Antoine De Rengervé

Raphael Braud
email: raphael.braud@ensea.fr

Pierre Andry
email: andry@ensea.fr

Philippe Gaussier
email: gaussier@ensea.fr

Behavior adaptation from negative social signal based on own goal awareness

Robots are expected to perform actions in a human environment where they will have to learn both how and when to act. Social human robot interaction could provide the robot with external feedback to guide them. In previous work, we have developed bio-inspired models for action planning which enables the system to adapt its representations and thus its behavior in the context of latent learning with rewards. In this paper the focus is put on using negative signals. It stresses an important feature of a cognitive system : it must be aware of its own objectives i.e. aware of what it is about to do. The model presented here allows the robot the awareness of its goal, and we show that such a knowledge enhances the behavior of a robot receiving an external negative signal.

I. INTRODUCTION

Robots are expected to enter in a closer and closer interaction with humans. They should be able to act on the world accordingly. Working in a human environment requires that robots can adapt to a changing environment i.e. with constraints on when and how to perform actions that can evolve. During human robot interaction, the robot is expected to follow human instructions. In pre-verbal stage, the feedback modulating the actions of the robot could be a simple social feedback like facial expressions. Social referencing [START_REF] Feinman | Social referencing in infancy[END_REF] corresponds to the observed fact that infants can use their parents' expression to valuate an object, a situation or an action. Social referencing was implemented on robots [START_REF] Thomaz | An embodied computational model of social referencing[END_REF] [START_REF] Boucenna | Autonomous Development of Social Referencing Skills[END_REF] using this social feedback to determine whether or not they could play with a given object. If an expression of joy is presented, the robot knows that it can reach for the object whereas an expression of anger or fear will make the robot avoid touching the object. The same feedback can also be used to modulate directly behaviors for instance by weighting some sensorimotor associations in navigation [START_REF] Hasson | Using Emotional Interactions for Visual Navigation Task Learning[END_REF].

Let us consider the case of an agent planning actions in its environment. Given a certain context, its behavior may unfold as different actions and specific goals that would terminate these sequences of actions. Basically, reaching the correct goal can give the agent a reward and usually changes the active context. For instance, the behavior could be navigating and getting resources like water at different places (goals) when the agent is thirsty. As several water resources may be available in the environment, the robot could reach any of them to satisfy it thirst. There is also a knowing agent (like a human) that can help the robot to decide where to go. The knowing agent can convey a negative signal when it sees that the robot is making wrong choices of action (e.g. going to a dried-out well). A negative feedback is usually given as soon as the human teacher notices that the robot is doing something wrong i.e. it will not wait until the end of the sequence to show the robot that it is making mistakes. The mistake may be the action and thus the performed action may be inhibited. However, if it is the goal that was incorrect, only inhibiting actions is not efficient to change the behavior of an agent that tries to reach a wrong goal (Fig. 1a). If the agent had access to the information of the pursued goal, it would be able to remove the activation of the incorrect goal. Then the agent would pursue another goal changing adequately its behavior (Fig. 1b). How can the robot update a context-goal association from an anticipatory given signal ? To do so, the robot will have to be aware of its goals and motivations in order to change them. Such a knowledge could help it to solve this immediate planning problems of inhibiting the target of a behavior, but it would more generally be useful for the agent to have a better control over its own behaviors.

In previous works, we developed models explaining how a robot can navigate and even plan its navigation. These models use place-cells, a particular type of neuron found in the Hippocampus that maximally fires when the robot is at a learned spatial position. At first, these place-cells can be directly associated with orientations i.e. direction to be heading to. Such simple sensorimotor associations can build attractors defining trajectories in the space [START_REF] Giovannangeli | Robust Mapless Outdoor Vision-based Navigation[END_REF] [START_REF] Giovannangeli | Interactive Teaching for Vision-Based Mobile Robots: A Sensory-Motor Approach[END_REF]. However, with such a model, action planning is limited to using reinforcement learning [START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF] that can be quite long to adapt to changes. A latent learning of a topological model of the environment can build faster representations of the possible actions and can adapt plans faster. Those topological maps, called cognitive maps, are based on encoded actions that are transitions between place-cells associated with orientations of movement [START_REF] Cuperlier | Transition Cells for Navigation and Planning in an Unknown Environment[END_REF]. An example of built cognitive map is given in Fig. 2. Cognitive maps can encode the possible sequences of transitions between place-cells as recurrent connections. A drive corresponding to a motivational context or an active physiological need (e.g. thirst) can be associated with one of the transitions in the cognitive map. With respect to the motivation, this transition then represents a goal for the system. As a result of the recurrent network encoding, a gradient of activities is diffused from the goal to the other transitions in the cognitive map. These propagated activities can give a bias on the transitions to be performed. The selected direction of the movement then enables the navigating robot to follow the shortest path toward its goal [START_REF] Giovannangeli | Autonomous Vision-Based Navigation: Goal-Oriented Action Planning by Transient States Prediction, Cognitive Map Building, and Sensory-Motor Learning[END_REF] [START_REF] Cuperlier | Transition Cells for Navigation and Planning in an Unknown Environment[END_REF]. The use of cognitive maps are not restricted to transitions between place-cells and navigational task. The categories encoded and used in the Hippocampus may correspond to multi-modal states [START_REF] Gaussier | From view cells and place cells to cognitive map learning: processing stages of the hippocampal system[END_REF]. This idea was implemented on real robots with a cognitive map based on proprioceptive states and color based motivational contexts that enabled a robot made of a robotic arm and a camera to sort colored cans [START_REF] De Rengerve | On-line learning and planning in a pick-and-place task demonstrated through body manipulation[END_REF]. Whatever the task is, the action planning with cognitive maps relies on a gradient ascent on the diffused activities. The selected actions lead the agent to the closest goal that is a local maximum of the gradient. The robot cannot know where is the local maximum before it is reached. As it does not have a direct access to the goal it is pursuing, how can an agent determine from the diffused gradient what its current goal is ?

