the damped longitudinal vibrations of a homogeneous flexible horizontal rod of length L when the end x = 0 is rigidly fixed while the other end x = L is free to move with an attached load. Thus she considered Dirichlet boundary condition at x = 0 and dynamic boundary conditions at x = L , namely

u tt (L, t) = -[u x + u tx ] (L, t), t > 0 . (3)
By rewriting the whole system within the framework of the abstract theories of the so-called Bevolution theory, the existence of a unique solution in the strong sense has been shown. An exponential decay result was also proved in [START_REF] Grobbelaar-Van Dalsen | On the initial-boundary-value problem for the extensible beam with attached load[END_REF] for a problem related to (2)-(3), which describe the weakly damped vibrations of an extensible beam. See [START_REF] Grobbelaar-Van Dalsen | On the initial-boundary-value problem for the extensible beam with attached load[END_REF] for more details. Subsequently, Zang and Hu [START_REF] Zhang | Energy decay for a nonlinear viscoelastic rod equations with dynamic boundary conditions[END_REF], considered the problem

u tt -p (u x ) xt -q (u x ) x = 0, x ∈ (0, 1) , t > 0 with u (0, t) = 0, p (u x ) t + q (u x ) (1, t) + ku tt (1, t) = 0, t ≥ 0.
By using the Nakao inequality, and under appropriate conditions on p and q, they established both exponential and polynomial decay rates for the energy depending on the form of the terms p and q.

Recently, the present authors have considered, in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF] and [START_REF] Gerbi | Asymptotic stability and blow up a semilinear damped wave equation with dynamic boundary conditions[END_REF], problem (1) with g = 0 and a nonlinear boundary damping of the form h (u t ) = |u t | m-2 u t . A local existence result was obtained by combining the Faedo-Galerkin method with the contraction mapping theorem. Concerning the asymptotic behavior, the authors showed that the solution of such problem is unbounded and grows up exponentially when time goes to infinity provided that the initial data are large enough and the damping term is nonlinear. The blow up result was shown when the damping is linear (i.e. m = 2). Also, we proved in [START_REF] Gerbi | Asymptotic stability and blow up a semilinear damped wave equation with dynamic boundary conditions[END_REF] that under some restrictions on the exponents m and p, we can always find initial data for which the solution is global in time and decays exponentially to zero. These results had been recently generalized for a wide range of nonlinearities in the equation and in the boundary term: the authors proved the local existence and uniqueness by a sophisticated application of the non linear semigroup theory, see [START_REF] Graber | Existence and asymptotic behavior of the wave equation with dynamic boundary conditions to appear in[END_REF].

In the absence of the strong damping α∆u t and for Dirichlet boundary conditions on the whole boundary ∂Ω, the question of blow up in finite time of problem (1) has been investigated by many authors. Messaoudi [START_REF] Messaoudi | Blow up and global existence in nonlinear viscoelastic wave equations[END_REF] showed that if the initial energy is negative and if the relaxation function g satisfies the following assumption

∞ 0 g(s)ds < (p/2) -1 (p/2) -1 + (1/2p) , (4)
then the solutions blow up in finite time. In fact this last condition has been assumed by other researchers. See for instance [START_REF] Kafini | A blow-up result for a viscoelastic system in R n[END_REF][START_REF] Kafini | A blow-up result in a Cauchy viscoelastic problem[END_REF][START_REF] Messaoudi | Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms[END_REF][START_REF] Messaoudi | Blow up in a nonlinearly damped wave equation[END_REF][START_REF] Song | Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal[END_REF][START_REF] Yu | Blow up for a Cauchy viscoelastic problem with a nonlinear dissipation of cubic convolution type[END_REF].

The main goal of this paper is to prove the local existence and to study the asymptotic behavior of the solution of problem [START_REF] Adams | Sobolev spaces[END_REF].

One of the main questions is to show a blow-up result of the solution. This question is a difficult open problem, since in the presence of the strong damping term, i.e. when α = 0, the problem has a parabolic structure, which means that the solution gains more regularity. However, in this paper, we give a partial answer to this question and show that for α = 0 and for large initial data, the solution is unbounded and grows exponentially as t goes to infinity. While for the case α = 0, the solution has been shown to blow up in finite time.

The main contribution of this paper in this blow up result is the following: the exponential growth and blow-up results hold without making the assumption (4). In fact the only requirement is that the exponent p has to be large enough which is a condition much weaker than condition (4). Moreover, unlike in the works of Messaoudi and coworkers, we do not assume any polynomial structure on the damping term h(u t ), to obtain an exponential growth of the solution or a blow up in finite time.

This paper is organized as follows: firstly, applying the Faedo-Galerkin method combined with the fixed point theorem, we show, in Section 2, the existence and uniqueness of a local in time solution. Secondly, under the smallness assumption on the initial data, we show, in Section 3, that the solution continues to exist globally in time. On the other hand, in Section 4, we prove that under some restrictions on the initial data and if the interior source dominates the boundary damping then the L p -norm of the solution grows as an exponential function. Lastly, in Section 5, we investigate the case when α = 0 and we prove that the solution ceases to exist and blows up in finite time.

Preliminary and local existence

In this section, we introduce some notations used throughout this paper. We also prove a local existence result of the solution of problem [START_REF] Adams | Sobolev spaces[END_REF].

We denote

H 1 Γ 0 (Ω) = u ∈ H 1 (Ω)/ u Γ 0 = 0 .
By (., .) we denote the scalar product in L 2 (Ω) i.e. (u, v)(t) = Ω u(x, t)v(x, t)dx. Also we mean by . q the L q (Ω) norm for 1 ≤ q ≤ ∞, and by . q,Γ 1 the L q (Γ 1 ) norm.

Let T > 0 be a real number and X a Banach space endowed with norm . X . L p (0, T ; X), 1 ≤ p < ∞ denotes the space of functions f which are L p over (0, T ) with values in X, which are measurable and f X ∈ L p (0, T ). This space is a Banach space endowed with the norm

f L p (0,T ;X) = T 0 f p X dt 1/p .
L ∞ (0, T ; X) denotes the space of functions f : ]0, T [ → X which are measurable and f X ∈ L ∞ (0, T ). This space is a Banach space endowed with the norm:

f L ∞ (0,T ;X) = ess sup 0<t<T f X .
We recall that if X and Y are two Banach spaces such that X ֒→ Y (continuous embedding), then

L p (0, T ; X) ֒→ L p (0, T ; Y ) , 1 ≤ p ≤ ∞.
We will also use the embedding (see [START_REF] Adams | Sobolev spaces[END_REF]Therorem 5.8]):

H 1 Γ 0 (Ω) ֒→ L p (Ω), 2 ≤ p ≤ p where p =    2N N -2 if N ≥ 3, +∞ if N = 1, 2 ,
and also

H 1 Γ 0 (Ω) ֒→ L q (Γ 1 ), 2 ≤ q ≤ q where q =    2(N -1) N -2 if N ≥ 3, +∞ if N = 1, 2.
For 2 ≤ m ≤ q, let us denote V = H 1 Γ 0 (Ω) ∩ L m (Γ 1 ). We assume that the relaxation functions g is of class C 1 on R and satisfies:

∀ s ∈ R , g (s) ≥ 0, and 1 - ∞ 0 g (s) ds = l > 0 . (5)
Moreover, we suppose that:

∀ s ≥ 0 , g ′ (s) ≤ 0. ( 6
)
The hypotheses on the function h are the following:

(H1) h is continuous and strongly monotone, i.e. for 2 ≤ m ≤ q, there exists a constant

m 0 > 0 such that (h(s) -h(v)) (s -v) ≥ m 0 |s -v| m , (7)
(H2) there exist two positive constants c m and C m such that

c m |s| m ≤ h(s)s ≤ C m |s| m , ∀s ∈ R . (8)
For a function u ∈ C [ 0, T ], H 1 Γ 0 (Ω) , let us introduce the following notation:

(g ⋄ u) (t) = t 0 g (t -s) ∇u (s) -∇u (t) 2 2 ds. Thus, when u ∈ C [ 0, T ], H 1 Γ 0 (Ω) ∩C 1 [ 0, T ], L 2 (Ω) such that u t ∈ L 2 0, T ; H 1 Γ 0 (Ω) , we have: d dt (g ⋄ u) (t) = t 0 g ′ (t -s) ∇u (s) -∇u (t) 2 2 ds + d dt ∇u (t) 2 2 t 0 g (s) ds -2 Ω t 0 g (t -s) ∇u (s) ∇u t (t) dsdx = g ′ ⋄ u (t) -2 Ω t 0 g (t -s) ∇u (s) ∇u t (t) dsdx (9)
+ d dt ∇u (t) 2 2 t 0 g (s) ds -g (t) ∇u (t) 2 2 .
This last identity implies:

Ω t 0 g (t -s) ∇u (s) ∇u t (t) dsdx = 1 2 g ′ ⋄ u (t) + 1 2 d dt ∇u (t) 2 2 t 0 g (s) ds - 1 2 g (t) ∇u (t) 2 2 - 1 2 d dt (g ⋄ u) (t) . (10)
For

u ∈ C [ 0, T ], H 1 Γ 0 (Ω) ∩C 1 [ 0, T ], L 2 (Ω) such that u t ∈ L 2 0, T ; H 1 Γ 0 (Ω)
, let us define the modified energy functional E by:

E (t, u, u t ) = E (t) = 1 2 u t (t) 2 2 + 1 2 u t (t) 2 2,Γ 1 + 1 2 1 - t 0 g (s) ds ∇u (t) 2 2 + 1 2 (g ⋄ u) (t) - 1 p u (t) p p . (11)
The following local existence result of the solution of problem ( 1) is closely related to the one we have proved for a slightly different problem in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF]Theorem 2.1], where no memory term was present. Let us sate it: Theorem 2.1. Assume that ( 5), ( 6) and ( 7) hold. Let 2 ≤ p ≤ q and max 2, q q+1-p ≤ m ≤ q. Then given u 0 ∈ H 1 Γ 0 (Ω) and u 1 ∈ L 2 (Ω), there exists T > 0 and a unique solution u of the problem (1) on (0, T ) such that

u ∈ C [ 0, T ], H 1 Γ 0 (Ω) ∩C 1 [ 0, T ], L 2 (Ω) , u t ∈ L 2 0, T ; H 1 Γ 0 (Ω) ∩L m ((0, T ) × Γ 1 ) .
Let us mention that Theorem 2.1 also holds for α = 0. The proof of Theorem 2.1 can be done along the same line as in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF]Theorem 2.1]. The main idea of the proof is based on the combination between the Fadeo-Galerkin approximations and the contraction mapping theorem. However, for the convenience of the reader we give only the outline of the proof here.

For

u ∈ C [0, T ], H 1 Γ 0 (Ω) ∩ C 1 [0, T ], L 2 (Ω)
given, let us consider the following problem:

                     v tt -∆v -α∆v t + t 0 g(t -s)∆v(s)ds = |u| p-2 u, x ∈ Ω, t > 0 , v(x, t) = 0, x ∈ Γ 0 , t > 0 , v tt (x, t) = - ∂v ∂ν (x, t) - t 0 g(t -s) ∂v ∂ν (x, s)ds + α∂v t ∂ν (x, t) + h (v t ) x ∈ Γ 1 , t > 0 , v(x, 0) = u 0 (x), v t (x, 0) = u 1 (x) x ∈ Ω. ( 12
) Definition 2.2. A function v(x, t) such that v ∈ L ∞ 0, T ; H 1 Γ 0 (Ω) , v t ∈ L 2 0, T ; H 1 Γ 0 (Ω) ∩ L m ((0, T ) × Γ 1 ) , v t ∈ L ∞ 0, T ; H 1 Γ 0 (Ω) ∩ L ∞ 0, T ; L 2 (Γ 1 ) , v tt ∈ L ∞ 0, T ; L 2 (Ω) ∩ L ∞ 0, T ; L 2 (Γ 1 ) , v(x, 0) = u 0 (x) , v t (x, 0) = u 1 (x) , is a generalized solution to the problem (12) if for any function ω ∈ H 1 Γ 0 (Ω) ∩ L m (Γ 1
) and ϕ ∈ C 1 (0, T ) with ϕ(T ) = 0, we have the following identity:

T 0 (|u| p-2 u, w)(t) ϕ(t) dt = T 0 (v tt , w)(t) + (∇v, ∇w)(t) - t 0 g(t -s)(∇v (s) , ∇w (t))ds +α(∇v t , ∇w)(t)ϕ(t)dt + T 0 ϕ(t) Γ 1 v tt (t)w dΓ + h (v t ) dΓ dt. Lemma 2.3. Let 2 ≤ p ≤ q and 2 ≤ m ≤ q. Let u 0 ∈ H 2 (Ω) ∩ V, u 1 ∈ H 2 (Ω)
, then for any T > 0, there exists a unique generalized solution (in the sense of Definition 2.2), v(t, x) of problem [START_REF] Gerbi | Asymptotic stability and blow up a semilinear damped wave equation with dynamic boundary conditions[END_REF].