In Section II, the model of cognitive maps that is used in this paper is briefly summarized. A specific focus is given on how the goals are encoded in the described implementation of the cognitive maps. In Section III, we detail how an agent that chooses actions on the basis of a gradient ascent can determine its own objectives. The mechanisms are first to select and inhibit one of the possible goals and then to monitor if it is related to the current behavior of the agent. In that case, the agent succeeded to estimate its goal and the result is the selected goal. In Section IV, a simulated agent goes through the different steps to built the representations for planning with motivational contexts. The goal awareness system is implemented and enables the robot to modify its behavior when an external negative signal is perceived. In Section V, we remind the biological relevancy of this model of cognitive map and we discuss its position in the development of planning capabilities.

II. COGNITIVE MAP AND GOAL PURSUIT

The cognitive map model relies on the computation of the performed transitions between different states. In the case of navigation, each state corresponds to a place-cell that fires maximally when the robot is at the location which is encoded by the cell. The predicted transitions are used to build the cognitive map. Each time a transition is performed, the cognitive map is updated to include this transition into the topological graph of the possible sequences of transitions. In the cognitive map, recurrent connections between neurons representing the different transitions are adapted as the robot behaves. The activity from recurrent network o rec j is the result of the competition between the different activities propagated through the recurrent connections.

The system is considered to be in an exclusive motivational state m given by the drive layer, also called the motivational context layer. In this layer, only one neuron (index m) can be different from null and equal to 1. In previous works [START_REF] Hirel | Biologically inspired neural networks for spatio-temporal planning in robotic navigation tasks[END_REF], the motivational contexts were directly associated with some neurons in the cognitive map implicitly defining the corresponding transition as a goal. In order to manipulate the goals more easily they are now recruited in a separate layer. The recruitment of a new learned goal L is done when a reward is received (R=1) (eq. ( 1)). A goal is directly related to the last performed transition L assumed to be the one that get the reward. The learning is based on a recruitment according to a vigilance threshold and a Hebbian like rule for the maximally activated neurone L J . The learning depends on a learning rate and a decay factor, α L , supposed equal to enable the convergence of the weights toward 1.

   L j = i w L ij • T L i ∆w L iJ = R • α L (T L i • L J -w L iJ ) (1)
The learned goals L are gated by the reception of a reward.

Only when a reward is received the correlation between an active drive M m and an active learned goal will be learned with the following Hebbian like rule [START_REF] Thomaz | An embodied computational model of social referencing[END_REF]. The resulting activities in the learning layer corresponds to the desire of performing this goal, called a desired goal D, given an active drive.

       D j = [ i w D ij • M i + 2R • H(L j ) -R] + ∆w D ij = α D j (M i • D j -•w D ij ) -λ D j • w D ij • M i with α D j = R • ε D • H(L j ) (2)
where H is the Heavyside function and ε D is the global learning rate and with a topological neuromodulation α D j of the learning given by the learned goals L and gated by the Changes of states are events that are predicted from memory delay memorizing the timing of the change. Based on these events, transitions are predicted. II.) When a reward is received, the last performed transition is encoded as a goal. The reward and the active goal supervise the association between the active drive with the goals. Desired goals encode the confidence in getting a reward related to the active drive. The desired goal layer projects these activities in the cognitive map. The cognitive map also learns the possible sequences of transitions. A gradient activity propagates from the active goal-transition to the previous transitions in the learned sequences. III.) The cognitive map activities can bias the selection of the transition to be performed. Doing so the system follows the gradient toward the topologically closest goal. reward R. The active decay λ D j can be used to unlearn the drive-goal association. It is always null except when a negative feedback is received (see eq. ( 14)). It must be noted that during reward reception there will only be one active desired goal. Thus, the reward only reinforces the association between the active drive and the goal corresponding to the current last performed transition (index k in the cognitive map). An intermediary layer G, receiving exciting connection from the desired goal layer D, must be introduced here. Currently it is only a copy of the activity in D. Its role appears clearly during the goal selection and inhibition during the goal detection process described in Section III. The connections from the goal layer G to the cognitive map are also learned [START_REF] Boucenna | Autonomous Development of Social Referencing Skills[END_REF].

∆w motiv ij = R(δ jk • ε motiv • D i -α motiv • w motiv ij ) (3)
In the cognitive map, only the neuron corresponding to the last performed transition (index k) is active. As the recruitment ensure that only one goal is associated to a given transition, a transition in the cognitive map can only be associated with this unique learned goal. A motivational context inputs an activity o motiv j in the cognitive map (4), considering the goals desired after inhibition G.

o motiv j = [max i (w motiv ij • G i )] + (4)
The output activities O j of the neurons in the cognitive map result from a competition between the computed activities from the recurrent connections o rec j and the activities o motiv j related to the motivations (drives) of the agent (5) 1 .

O j = [max(o motiv j , o rec j )] + (5)
The motivational activities are diffused from the associated transitions to the previous transitions and so on with decreasing activities as the recurrent connective weights are lower than 1. The activities in the cognitive map come to bias the selection of the transition T W b that determines the motor commands (eq. 6).

     T W j = δ jb • H(T s j ) b = argmax j (T s j )) T s j = max((T P j -1) + O j ) (6)
with T P the possible transitions in a given state. As a result of the different competitions, the activity of each neuron in the cognitive map corresponds to only one gradient, resulting of the activity of the different goals.

III. DETECTING OWN OBJECTIVES FROM GRADIENT PROPAGATION IN A LEARNED COGNITIVE MAP

The principle of the goal detection is to modify the diffused gradients by modifying goals activities, one after another, and to monitor if the modifications are propagated to the activity of the selected transition T W (Figure 4). The desired goal activities can be modulated by the research of the current followed goal. Goals are successively selected and tested.