The proof of Lemma 2.3 is essentially based on the Fadeo-Galerkin approximations combined with the compactness method and can be done along the same line as in [11, Lemma 2.2], we omit the details.

In the following lemma we state a local existence result of problem [START_REF] Gerbi | Asymptotic stability and blow up a semilinear damped wave equation with dynamic boundary conditions[END_REF].

Lemma 2.4. Let 2 ≤ p ≤ q and max 2, q q+1-p ≤ m ≤ q. Then given u 0 ∈ H 1 Γ 0 (Ω) , u 1 ∈ L 2 (Ω) there exists T > 0 and a unique solution v of the problem (12) on (0, T ) such that

v ∈ C [0, T ], H 1 Γ 0 (Ω) ∩ C 1 [0, T ], L 2 (Ω) , v t ∈ L 2 0, T ; H 1 Γ 0 (Ω) ∩L m ((0, T ) × Γ 1 )
and satisfies the energy inequality:

1 2 u t (t) 2 2 + u t (t) 2 2,Γ 1 + 1 - t 0 g (s) ds ∇u (t) 2 2 + (g ⋄ u) (t) t s +α t s ∇v t (τ ) 2 2 dτ + t s Γ 1 h(v t (σ, τ ))dσdτ ≤ t s Ω |u(τ )| p-2 u(τ )v t (τ )dτ dx for 0 ≤ s ≤ t ≤ T . Proof. We first approximate u ∈ C([0, T ], H 1 Γ 0 (Ω)) ∩ C 1 [0, T ], L 2 (Ω) endowed with the standard norm u = max t∈[0,T ] u t (t) 2 + u(t) H 1 (Ω) , by a sequence (u k ) k∈N ⊂ C ∞ ([0, T ] × Ω)
by a standard convolution arguments (see [START_REF] Brezis | Analyse fonctionnelle[END_REF]). Next, we approximate the initial data

u 1 ∈ L 2 (Ω) by a sequence (u k 1 ) k∈N ⊂ C ∞ 0 (Ω). Finally, since the space H 2 (Ω) ∩ V ∩ H 1 Γ 0 (Ω) is dense in H 1 Γ 0 (Ω) for the H 1 norm, we approximate u 0 ∈ H 1 Γ 0 (Ω) by a sequence (u k 0 ) k∈N ⊂ H 2 (Ω) ∩ V ∩ H 1 Γ 0 (Ω).
We consider now the set of the following problems:

                     v k tt -∆v k -α∆v k t + t 0 g(t -s)∆v k (s)ds = |u k | p-2 u k , x ∈ Ω, t > 0, v k (x, t) = 0, x ∈ Γ 0 , t > 0, v k tt (x, t) = - ∂v k ∂ν (x, t) - t 0 g(t -s) ∂v k ∂ν (x, s)ds + α∂v k t ∂ν (x, t) + h v k t , x ∈ Γ 1 , t > 0, v k (x, 0) = u k 0 , v k t (x, 0) = u k 1 , x ∈ Ω. ( 13
) Since every hypothesis of Lemma 2.3 are verified, we can find a sequence of unique solution (v k ) k∈N of the problem [START_REF] Graber | Existence and asymptotic behavior of the wave equation with dynamic boundary conditions to appear in[END_REF]. Our goal now is to show that (v k , v k t ) k∈N is a Cauchy sequence in the space

Y T = (v, v t )/v ∈ C [0, T ] , H 1 Γ 0 (Ω) ∩ C 1 [0, T ] , L 2 (Ω) , v t ∈ L 2 0, T ; H 1 Γ 0 (Ω) ∩ L m ((0, T ) × Γ 1 )
endowed with the norm

(v, v t ) 2 Y T = max 0≤t≤T v t 2 2 + l ∇v 2 2 + v t 2 L m ((0,T )×Γ 1 ) + t 0 ∇v t (s) 2 2 ds . ( 14
)
For this purpose, we set

U = u k -u k ′ , V = v k -v k ′ . It is straightforward to see that V satisfies:                      V tt -∆V -α∆V t + t 0 g(t -s)∆V (s)ds = |u k | p-2 u k -|u k ′ | p-2 u k ′ x ∈ Ω, t > 0 , V (x, t) = 0 x ∈ Γ 0 , t > 0 , V tt (x, t) = - ∂V ∂ν (x, t) - t 0 g(t -s) ∂V ∂ν (x, s)ds + α∂V t ∂ν (x, t) + h(v k t ) -h(v k ′ t ) x ∈ Γ 1 , t > 0 , V (x, 0) = u k 0 -u k ′ 0 , V t (x, 0) = u k 1 -u k ′ 1 x ∈ Ω.
We multiply the above differential equations by V t , we integrate over (0, t) × Ω, we use integration by parts and the identity (9) to obtain:

1 2 V t (t) 2 2 + V t (t) 2 2,Γ 1 + 1 - t 0 g (r) dr ∇V (t) 2 2 +α t 0 ∇V t 2 2 ds + t 0 Γ 1 h(v k t (x, τ )) -h(v k ′ t (x, τ )) v k t (x, τ ) -v k ′ t (x, τ ) dΓdτ - 1 2 t 0 g ′ ⋄ V (s) ds + 1 2 t 0 g (s) ∇V (s) 2 2 ds = 1 2 V t (0) 2 2 + ∇V (0) 2 2 + V t (0) 2 2,Γ 1 + t 0 Ω |u k | p-2 u k -|u k ′ | p-2 u k ′ v k t -v k ′ t dxds, ∀t ∈ (0, T ) . (15)
Consequently, the above inequality together with ( 5), ( 6) and ( 7) gives

1 2 V t (t) 2 2 + V t (t) 2 2,Γ 1 + l ∇V (t) 2 2 + α t 0 ∇V t 2 2 ds + m 0 t 0 V t m m,Γ 1 ds ≤ 1 2 V t (0) 2 2 + ∇V (0) 2 2 + V t (0) 2 2,Γ 1 + t 0 Ω |u k | p-2 u k -|u k ′ | p-2 u k ′ v k t -v k ′ t dxds, ∀t ∈ (0, T ) . (16)
Following the same method as in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF], we deduce that there exists C depending only on Ω and p such that:

V Y T ≤ C V t (0) 2 2 + ∇V (0) 2 2 + V t (0) 2 2,Γ 1 + CT U Y T . Since (u k 0 ) k∈N is a converging sequence in H 1 Γ 0 (Ω), (u k 1 ) k∈N is a converging sequence in L 2 (Ω) and u k k∈N is a converging sequence in C [0, T ] , H 1 Γ 0 (Ω) ∩ C 1 [0, T ] , L 2 (Ω) (so in Y T also), we conclude that (v k , v k t ) k∈N is a Cauchy sequence in Y T . Thus (v k , v k t ) converges to a limit (v, v t ) ∈ Y T .
Now by the same procedure used by Georgiev and Todorova in [START_REF] Georgiev | Existence of a solution of the wave equation with nonlinear damping and source term[END_REF], we prove that this limit is a weak solution of the problem [START_REF] Gerbi | Asymptotic stability and blow up a semilinear damped wave equation with dynamic boundary conditions[END_REF]. This completes the proof of the Lemma 2.4.

Proof of Theorem 2.1. In order to prove Theorem 2.1, we use the contraction mapping theorem. Indeed, for T > 0, let us define the convex closed subset of Y T :

X T = {(v, v t ) ∈ Y T such that v(0) = u 0 , v t (0) = u 1 } . Let us denote: B R (X T ) = {v ∈ X T ; v Y T ≤ R} ,
the ball of radius R in X T . Then, Lemma 2.4 implies that for any u ∈ X T , we may define v = Φ (u) the unique solution of (12) corresponding to u. Our goal now is to show that for a suitable T > 0, Φ is a contractive map satisfying Φ (B R (X T )) ⊂ B R (X T ).

Let u ∈ B R (X T ) and v = Φ (u). Then for all t ∈ [0, T ] we have as in [START_REF] Grobbelaar-Van Dalsen | On the initial-boundary-value problem for the extensible beam with attached load[END_REF]:

v t 2 2 + l ∇v 2 2 + v t 2 2,Γ 1 + 2α t 0 ∇v t 2 2 ds + c t 0 v t m m,Γ 1 ds ≤ u 1 2 2 + ∇u 0 2 2 + u 1 2 2,Γ 1 + 2 t 0 Ω |u (τ ) | p-2 u (τ ) v t (τ ) dx dτ. (17)
Using Hölder's inequality, we can control the last term in the right hand side of the inequality [START_REF] Kafini | A blow-up result in a Cauchy viscoelastic problem[END_REF] as follows:

t 0 Ω |u (τ ) | p-2 u (τ ) v t (τ ) dxdτ ≤ t 0 u (τ ) p-1 2N/(N -2) v t (τ ) 2N / 3N -N p+2(p-1) dτ Since p ≤ 2N N -2 , we have: 2N 3N -N p + 2(p -1) ≤ 2N N -2 .
Thus, by Young's and Sobolev's inequalities, we get for all δ > 0 there exists C(δ) > 0, such that for all t ∈ (0, T )

t 0 Ω |u (τ ) | p-2 u (τ ) v t (τ ) dxdτ ≤ C(δ)tR 2(p-1) + δ t 0 ∇v t (τ ) 2 2 dτ.
Inserting the last estimate in the inequality [START_REF] Kafini | A blow-up result in a Cauchy viscoelastic problem[END_REF] and choosing δ small enough such that: 1) .

v 2 Y T ≤ 1 2 R 2 + CT R 2(p-
Thus, for T sufficiently small, we have v Y T ≤ R. This shows that v ∈ B R (X T ).

To prove that Φ is a contraction, we have to follow the same steps (up to minor changes) as in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF]. We omit the details. Thus the proof of Theorem 2.1 is finished. Remark 2.5. Let us say that the hypothesis on m, max 2, q q+1-p ≤ m ≤ q, is made to pass to the limit in the nonlinear term, by the same way we have used in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF]Equation (2.28)].

Global existence

In this section, we show that, under some restrictions on the initial data, the local solution of problem (1) can be continued in time and the lifespan of the solution will be [0, ∞).

Definition 3.1. Let 2 ≤ p ≤ q, max 2, q q+1-p ≤ m ≤ q, u 0 ∈ H 1 Γ 0 (Ω) and u 1 ∈ L 2 (Ω)
. We denote by u the solution of (1). We define:

T max = sup T > 0 , u = u(t) exists on [0, T ] .
Since the solution u ∈ Y T (the solution is "regular enough"), from the definition of the norm given by ( 14), let us recall that if T max < ∞, then

lim t→Tmax t<Tmax ∇u(t) 2 + u t (t) 2 = +∞.
If T max < ∞, we say that the solution of (1) blows up and that T max is the blow up time. If T max = ∞, we say that the solution of ( 1) is global.

In order to study the blow up phenomenon or the global existence of the solution of ( 1), for all 0 ≤ t < T max , we define:

I(t) = I(u(t)) = 1 - t 0 g (s) ds ∇u (t) 2 2 + (g ⋄ u) (t) -u p p , (18)
J(t) = J(u(t)) = 1 2 1 - t 0 g (s) ds ∇u (t) 2 2 + 1 2 (g ⋄ u) (t) - 1 p u p p . (19)
Thus the energy functional defined in ( 11) can be rewritten as

E(u(t)) = E(t) = J(t) + 1 2 u t 2 2 + 1 2 u t 2 2,Γ 1 . (20)
As in [START_REF] Gazzola | Global solutions and finite time blow up for damped semilinear wave equations[END_REF][START_REF] Vitillaro | Global existence theorems for a class of evolution equations with dissipation[END_REF], we denote by B the best constant in the Poincaré-Sobolev embedding H 1 Γ 0 (Ω) ֒→ L p (Ω) defined by:

B -1 = inf ∇u 2 : u ∈ H 1 Γ 0 (Ω), u p = 1 . (21)
For u 0 ∈ H 1 Γ 0 (Ω) , u 1 ∈ L 2 (Ω), we define:

E(0) = 1 2 u 1 2 2 + 1 2 u 1 2 2,Γ 1 + 1 2 ∇u 0 2 2 - 1 p u 0 p p .
The first goal is to prove that the above energy E (t) defined in ( 11) is a non-increasing function along the trajectories. More precisely, we have the following result: Lemma 3.2. Let 2 ≤ p ≤ q, max 2, q q+1-p ≤ m ≤ q, and u be the solution of (1). Then, for all t > 0, we have

dE (t) dt = 1 2 g ′ ⋄ u (t) - 1 2 g (t) ∇u (t) 2 2 -α ∇u t 2 2 - Γ 1 h (u t ) u t dΓ ≤ 1 2 g ′ ⋄ u (t) -α ∇u t 2 2 - Γ 1 h (u t ) u t dΓ, ∀t ∈ [0, T max ). ( 22
)
Proof. Multiplying the first equation in (1) by u t , integrating over Ω, using integration by parts we get:

d dt 1 2 u t 2 2 + 1 2 u t 2 2,Γ 1 + 1 2 ∇u 2 2 - 1 p u p p - Ω t 0 g (t -s) ∇u (s) ∇u t (t) dsdx = -α ∇u t 2 2 - Γ 1 h (u t ) u t dΓ. (23)
A simple use of the identity [START_REF] Georgiev | Existence of a solution of the wave equation with nonlinear damping and source term[END_REF] gives [START_REF] Messaoudi | Blow up in a nonlinearly damped wave equation[END_REF]. This completes the proof of Lemma 3.2.