An internally built "keep goal" signal K supervises the goal checking by gating the selection of a new goal. When it is null, a new goal can be selected to modulate the desired goal activities. Otherwise the K signal is equal to 1, and it maintains the selected goal until the checking processed is finished or as long as required to keep the result when the detection is successful. The goal checking process depends on the propagation of modifications of the gradients in the cognitive map. Once the running propagation signal P (eq. ( 7)) . The model of cognitive map is extended to let the agent be aware its goals. The thick lines blocks of the gray area in Figure 3 are displayed with the added blocks. One of the desired goals is selected to briefly inhibit one of the goal activities G to be input in the cognitive map. The wave of modulation of activities propagates in the cognitive map to any neurons representing a transition related to the selected goal. If the variation monitoring V matches the current selected transition T W , the selected goal effectively determines to the current behavior. If the activities in the cognitive map converge (no more modulation propagation) without successful detection, then another goal is selected to be checked. When the goal is detected, the result is in the selected goal layer. When a negative signal is received, the connection between the active drive and this selected goal is decayed reducing the activity of this goal in the desired goals layer.

stops detecting changes in the cognitive map, the K signal can become null again unless the goal detection is successful (A = 1) (eq. ( 8)).

P = j H(O j (t -1) -O j (t)) + H(O j (t) -O j (t -1)) (7) K = H(A(t -1) + P (t -1) -reset) (8)
A reset signal can also be applied to force a new a goal detection 2 . The activities in the selected goal layer S is modified only when K is null. A new goal (index a) is selected depending on the current possible goals estimated from the desired drives D (binary values) and some noise η (eq. 9).

             S j = δ ja • H(s j ) a = argmax j (s j ) s j = [ K • S j (t -1) +(1 -K)( (H(D j ) -1) +A(t -1) • S j (t -1) + η ) ] + (9)
In order to perform the goal detection, the selected goal are inhibited one after the other. The inhibition goal layer I receives the selected goal S activities and some inhibition from the reward R and the keep goal signals K (eq. ( 10)). The reward signal R can prevent the modulation in order to avoid perturbing the learning of the goal to cognitive map associations.

I j = H(S j -K -R) (10)
The goal layer G contains the desired goal D activities modulated by the goal inhibition I.

G j = D j -0.5I j (11)
The success of the detection is stored in the goal detected signal A meaning the detection have been achieved (eq. ( 13)). From (9), a selected goal a is detected as the current goal if it generates the propagated gradient that gives the activity of the current transition to be performed. Neurons in the layer V are dedicated to detecting strong negative variations of the propagated activities in the cognitive map. The layer keeps the activations in memory as long as no new goal is checked [START_REF] Hirel | Biologically inspired neural networks for spatio-temporal planning in robotic navigation tasks[END_REF]. The goal-detected signal is activated only if one of the active neurons in the variation detection layer corresponds to the selected transition in T W .

V j = H(O j (t -1) -O j (t)) + V j (t -1) • K (12) A = H( j V j • T W j ) (13)
Thereby, the own goal evaluation is based on simple mechanisms: selecting a goal, modulating its propagation and monitoring if it influences the propagated activity at the level of the selected transition. The information of which goal is pursued is important to let the agent have a better control over its own behavior. For instance, the information of the current goal can be used to reduce the desire for this goal when a negative signal is received. In the equation of the drivegoal association learning (eq. ( 2)), a topological active decay term λ A j is introduced. This term can be modulated to ensure that the association is unlearned when a negative feedback is received (eq. 14).

λ A j = λ A • N • A • S j with λ A = 0.5 (14)
with λ A a global decay factor. If a negative feedback N is received while the goal detection is successful (A = 1), then the detected goal present in S will enable the decay of the connection between this goal and the active drive. As the necessary signals are already present, the mechanism to inhibit the behavior is then very simple.

IV. BEHAVIOR INHIBITION IN AN AGENT AWARE OF ITS

GOALS

The model for goal awareness was tested in a simulation of an autonomous agent navigating in a Cartesian 2D space. The basis of the simulation is a quite classic paradigm of autonomous motivated navigation. The agent is to build the corresponding action representations. Then, some interactions with the agents will be used to modify the behavior of the robot with the use of the goal awareness system. The virtual agent needs are food and water. In the environment, two water Fig. 6. Left Representation of the goal detected by the agent after the cognitive map is built. For each position, the agent estimates the current goal it would pursue. Red dots correspond to the goal on the right (W2) and blue squares are for the goal on the left (W1). The black thick line separates the two areas related to each goal. Right Initial behavior of the agent. The agent exploits the two resources on the right (Water 2 and Food 2). The agent always estimates the goal it is going to. The color and the shape of the points of the trajectory correspond to this goal estimation in the case of thirst motivation. Red dots are for W2, blue squares for W1 and black diamond for goals that are not associated with thirst (F1 and F2). . Left Representation of the goals detected by the agent after the negative signal is received. The association between thirst and the goal W2 was reduced by half. Thus, the propagated gradient now only corresponds to the goal W1 (blue squares). Right A negative signal is given while the robot moves toward W2. As the propagated gradients changed also does the target of the agent. The change of goal is visible in the goal estimation displayed for each point of the trajectory (blue squares replace red dots). The consecutive behavior of the agent is effectively adapted. It then aims to the goal W1 and keeps exploiting the two goals W1 and F1.

sources and two food sources (all infinite) can be available (see Fig. 5b). As time passes, food and water drives increase. When they are still low, the behavior of the agent is exploratory. When one of them is over a given threshold, a competition selects the most activated drive to motivate the behavior of the agent. Depending on its representation of the environment and the possible actions in the agent will reach for the closest adequate resource. When the agent is at the spot, the satisfied drive is reset letting the agent pursue the other drive, if activated enough, or resume its exploratory behavior. In the first phase, the agent explores its environment randomly. Place-cells like categories are recruited when the position of the robot is too different from any already encoded position with respect to a vigilance threshold. In the experiments, a simplification of the visuo-motor based architecture [START_REF] Giovannangeli | Robust Mapless Outdoor Vision-based Navigation[END_REF] is used. Place-cells are not built on landmark-azimuth couples but on Cartesian position (x, y). Also, the movements of the agent are not directly encoded as an orientation neural field but as the position of the next place-cell to be reached. The goal estimation mechanisms does not depends on the kind of states used but on the goal inhibition and the activity propagation in a topological map. As the agent moves, the maximally recognized category can change. Consecutive categories are then coded as transitions. The possibles successions of transitions are encoded in the cognitive map. The resulting representation of the possible actions in the environment is displayed in Figure 5c.