Lemma 3.3. Let 2 ≤ p ≤ q, max 2, q q+1-p ≤ m ≤ q. Assume that ( 5) and ( 6) hold. Then given

u 0 ∈ H 1 Γ 0 (Ω) , u 1 ∈ L 2 (Ω) satisfying      β = B p l 2p l (p -2) E(0) (p-2)/2 < 1, I (u 0 ) > 0, (24)
we have:

I (u (t)) > 0, ∀t ∈ [0, T max ).
Proof. Since I (u 0 ) > 0, then by continuity, there exists T * < T max , such that

I (t) > 0, ∀t ∈ [0, T * ]
which implies that for all t ∈ [0, T * ],

J (t) = 1 p I (t) + p -2 2p 1 - t 0 g (s) ds ∇u (t) 2 2 + (g ⋄ u) (t) ≥ p -2 2p 1 - t 0 g (s) ds ∇u (t) 2 2 + (g ⋄ u) (t) . ( 25
)
By using ( 5), ( 6), ( 20) and ( 22), we easily get, for all t ∈ [0,

T * ] l ∇u(t) 2 2 ≤ 2p p -2 J (t) , ≤ 2p p -2 E (t) ≤ 2p p -2 E (0) . (26)
From the definition of the constant B in [START_REF] Messaoudi | Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms[END_REF], we first get: 26) and ( 24), we obtain, for all t ∈ [0, T * ]:

∀t ∈ [0, T * ] , u (
u(t) p p ≤ βl ∇u(t) 2 2 ≤ β 1 - t 0 g (s) ds ∇u(t) 2 2 < 1 - t 0 g (s) ds ∇u(t) 2 2 .
Therefore, by using ( 18), we conclude that

I (t) > 0, ∀t ∈ [0, T * ].
Using the fact that E is decreasing along the trajectory, we get:

∀ 0 ≤ t < T max , B p l 2p l (p -2) E (t) (p-2)/2 ≤ β < 1 .
By repeating this procedure, T * is extended to T max .

Proof. We first note that, by (30), we have:

E(t) ≥ 1 2 γ (t) - 1 p u (t) p p ≥ 1 2 γ (t) - B p 1 p (l ∇u (t) 2 ) p ≥ 1 2 γ (t) - B p 1 p (γ (t)) p/2 (34) = 1 2 α 2 - B p 1 p α p = F (α) ,
where α = (γ (t)) 1/2 . It is easy to verify that F is increasing for 0 < α < α 1 , decreasing for α > α 1 , F (α) → -∞ as α → +∞, and

F (α 1 ) = E 1 ,
where α 1 is given in [START_REF] Yu | Blow up for a Cauchy viscoelastic problem with a nonlinear dissipation of cubic convolution type[END_REF]. Therefore, since E(0) < E 1 , there exists

α 2 > α 1 such that F (α 2 ) = E (0) .
If we set α 0 = (γ (0)) 1/2 , then by (34) we have:

F (α 0 ) ≤ E (0) = F (α 2 ) ,
which implies that α 0 ≥ α 2 . Now to establish (32), we suppose by contradiction that:

(γ (t 0 )) 1/2 < α 2 ,
for some t 0 > 0 and by the continuity of γ (.) , we may choose t 0 such that (γ (t 0 )) 1/2 > α 1 .

Using again (34) leads to:

E (t 0 ) ≥ F γ (t 0 ) 1/2 > F (α 2 ) = E (0) .
But this is impossible since for all t > 0, E(t) ≤ E (0). Hence (32) is established.

To prove (33), we use (30) to get:

1 2 γ (t) ≤ E (0) + 1 p u (t) p p .
Consequently, using (32) leads to:

1 p u (t) p p ≥ 1 2 γ (t) -E (0) ≥ 1 2 α 2 2 -E (0) .
But we have:

1 2 α 2 2 -E (0) = 1 2 α 2 2 -F (α 2 ) = B p 1 p α p 2 .
Therefore (33) holds. This finishes the proof of Lemma 4.1.

The exponential growth result reads as follows:

Theorem 4.2. Suppose that ( 5), (6) (8) hold. Assume that 2 ≤ m and max (m, 2/l) < p ≤ p.

Then, the solution of (1) satisfying

E (0) < E 2 and ∇u 0 2 ≥ α 1 , (35)
grows exponentially in the L p norm.

Remark 4.3. It is obvious that for g = 0, we have E 1 = E 2 , and Theorem 4.2 reduces to Theorem 3.1 in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF].

Remark 4.4. In Theorem 4.2, the condition [START_REF] Kafini | A blow-up result for a viscoelastic system in R n[END_REF][START_REF] Kafini | A blow-up result in a Cauchy viscoelastic problem[END_REF][START_REF] Messaoudi | Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms[END_REF][START_REF] Messaoudi | Blow up in a nonlinearly damped wave equation[END_REF][START_REF] Messaoudi | Blow up and global existence in nonlinear viscoelastic wave equations[END_REF][START_REF] Song | Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal[END_REF][START_REF] Yu | Blow up for a Cauchy viscoelastic problem with a nonlinear dissipation of cubic convolution type[END_REF] is unnecessary and our result holds without it.