The capability of the agent to recognize its goals entirely relies on the previous learning and encoding of the cognitive map. After the phase of learning with the resources, the representation given by the own goal detection is evaluated. For different positions in the environment, the agent estimates the goal it expects to reach (Fig. 6). The result stresses the fact that what is important is the state in which the robot is.

In each state only one goal is ever expected.

In Fig. 7, a negative feedback is received while the agent is exploiting the resources. It is directly converted into a decay of the drive-goal association reducing it by half. As a result, the diffused gradient is modified and the agent changes its goal and thus its behavior. The resulting goal estimation for each position is given in Fig. 7a. With such a strong decay, the former goal does not propagate anymore because its related motivational activity is lower than the gradient propagated by the other strongly activated goal.

The choice of the action to be performed by the robot is given by both the drive and the state in which the robot is. Depending on its state, some transitions will be possible or not and the propagated gradient may not come from the same goal. The most important effect of the negative feedback is not to reduce the desirability of the goal transition but to modify the area of domination for each goal. In comparison with our previous work [START_REF] Hirel | Biologically inspired neural networks for spatio-temporal planning in robotic navigation tasks[END_REF], goals are now stressed as a major actor of the planning process. They are explicitly coded and can be used to make hypotheses ("Is this one the current goal?") and modulate them in order to find and select the current followed goal.

V. DISCUSSION

In this paper, we studied how negative signals can induce behavior adaptation in the case of action planning based on cognitive maps. With cognitive maps, behaviors rely not only actions but also goals that will generate an activity propagated from action to action in order to trigger specific sequences. Adapting the behavior may not only be changing what action should be done but also changing what sequence is to be done. Determining which goal and thus which sequence is followed needs a particular processing due to the properties of the used cognitive maps. Potential goals are selected to estimate whether or not they are related to the current behavior of the robot. The selected goal is inhibited and then can be determined as the current goal if this inhibition eventually modifies the value of the propagated gradient that biases the activity of the selected action.

The architecture for planning with a cognitive map mainly relies on models of the Hippocampus, the Prefrontal Cortex and Parieto-temporal Cortices. The encoded low level actions are transitions corresponding to changes between two placecells. The Hippocampus with the Entorhinal Cortex, known as a novelty detector can detect these changes of states. The cells of the Dendate Gyrus (DG) provide a time basis to the cells of the CA3 to predict the place-cell activities and thus events like changes of most recognized place-cell. These predictions are then used to predict transitions (in CA1) [START_REF] Hirel | Biologically inspired neural networks for spatio-temporal planning in robotic navigation tasks[END_REF]. A competition between the possible transitions is performed at the level of the Basal Ganglia giving the action selection. It can be biased by the Prefrontal activities from the propagation in the cognitive maps.

This described system is not the only solution to explain how behavior can be adapted from negative feedback. It more likely corresponds to a last developmental stage of action planning. At first the brain can directly use simple sensorimotor actions valuated by a reinforcement learning process [START_REF] Sutton | Reinforcement Learning: An Introduction[END_REF] occurring in the Basal Ganglia. In order to capture the correct properties of the task to be performed, this process must slowly adapt the encoding and thus the behavior. The Basal Ganglia can count on the frontal cortex to solve this problem. There exist several cortico-striatal loops involving the Basal Ganglia and the frontal cortex with different functional levels [START_REF] Alexander | Parallel organization of functionally segregated circuits linking basal ganglia and cortex[END_REF] [START_REF] Lawrence | Cognitive functions and corticostriatal circuits: insights from Huntington's disease[END_REF]. The simple action planning directly based on reinforcement learning correspond to the motor loop represented in the frontal cortex by the Supplementary motor areas (SMA), the Premotor Cortex (PMC) and the Somatosensory area (SSA). When a negative signal is received, working memories [START_REF] Dehaene | Neuronal models of prefrontal cortical functions[END_REF] present in the frontal cortex could come to temporarily inhibit the incorrect actions. As a result, the behavior is adapted fast while the reinforcement learning process learn what to do at its own speed. A more cognitive loop (called spatial loop in [START_REF] Lawrence | Cognitive functions and corticostriatal circuits: insights from Huntington's disease[END_REF]) includes the dorso-lateral cortex (DLC) and the posterior parietal cortex (PPC). This loop correspond to the cognitive map model. The goals would be in the dorsolateral cortex whereas the cognitive map would be encoded in the recurrent connections of the posterior parietal cortex.

The nodes corresponding to motor actions in the motor loop find their homologous in the goals of the spatial loop. The difference is that these goals can propagate activities in networks (cognitive maps) thus encapsulating complete sequences of actions. Considering that the spatial loop are a development of the original motor loop, the same inhibition process can occur. Depending on the reception of a negative the working memories can come to inhibit goals as well as actions providing the basis for enabling an agent to detect its own goal while planning with the cognitive map. In [START_REF] Dehaene | Neuronal models of prefrontal cortical functions[END_REF], the authors showed that cognitive tasks (Wisconsin test, Tower of London test) could be solved by neural network models of the prefrontal functions based on testing and selecting code-rule clusters (goals).

Implementing the motor and spatial loops in parallel can be used to get the best of the two strategies [START_REF] Dollé | Path planning versus cue responding: a bio-inspired model of switching between navigation strategies[END_REF]. However their interactions may not be restricted to selecting which strategy is the best at a given moment. As the cognitive map can plan sequences of actions, such sequences could be reencoded as action primitives. The reinforcement learning process and the motor loop could directly process such primitives. As these primitives are new possible actions, they should be integrated in the cognitive representations of the possible actions i.e. in the cognitive maps. Then, the spatial loop could build sequences including the more complex action primitives. The development of more and more complex behaviors would not be performed by one superior structure but rather from the recurrent interactions between the quite simple motor loop and spatial loop. In order to handle these primitives and complex sequences, the nodes (representing goals or actions) should be reencoded as chunks merging adequately many different sensory signal [START_REF] Miller | The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information[END_REF]. An adequate extraction of the relevant features to be encoded is the challenge to be tackled [START_REF] Hanoune | An architecture for online chunk learning and planning in complex navigation and manipulation tasks[END_REF]. Finally, current ongoing work also focuses on implementing and validating the own goal detection model on real robot (Fig. 8). Correctly taking negative feedback into account should improve how natural interactions can be, including in non-navigational tasks like arm control.

Fig. 1 .

 1 Fig. 1. Modifying an agent behavior from negative signals. A motivated agent navigates in an environment with two resources placed at Goal1 and Goal2. The motivational context (drive) for reaching one of the resources is active. A knowing agent conveys negative signals to prevent the motivated agent from going to Goal2 (trajectory in dash line). Left The negative signals trigger the inhibition of the directions of movement. As the agent still considers the Goal2 as a correct goal, many negative signals are needed to change the target of the agent. Right The negative signal removes the attractiveness of the second goal. The agent now aims directly for Goal1.

Fig. 2 .

 2 Fig. 2. Cognitive map built on-line during visual navigation task with Robulab (Robosoft) mobile platform.

Fig. 3 .

 3 Fig. 3. Cognitive map based motor control and own goal estimation. I.) States representing the places are recruited with respect to a vigilance threshold.Changes of states are events that are predicted from memory delay memorizing the timing of the change. Based on these events, transitions are predicted. II.) When a reward is received, the last performed transition is encoded as a goal. The reward and the active goal supervise the association between the active drive with the goals. Desired goals encode the confidence in getting a reward related to the active drive. The desired goal layer projects these activities in the cognitive map. The cognitive map also learns the possible sequences of transitions. A gradient activity propagates from the active goal-transition to the previous transitions in the learned sequences. III.) The cognitive map activities can bias the selection of the transition to be performed. Doing so the system follows the gradient toward the topologically closest goal.

Fig. 4

 4 Fig.[START_REF] Hasson | Using Emotional Interactions for Visual Navigation Task Learning[END_REF]. The model of cognitive map is extended to let the agent be aware its goals. The thick lines blocks of the gray area in Figure3are displayed with the added blocks. One of the desired goals is selected to briefly inhibit one of the goal activities G to be input in the cognitive map. The wave of modulation of activities propagates in the cognitive map to any neurons representing a transition related to the selected goal. If the variation monitoring V matches the current selected transition T W , the selected goal effectively determines to the current behavior. If the activities in the cognitive map converge (no more modulation propagation) without successful detection, then another goal is selected to be checked. When the goal is detected, the result is in the selected goal layer. When a negative signal is received, the connection between the active drive and this selected goal is decayed reducing the activity of this goal in the desired goals layer.

Fig. 5 .

 5 Fig.5. Left The agent explore its environment and encoded it as states (place-cells), transitions and sequences of transitions. Dots are place-cells, arrows are learned oriented transitions and the dash-gray lines are the trajectories of the robot during exploration. Center Infinite resources (water: W 1, W 2 and food: F 1, F 2) are added to the environment. The agent continue to explore and learn the rewarded goals. Right Resulting cognitive map built during these two first phases. The arrows and their thicknesses and colors represent respectively the possible transitions and their activities in the cognitive map (the bigger and darker for higher values). The activities correspond to the propagated gradient when the first drive (thirst) is active.

Fig. 7

 7 Fig. 7. Left Representation of the goals detected by the agent after the negative signal is received. The association between thirst and the goal W2 was reduced by half. Thus, the propagated gradient now only corresponds to the goal W1 (blue squares). Right A negative signal is given while the robot moves toward W2. As the propagated gradients changed also does the target of the agent. The change of goal is visible in the goal estimation displayed for each point of the trajectory (blue squares replace red dots). The consecutive behavior of the agent is effectively adapted. It then aims to the goal W1 and keeps exploiting the two goals W1 and F1.

Fig. 8 .

 8 Fig. 8.Can sorting based on a cognitive map model: Pick and place experiment with a Katana (Neuronics AG) robotic arm.

In the following description of the model (Fig.4), the equations are given for discrete time.

Each signal is noted without the time index when it correspond to the current iteration. The time index is only given when it differs from the current iteration like (t -1) for previous iteration.

ACKNOWLEDGMENT

This work was supported by the INTERACT french project referenced ANR-09-CORD-014 and the NEUROBOT french ANR project.


Elissar Nasreddine
email: elissar.nasreddine@math.univ-toulouse.fr

Well-posedness for a model of individual clustering

Keywords: Elliptic system, local existence, global existence, steady state, compactness method

Well-posedness for a model of individual clustering

Elissar

Introduction

In [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF], a model for the dispersal of individuals with an additional aggregation mechanism is proposed. More precisely, classical models for the spatial dispersion of biological populations read ∂ t u = ∆(Φ(u)) + f (u, t, x).

(

) 1
where u(t, x) denotes the population density at location x and time t, and f (u, t, x) represents the population supply, due to births and deaths. The dispersal of individuals is either due to random motion with Φ(u) = u or rests on the assumption that individuals disperse to avoid crowding and Φ satisfies Φ(0) = 0, and Φ ′ (u) > 0, for u > 0.