∞ 0 g(s)ds < (p/2) -1 (p/2) -1 + (1/2p) used in
Remark 4.5. Let us denote c 1 = l -2 p -2E 2 (B 1 α 2 ) -p .
Since we have seen that α 2 > α 1 , using the definition of E 2 , we easily get c 1 > 0. This constant will play an important role in the proof of Theorem 4.2 Proof of Theorem 4.2. We implement the so-called Georgiev-Todorova method (see [START_REF] Georgiev | Existence of a solution of the wave equation with nonlinear damping and source term[END_REF][START_REF] Messaoudi | Blow up in a nonlinearly damped wave equation[END_REF] and also [START_REF] Messaoudi | Global non-existence of solutions of a class of wave equations with non-linear damping and source terms[END_REF]). So, we suppose that the solution exists for all time and we will prove an exponential growth. For this purpose, we set:

H (t) = E 2 -E (t) . (36)
Of course by (31) and [START_REF] Messaoudi | Blow up in a nonlinearly damped wave equation[END_REF] and since E 2 < E 1 , we deduce that H is a non-decreasing function. So, by using (30) and, (36) we get successively:

0 < H (0) ≤ H (t) ≤ E 2 -E (t) ≤ E 1 - 1 2 γ(t) + 1 p u (t) p p .
On one hand as F (α 1 ) = E 1 and ∀ t > 0 , γ(t) ≥ α 2 2 > α 2 1 , we obtain:

E 1 - 1 2 γ(t) < F (α 1 ) - 1 2 α 2 1
On the other hand, since

F (α 1 ) - 1 2 α 2 1 = - B p 1 p α p 1 ,
we obtain the following inequality:

0 < H (0) ≤ H (t) ≤ 1 p u (t) p p , ∀t ≥ 0. ( 37
)
For ε small to be chosen later, and inspired by the ideas of the authors in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF], we then define the auxiliary function:

L (t) = H (t) + ε Ω u t udx + ε Γ 1 u t udΓ + εα 2 ∇u 2 2 . (38)
Let us remark that L is a small perturbation of the energy. By taking the time derivative of (38), using problem (1), we obtain:

dL (t) dt = α ∇u t 2 2 + Γ 1 h (u t ) u t dΓ + ε u t 2 2 -ε ∇u 2 2 +ε u p p + ε u t 2 2,Γ 1 -ε Γ 1 h (u t ) u(x, t)dσ + Ω ∇u (t) t 0 g (t -s) ∇u (s) dsdx. (39)
By making use of ( 8) and the following Young's inequality

XY ≤ λ µ X µ µ + λ -ν Y ν ν , (40)
X, Y ≥ 0, λ > 0, µ, ν ∈ R + such that 1/µ + 1/ν = 1, then we get

Γ 1 h (u t ) udΓ ≤ C m Γ 1 |u t | m-2 u t udΓ ≤ C m λ m m u m m,Γ 1 + C m m -1 m λ -m/(m-1) u t m m,Γ 1 . (41)
Now, the term involving g on the right-hand side of (39) can be written as

Ω ∇u (t, x) t 0 g (t -s) ∇u (s, x) dsdx = ∇u (t) 2 2 t 0 g (s) ds (42)
+

Ω ∇u (t, x) t 0 g (t -s) (∇u (s, x) -∇u (t, x)) dsdx.
On the other hand, by using Hölder's and Young's inequalities, we infer that for all µ > 0, we get

Ω ∇u (t, x) t 0 g (t -s) (∇u (s, x) -∇u (t, x)) dsdx ≤ t 0 g (t -s) ∇u (t) 2 ∇u (s) -∇u (t) 2 ds ≤ µ (g ⋄ u) (t) + 1 4µ t 0 g (s) ds ∇u (t) 2 2 . (43)
Inserting the estimates (41) and ( 42) into (39), taking into account the inequality (43) and making use of [START_REF] Conrad | Stabilization of a flexible beam with a tip mass[END_REF], we obtain by choosing µ = 1/2 and multiplying by l

lL ′ (t) ≥ αl ∇u t 2 2 + l c m -C m ε m -1 m λ -m/(m-1) u t m m,Γ 1 + εl u t 2 2 +εl u p p + εl u t 2 2,Γ 1 -C m εl λ m m u m m,Γ 1 (44) - εl 2 (g ⋄ u) (t) -εl ∇u (t) 2 2 .
We want now to estimate the term involving u m m,Γ 1 in (44). We proceed as in [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF]. Then, we have

u m,Γ 1 ≤ C u H s (Ω) ,
which holds for:

m ≥ 1 and 0 < s < 1, s ≥ N 2 - N -1 m > 0,
where C here and in the sequel denotes a generic positive constant which may change from line to line.

Recalling the interpolation and Poincaré's inequalities (see [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF])

u H s (Ω) ≤ C u 1-s 2 ∇u s 2 , ≤ C u 1-s p ∇u s 2 ,
we finally have the following inequality:

u m,Γ 1 ≤ C u 1-s p ∇u s 2 . ( 45
)
If s < 2/m, using again Young's inequality, we get:

u m m,Γ 1 ≤ C u p p m(1-s)µ p + ∇u 2 2 msθ 2 (46)
for 1/µ + 1/θ = 1. Here we choose θ = 2/ms, to get µ = 2/ (2 -ms). Therefore the previous inequality becomes:

u m m,Γ 1 ≤ C u p p m(1-s)2
(2-ms)p + ∇u 2 2 .

(47) Now, choosing s such that:

0 < s ≤ 2 (p -m) m (p -2) , we get: 2m (1 -s) (2 -ms) p ≤ 1. (48)
Once the inequality (48) is satisfied, we use the classical algebraic inequality:

z ν ≤ (z + 1) ≤ 1 + 1 ω (z + ω) , ∀z ≥ 0 , 0 < ν ≤ 1 , ω ≥ 0, (49)
to obtain the following estimate:

u p p m(1-s)2 (2-ms)p ≤ d u p p + H (0) ≤ d u p p + H (t) , ∀t ≥ 0, (50)
where we have set d = 1 + 1/H (0). Inserting the estimate (50) into (46) we obtain the following important inequality:

u m m,Γ 1 ≤ C u p p + l ∇u 2 2 + H (t) . (51)
Keeping in mind that l = 1 -∞ 0 g (s) ds, in order to control the term ∇u 2 2 in equation ( 44), we preferely use (as H (t) > 0), the following estimate:

u m m,Γ 1 ≤ C u p p + l ∇u 2 2 + 2H (t)
. which gives finally:

u m m,Γ 1 ≤ C 2E 2 + 1 + 2 p u p p -u t 2 2 -u t 2 2,Γ 1 + l -1 - t 0 g (s) ds ∇u 2 2 -(g ⋄ u) (t) . (52)
Since 1 -t 0 g (s) ds ≥ l, then we obtain from above

u m m,Γ 1 ≤ C 2E 2 + 1 + 2 p u p p -u t 2 2 -u t 2 2,Γ 1 -(g ⋄ u) (t) . (53)
Now, inserting (53) into (44), then we infer that:

lL ′ (t) ≥ αl ∇u t 2 2 + l c m -C m ε m -1 m λ -m/(m-1) u t m m,Γ 1 +ε l + lC m λ m m C u t 2 2 + ε l + lC m λ m m C u t 2 2,Γ 1 +ε l -C m l λ m m C 1 + 2 p u p p (54) +ε C m l λ m m C - l 2 (g ⋄ u) (t) -εl ∇u (t) 2 2 -2C m εl λ m m CE 2 .
From (36), we get

H (t) ≤ E 2 - 1 2 1 - t 0 g (s) ds ∇u (t) 2 2 - 1 2 (g ⋄ u) (t) + 1 p u (t) p p ≤ E 2 - l 2 ∇u (t) 2 2 - 1 2 (g ⋄ u) (t) + 1 p u (t) p p .
This last inequality gives

-l ∇u (t) 2 2 ≥ 2 H (t) -E 2 + 1 2 (g ⋄ u) (t) - 1 p u (t) p p . (55)
Consequently, (54) takes the form

lL ′ (t) ≥ αl ∇u t 2 2 + l c m -C m ε m -1 m λ -m/(m-1) u t m m,Γ 1 +ε l + lC m λ m m C u t 2 2 + ε l + lC m λ m m C u t 2 2,Γ 1 +ε l - 2 p -C m l λ m m C 1 + 2 p u p p -2εE 2 (56) +ε C m l λ m m C - l 2 + 1 (g ⋄ u) (t) + 2εH (t) -2C m εl λ m m CE 2 .
Now to estimate the terms involving u p p and E 2 in (56), we simply write:

l - 2 p u p p -2E 2 = l - 2 p u p p -2E 2 u p p u p p .
Then by using (33), we get:

l - 2 p u p p -2E 2 ≥ c 1 u p p ,
where c 1 > 0 is defined in Remark 4.5. Thus, (56) becomes:

lL ′ (t) ≥ αl ∇u t 2 2 + l c m -C m ε m -1 m λ -m/(m-1) u t m m,Γ 1 +ε l + lC m λ m m C u t 2 2 + ε l + lC m λ m m C u t 2 2,Γ 1 +ε c 1 -C m l λ m m C 1 + 2 p -4C m εl λ m m CE 2 (B 1 α 2 ) -p u p p (57) +ε C m l λ m m C - l 2 + 1 (g ⋄ u) (t) + 2ε H (t) + C m l λ m m CE 2 .
Notice that since l < 1, we first have :

∀ λ > 0 , C m l λ m m C - l 2 + 1 > 0 .
At this point, we pick λ small enough such that:

c 1 -C m l λ m m C 1 + 2 p -4C m εl λ m m CE 2 (B 1 α 2 ) -p > 0 .
Once λ is fixed, we may choose ε small enough such that

   c m -C m ε m -1 m λ -m/(m-1) > 0, L (0) > 0.
Consequently, we end up with the estimate:

L ′ (t) ≥ η 1 u t 2 2 + u t 2 2,Γ 1 + u p p + H (t) + E 2 , ∀t ≥ 0 . (58)
Next, it is clear that, by Young's and Poincaré's inequalities, we have:

L (t) ≤ γ H (t) + u t 2 2 + u t 2 2,Γ 1 + ∇u 2 2
for some γ > 0.

Since H (t) > 0, then for all t ≥ 0, we have:

l 2 ∇u 2 2 ≤ 1 p u p p + E 2 , (60)
To do so, we define the functional L as follows:

L (t) = H 1-σ (t) + ǫ Ω u t udx + ǫ Γ 1 u t udΓ, (67)
where the functional H is defined in (36), σ is satisfying

0 < σ ≤ min p -m p (m -1) , p -2 2p , m -2 2m , σ , (68)
where σ is defined later in (71) and ǫ is a small positive constant to be chosen later. Taking the time derivative of L(t) and following the same steps as in the proof of Theorem 4.2, we get (instead of inequality (44)), for all λ > 0,

l L′ (t) ≥ lc m (1 -σ) H -σ (t) u t m m,Γ 1 -C ǫ m -1 m λ -m/(m-1) u t m m,Γ 1 + ǫl u t 2 2 +ǫl u p p + ǫl u t 2 2,Γ 1 -C m ǫl λ m m u m m,Γ 1 (69) - ǫl 2 (g ⋄ u) (t) -ǫl ∇u (t) 2 2 .
Next, for large positive M , we select λ -m/(m-1) = M H -σ (t). Then the estimate (69) takes the form:

l L′ (t) ≥ lc m (1 -σ) -M C m ǫ m -1 m H -σ (t) u t m m,Γ 1 + ǫl u t 2 2 +ǫl u p p + ǫl u t 2 2,Γ 1 -C m ǫl M -(m-1) m H σ(m-1) u m m,Γ 1 (70) - ǫl 2 (g ⋄ u) (t) -ǫl ∇u (t) 2 2 .
Exploiting (37) and (45), we get:

H σ(m-1) u m m,Γ 1 ≤ C u (1-s)m+σp(m-1) p ∇u sm 2 .
Thus, as in section 4, we have

u (1-s)m+σp(m-1) p ∇u sm 2 ≤ C u p p m(1-s) p +σ(m-1) µ + ∇u 2 2 msθ 2 .
Choosing µ, θ, and s exactly as in section 4 (with strict inequalities), we choose σ that verifies:

σ ≤ 2 -ms 2 (m -1) 1 - 2m (1 -s) (2 -ms) p = σ. (71)
The hypotheses on m and p ensure to have 0 < σ < 1. Consequently, we get from above: for some η > 0.