(

) 2
No aggregation mechanism is present in this model though, as discussed in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF], the onset of clustering of individuals in a low density region might balance the death and birth rates and guarantee the survival of the colony. To account for such a phenomenon, a modification of the population balance (1) is proposed in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF] and reads

∂ t u = -∇ • (u V (u, t, x)) + u E(u, t, x). ( 3
)
where V is the average velocity of individuals, and E is the net rate of reproduction per individual at location x and time t. To complete the model, we must specify how V is related to u and E. Following [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF], we assume that each individual disperses randomly with probability δ ∈ (0, 1) and disperses deterministically with an average velocity ω so as to increase his expected rate of reproduction with probability 1 -δ. The former is accounted for by a usual Fickian diffusion ∇u u while the latter should be in the direction of increasing E(u, t, x), say, of the form λ ∇E(u, t, x) with λ > 0. A slightly different choice is made in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF] and results in the following system

∂ t u = δ ∆u -(1 -δ) ∇ • (u ω) + u E(u, t, x) -ε ∆ω + ω = λ∇E(u, t, x). (4)
After a suitable rescaling, and assuming that the environment is homogeneous, (4) becomes

∂ t u = δ ∆u -∇ • (u ω) + r u E(u) -ε ∆ω + ω = E ′ (u) ∇u, (5)
for x ∈ Ω and t ≥ 0, where Ω is an open bounded domain of R N , 1 ≤ N ≤ 3. We supplement [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF] with no-flux boundary conditions

n • ∇u = n • ω = 0 x ∈ ∂Ω, t ≥ 0, (6)
as suggested in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF]. However, the previous boundary conditions [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF] are not sufficient for the well-posedness of the elliptic system verified by ω in several space dimensions and we must impose the following additional condition given in [START_REF] Dias | A simplified variational model for the bidimensional coupled evolution equations of a nematic liquid crystal[END_REF][START_REF] Dias | Un problème aux limites pour un système d'équations non linéaires tridimensionnel[END_REF][START_REF] Schoenauer | Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées[END_REF]:

∂ n ω × n = 0, x ∈ ∂Ω, t > 0. ( 7
)
As usual, v × ω is the number v 1 ω 1 + v 2 ω 2 if N = 2 and the vector field (v 2 ω 3 -

v 3 ω 2 , -v 1 ω 3 + v 3 ω 1 , v 1 ω 2 -v 2 ω 1 ) if N = 3.
We note that the boundary condition [START_REF] Langlais | Stabilization of solutions of nonlinear and degenerate evolution equations[END_REF] is useless if N = 1.

Summarizing, given a sufficiently smooth function E, parameters δ > 0, ε ≥ 0 and r ≥ 0, our aim in this paper is to look for (u, ω) solving the problem

           ∂ t u = δ ∆u -∇ • (u ω) + r u E(u), x ∈ Ω, t > 0 -ε ∆ω + ω = ∇E(u), x ∈ Ω, t > 0 ∂ n u = 0 , ω • n = 0, x ∈ ∂Ω, t > 0 ∂ n ω × n = 0, x ∈ ∂Ω, t > 0 u(0, x) = u 0 (x), x ∈ Ω. (8)
In the first part of this paper, we show that, for p > N , the system (8) has a maximal solution u in the sense of Definition 2.1 where u ∈ C [0, T max ), W 1,p (Ω) ∩C (0, T max ), W 2,p (Ω) , see Theorem 2.2.

In the second part, we turn to the global existence issue and focus on space dimension 1, and two specific forms of E suggested in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF]: the "bistable case" where E(u) = (1-u)(u-a) for some a ∈ (0, 1), see Theorem 2.3, and the "monostable case" E(u) = 1 -u. In both cases, we prove the global existence of solution. In addition, in the monostable case, i.e E(u) = 1 -u, thanks to the Liapunov functional

L(u) = 1 -1 (u log u -u + 1) dx,
we can study the asymptotic behaviour of solutions for t large, and show that the solution u converges, when t goes to ∞, to a steady state in L 2 (-1, 1), see Theorem 2.4.

In the third part, we investigate the limiting behaviour as ε → 0. Heuristically, when ε goes to zero, the velocity ω becomes sensitive to extremely local fluctuations in E(u), and the system (8) reduces to the single equation

∂ t u = ∇ • (δ -u E ′ (u)) ∇u + r u E(u). (9)
Clearly ( 9) is parabolic only if δ -u E ′ (u) ≥ 0 for all u > 0. This is in particular the case when E(u) = 1 -u, see Theorem 2.6. But this limit is not well-posed in general. As a result the population distribution may become discontinuous when neighbouring individuals decide to disperse in opposite direction, that is in particular the case when

E(u) = (1 -u)(u -a).

Main results

Throughout this paper and unless otherwise stated, we assume that

E ∈ C 2 (R), δ > 0, ε > 0, r ≥ 0.
We first define the notion of solution to [START_REF] Schoenauer | Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées[END_REF] to be used in this paper.

Definition 2.1. Let T > 0, p > N , and an initial condition

u 0 ∈ W 1,p (Ω) . A strong solution to (8) on [0, T ) is a function u ∈ C [0, T ), W 1,p (Ω) ∩ C (0, T ), W 2,p (Ω) , such that    ∂ t u = δ ∆u -∇ • (u ω u ) + r u E(u), a.e. in [0, T ) × Ω u(0, x) = u 0 (x), a.e. in Ω ∂ n u = 0, a.e. on [0, T ) × ∂Ω, (10)
where, for all t ∈ [0, T ), ω u (t) is the unique solution in W 2,p (Ω) of

-ε∆ω u (t) + ω u (t) = ∇E(u(t)) a.e. in Ω ω u (t) • n = ∂ n ω u (t) × n = 0 a.e. on ∂Ω (11)
Our first result gives the existence and uniqueness of a maximal solution of (8) in the sense of Definition 2.1. Theorem 2.2. Let p > N and a nonnegative function u 0 ∈ W 1,p (Ω). Then there is a unique maximal solution u ∈ C [0, T max ), W 1,p (Ω) ∩ C (0, T max ), W 2,p (Ω) to (8) in the sense of Definition 2.1, for some T max ∈ (0, ∞]. In addition, u is nonnegative. Moreover, if for each T > 0, there is C(T ) such that

||u(t)|| W 1,p ≤ C(T ), for all t ∈ [0, T ] ∩ [0, T max ), then T max = ∞.
The proof of the previous theorem relies on a contraction mapping argument.

We now turn to the global existence issue and focus on the one dimensional case, where E(u) has the structure suggested in [START_REF] Grindrod | Models of individual aggregation or clustering in single and multi-species communities[END_REF].

In the following theorem we give the global existence of solution to [START_REF] Schoenauer | Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées[END_REF] in the bistable case, that is when E(u) = (1 -u)(u -a), for some a ∈ (0, 1). Theorem 2.3. Assume that u 0 is a nonnegative function in W 1,2 (-1, 1), and E(u) = (1 -u)(u -a) for some a ∈ (0, 1). Then (8) has a global nonnegative solution u in the sense of Definition 2.1.

The proof relies on a suitable cancellation of the coupling terms in the two equations which gives an estimate for u in L ∞ (L 2 ) and for ω in L 2 W 1,2 .

Next, we can prove the global existence of a solution to [START_REF] Schoenauer | Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées[END_REF] in the monostable case, that is, when E(u) = 1 -u, and we show that the solution converges as t → ∞ to a steady state. More precisely, we have the following theorem Theorem 2.4. Assume that u 0 is a nonnegative function in W 1,2 (-1, 1), and E(u) = 1 -u. There exists a global nonnegative solution u of (8) in the sense of Definition 2.1 which belongs to L ∞ [0, ∞); W 1,2 (-1, 1) . In addition, if r = 0,

lim t→∞ u(t) - 1 2 1 -1 u 0 dx 2 = 0,
and if r > 0 the solution u(t) converges either to 0 or to 1 in L 2 (-1, 1) as t → ∞.

In contrast to the bistable case, it does not seem to be possible to begin the global existence proof with a L ∞ (L 2 ) estimate on u. Nevertheless, there is still a cancellation between the two equations which actually gives us an L ∞ (L log L) bound on u and a L 2 bound on ∂ x √ u.

Remark 2.5. We note that when N = 1, there is a relation between our model when E(u) = 1 -u and r = 0, and the following chemorepulsion model studied in [START_REF] Cieślak | Global existence and convergence to steady states in a chemorepulsion system[END_REF]   

∂ t u = δ ∂ 2 xx u + ∂ x (u ∂ x ψ), in (0, ∞) × (-1, 1) -ε ∂ 2 xx ψ + ψ = u, in (0, ∞) × (-1, 1) ∂ x u(t, ±1) = ∂ x ψ(t, ±1) = 0 on (0, ∞). (12)
Indeed, define ϕ = -∂ x ψ, and substitute it into (12). Then differentiating the second equation in (12) we find

   ∂ t u = δ ∂ 2 xx u -∂ x (u ϕ), in (0, ∞) × (-1, 1) -ε ∂ 2 xx ϕ + ϕ = -∂ x u = ∂ x E(u), in (0, ∞) × (-1, 1) ∂ x u(t, ±1) = ϕ(t, ±1) = 0 on (0, ∞). (13)
So that u is a solution to our model.

When E(u) = 1 -u, the limit ε → 0 is formally justified and (8) takes the qualitative form of (1) with Φ(u) = δu + 1 2 u 2 . In this example though, since E ′ < 0, the individuals dispersing so as to maximise E would seek isolation, and there is clearly no mechanism capable of producing aggregation of individuals. This observation is actually consistent with Remark 2.5.

Theorem 2.6. Assume that u 0 is a nonnegative function in W 1,2 (-1, 1), and that E(u) = 1 -u. For ε > 0 let u ε be the global solution to [START_REF] Schoenauer | Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées[END_REF] given by Theorem 2.4. Then, for all T > 0,

lim ε→0 T 0 ||u ε (t) -u(t)|| 2 2 dt = 0, ( 14
)
where u is the unique solution to

   ∂ t u = ∂ 2 xx (δ u + 1 2 u 2 ) + r u (1 -u), x ∈ (-1, 1), t > 0, u(0, x) = u 0 (x), x ∈ (-1, 1), ∂ x u(t, ±1) = 0, t > 0. ( 15
)
Since δ + u > 0 for u ≥ 0, the previous equation ( 15) is uniformly parabolic and has a unique solution u, see [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF] for instance. The proof of Theorem 2.6 is performed by a compactness method.

Preliminaries

We first recall some properties of the following system,

   -ε ∆ω + ω = f, in Ω, ω • n = 0, on ∂Ω, ∂ n ω × n = 0, on ∂Ω, (16)
where f ∈ (L p (Ω)) N and Ω is a bounded open subset of R N , N = 2, 3. Let us first consider weak solutions of (16). For that purpose, we define

W 1 = {v ∈ (H 2 (Ω)) N ; v • n = 0, and ∂ n v × n = 0 on ∂Ω} and take W as the closure of W 1 in (H 1 (Ω)) N . If f ∈ (L 2 (Ω)) N , the weak formulation for (16) is    ε Ω ∇ω • ∇v + Ω ω • v = Ω f • v, for all v ∈ W ω ∈ W (17) where ∇ω • ∇v = i ∇ω i • ∇v i and ω • v = i ω i v i .
We recall some results about the existence, regularity and uniqueness of solution for (17), see [START_REF] Dias | A simplified variational model for the bidimensional coupled evolution equations of a nematic liquid crystal[END_REF][START_REF] Dias | Un problème aux limites pour un système d'équations non linéaires tridimensionnel[END_REF].

Theorem 3.1. If f ∈ (L 2 (Ω)) N , (17) has a unique solution in W and there is C = C(Ω, N ) such that ||ω|| W ≤ C ||f || 2 .
We next consider strong solutions of (16), that is, solutions solving (16) a.e. in Ω. In this direction the existence and uniqueness of the strong solution to (16) is proved in [START_REF] Schoenauer | Quelques résultats de régularité pour un système elliptique avec conditions aux limites couplées[END_REF]:

Theorem 3.2. If f ∈ (L p (Ω)) N with 1 < p < ∞, (16) has a unique solution in (W 2,p (Ω)) N and ||ω|| W 2,p ≤ K(p) ε ||f || p , (18)
where K(p) = K(p, Ω, N ).