H σ(m-
On the other hand, it is clear from the definition (67), we have:

L 1 1-σ (t) ≤ C (ǫ, σ) H (t) + Ω u t udx 1 1-σ + Γ 1 u t udΓ 1 1-σ . ( 77
)
By the Cauchy-Schwarz inequality and Hölder's inequality, we have:

Ω u t udx ≤ Ω u 2 t dx 1 2 Ω u 2 dx 1 2 ≤ C Ω u 2 t dx 1 2 Ω |u| p dx 1 p
, where C is the positive constant which comes from the embedding L p (Ω) ֒→ L 2 (Ω). This inequality implies that there exists a positive constant C 1 > 0 such that:

Ω u t udx 1 1-σ ≤ C 1 Ω |u| p dx 1 (1-σ)p Ω u 2 t dx 1 2(1-σ)
. Applying Young's inequality to the right hand-side of the preceding inequality, there exists a positive constant also denoted C > 0 such that:

Ω u t udx 1 1-σ ≤ C Ω |u| p dx τ (1-σ)p + Ω u 2 t dx θ 2(1-σ) , (78)
for 1/τ + 1/θ = 1. We take θ = 2(1 -σ), hence τ = 2 (1 -σ) / (1 -2σ), to get:

Ω u t udx 1 1-σ ≤ C Ω |u| p dx 2 (1-2σ)p + Ω u 2 t dx .
Using the algebraic inequality (49) with z = u p p , d = 1 + 1/H (0), ω = H (0) and ν = 2 p (1 -2σ) (the condition (68) on σ ensuring that 0 < ν ≤ 1) we get:

z ν ≤ d (z + H (0)) ≤ d (z + H (t)) .
Therefore, there exists a positive constant denoted C 2 such that for all t ≥ 0,

Ω u t udx 1 1-σ ≤ C 2 H (t) + u (t) p p + u t (t) 2 2 . ( 79
)
Following the same method as above, we can show that there exists C 3 > 0 such that:

Γ 1 u t udΓ 1 1-σ ≤ C 3 H (t) + u (t) m m,Γ 1 + u t (t) 2 2,Γ 1 .
Applying the inequality (51), we get:

Γ 1 u t udΓ
for some η1 > 0. Combining (76) and (81), then, there exists a positive constant ξ > 0, as small as ǫ, such that for all t ≥ 0, L′ (t) ≥ ξ L 1 1-σ (t).

(82) Thus, inequality (66) holds. Therefore, L(t) blows up in a finite time T * . On the other hand, from the definition of the function L(t) and using inequality (37), for small values of the parameter ε, it follows that:

L(t) ≤ κ u (t) p p 1-σ , ( 83
)
where κ is a positive constant. Consequently, from the inequality (83) we conclude that the norm u (t) p of the solution u, blows up in the finite time T * , which implies the desired result. This completes the proof of Theorem 5.1.

  ≤ C 4 H (t) + u (t) p p + l ∇u (t) 2 2 + u t (t) 2 2,Γ 1 .

		1
		1-σ
	Furthermore, inequality (60) leads to:
			1
	Γ 1	u t udΓ	1-σ	≤ C 5 H (t) + u (t) p p + u t (t) 2 2,Γ 1 + E 2 .	(80)
	Collecting (77), (79) and (80), we obtain:
	L 1 1-σ (t) ≤ η1 u t (t) 2 2 + u t	2 2,Γ

1 + H (t) + u (t) p p + E 2 , ∀t ≥ 0,

Now, we are able to state the global existence theorem. Theorem 3.4. Let 2 ≤ p ≤ q, max 2, q q+1-p ≤ m ≤ q. Assume that ( 5) and ( 6) hold. Then given u 0 ∈ H 1 Γ 0 (Ω) , u 1 ∈ L 2 (Ω) satisfying [START_REF] Meurer | Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator[END_REF]. Then the solution of ( 1) is global and bounded. Proof. To prove Theorem 3.4, using the definition of T max , we have just to check that

is uniformly bounded in time. To achieve this, we use [START_REF] Littman | Stabilization of a hybrid system of elasticity by feedback boundary damping[END_REF], [START_REF] Messaoudi | Global non-existence of solutions of a class of wave equations with non-linear damping and source terms[END_REF], [START_REF] Messaoudi | Blow up in a nonlinearly damped wave equation[END_REF] and [START_REF] Song | Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal[END_REF] to get

Therefore,

where C is a positive constant, which depends only on p.

Exponential growth for α > 0

In this section we will prove that when the initial data are large enough, the energy of the solution of problem (1) defined by [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF] grows exponentially and thus so the L p norm.

In order to state and prove the exponential growth result, we introduce the following constants:

Let us first mention that E 2 < E 1 .

The following Lemma will play an essential role in the proof of the exponential growth result, and it is inspired by the work in [START_REF] Cavalcanti | Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction[END_REF] where the authors proved a similar lemma for the wave equation.

First, we define the function

Let us rewrite the energy functional E defined by [START_REF] Gerbi | Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions[END_REF] as:

Let u be the solution of [START_REF] Adams | Sobolev spaces[END_REF]. Assume that

Then there exists a constant α 2 > α 1 such that

Thus, the inequality (59) becomes:

From the two inequalities (58) and (61), we finally obtain the differential inequality:

An integration of the previous differential inequality (62) between 0 and t gives the following estimate for the function L :

On the other hand, from the definition of the function L , from inequality (37) and for small values of the parameter ε, it follows that:

From the two inequalities ( 63) and ( 64) we conclude the exponential growth of the solution in the L p -norm.

5 Blow up in finite time for α = 0

In this section, we prove that in the absence of the strong damping -∆u t , (i.e. α = 0), the solution of problem (1) blows up in finite time that is it exists 0

The blow up result reads as follows:

Theorem 5.1. Suppose that (5), ( 6) and ( 8) hold. Assume that 2 < m and max (m, 2/l) < p ≤ p.

Then, the solution of (1) satisfying

blows up in finite time. That is u (t) p → ∞ as t → T * for some 0 < T * < ∞.

Remark 5.2. The requirement m > 2 in Theorem 5.1 is technical but it seems necessary in our proof. The case m = 2 cannot be handled with the method we use here. But the same result can be shown for m = 2 by using the concavity method. See [START_REF] Gerbi | Asymptotic stability and blow up a semilinear damped wave equation with dynamic boundary conditions[END_REF] for more details.

Proof of Theorem 5.1. To prove Theorem 5.1, we suppose that the solution exists for all time and we reach to a contradiction. Following the idea introduced in [START_REF] Georgiev | Existence of a solution of the wave equation with nonlinear damping and source term[END_REF] and developed in [START_REF] Messaoudi | Global non-existence of solutions of a class of wave equations with non-linear damping and source terms[END_REF] and [START_REF] Vitillaro | Global existence theorems for a class of evolution equations with dissipation[END_REF], we will define a function L which is a perturbation of the total energy of the system and which will satisfy the differential inequality d L (t) dt ≥ ξ L1+ν (t) ,

where ν > 0. Inequality (66) leads to a blow up of the solution in finite time T * ≥ L (0) -ν ξ -1 ν -1 , provided that L (0) > 0.