In other words, the strong solution has the same regularity as elliptic equations with classical boundary conditions.

We finally recall some functional inequalities: in several places we shall need the following version of Poincaré's inequality

||u|| W 1,p ≤ C (||∇u|| p + ||u|| q ) , u ∈ W 1,p (Ω) (19)
with arbitrary p ≥ 1 and q ∈ [1, p]. Also, we will frequently use the Gagliardo-Nirenberg inequality

||u|| p ≤ C ||u|| θ W 1,2 ||u|| 1-θ q , with θ = N q -N p 1 -N 2 + N q , u ∈ W 1,2 (Ω) (20)
which holds for all p ≥ 1 satisfying p (N -2) < 2 N and q ∈ [1, p).

Local well-posedness

Throughout this section, we assume that

E ∈ C 2 (R), and set Ẽ(z) = z E(z) for z ∈ R. ( 21
)
Proof of Theorem 2.2. We fix p > N , R > 0, and define for T ∈ (0, 1) the set

X R (T ) := u ∈ C [0, T ]; W 1,p (Ω) , sup t∈[0,T ] ||u(t)|| W 1,p ≤ R ,
which is a complete metric space for the distance

d X (u, v) = sup t∈[0,T ] ||u(t) -v(t)|| W 1,p , (u, v) ∈ X R (T ) × X R (T ).
For u ∈ X R (T ), and t ∈ [0, T ], the embedding of

W 1,p (Ω) in L ∞ (Ω) ensures that ∇E(u(t)) ∈ L p (Ω) so that (16) with f = ∇E(u) has a unique solution ω u ∈ W 2,p (Ω) N .
We then define Λ(u) by

Λu(t, x) = (e t (δ ∆) u 0 )(x) + t 0 e (t-s) (δ ∆) -∇ • (u ω u ) + r Ẽ(u) (s, x) ds, (22) for
(t, x) ∈ [0, T ] × Ω
, where e t (δ ∆) denotes the semigroup generated in L p (Ω) by δ ∆ with homogeneous Neumann boundary conditions. We now aim at showing that Λ maps X R (T ) into itself, and is a strict contraction for T small enough. In the following, (C i ) i≥1 and C denote positive constants depending only on Ω, δ, r, ε, E, p and R.

• Step 1. Λ maps X R (T ) into itself.
We first recall that there is

C 1 > 0 such that ||v|| ∞ ≤ C 1 ||v|| W 1,p , (23)
and

||e t(δ ∆) v|| W 1,p ≤ C 1 ||v|| W 1,p , and ||∇e t (δ ∆) v|| p ≤ C 1 δ -1 2 t -1 2 ||v|| p , (24)
for all v ∈ W 1,p (Ω). Indeed, (23) follows from the continuous embedding of

W 1,p (Ω) in L ∞ (Ω) due to p > N while (24) is a consequence of the regularity properties of the heat semigroup. Consider u ∈ X R (T ), and t ∈ [0, T ]. It follows from (24) that ||Λu(t)|| p ≤ C 1 ||u 0 || p + t 0 ||∇e (t-s)(δ ∆) (u ω u )(s) || p ds + r t 0 ||e (t-s)(δ ∆) Ẽ(u)(s)|| p ds ≤ C 1 ||u 0 || p + C 1 δ -1 2 t 0 (t -s) -1 2 ||u ω u (s)|| p ds + C 1 r t 0 || Ẽ(u)(s)|| p ds.
Thanks to (23), we have

||u(t)|| ∞ ≤ C 1 ||u(t)|| W 1,p ≤ R C 1 ≤ C 2 . (25)
Therefore, using elliptic regularity (see Theorem 3.2) and (25), we obtain

||ω u (t)|| W 2,p ≤ K(p) ε ||∇E(u(t))|| p ≤ C ||E ′ || L ∞ (-C 2 ,C 2 ) R ≤ C. (26)
Using again (25) along with (26) we find

||Λu(t)|| p ≤ C 1 ||u 0 || p + C t 0 (t -s) -1 2 ||u(s)|| ∞ ||ω u (s)|| p ds + r t 0 ||u(s)|| p ||E(u(s))|| ∞ ds ≤ C 1 ||u 0 || p + C t 0 (t -s) -1 2 ds + r T R ||E|| L ∞ (-C 2 ,C 2 ) ≤ C 1 ||u 0 || p + C t 1 2 + C T ≤ C 1 ||u 0 || p + C 3 T 1 2 (27)
(recall that T ≤ 1). On another hand, by (24) we have

||∇Λu(t)|| p ≤ C 1 ||∇u 0 || p + t 0 ||∇e (t-s) (δ ∆) ∇ • (u ω u )(s)|| p ds + r t 0 ||e (t-s)(δ ∆) ∇(u E(u))(s)|| p ds ≤ C 1 ||∇u 0 || p + δ -1 2 t 0 (t -s) -1 2 ||(∇u • ω u + u ∇ • ω u )(s)|| p ds + r t 0 ||∇( Ẽ(u))(s)|| p ds.
Since u ∈ X R (T ), using (25) we can see that

r ||∇( Ẽ(u))|| p ≤ r || Ẽ′ (u) ∇u|| p ≤ r || Ẽ′ || L ∞ (-C 2 ,C 2 ) ||∇u|| p ≤ C 4 ,
which gives that

||∇Λu(t)|| p ≤ C 1 ||∇u 0 || p + δ -1 2 t 0 (t -s) -1 2 ||ω u || ∞ ||∇u|| p ds + δ -1 2 t 0 (t -s) -1 2 (||∇ • ω u || p ||u|| ∞ ) ds + C 4 T. Since ||ω u || ∞ ≤ C 1 ||ω u || W 1,p ≤ C 5 , (28)
by ( 26) and (23), we use once more (25) and obtain that

||∇Λu(t)|| p ≤ C 1 ||∇u 0 || p + C t 0 (t -s) -1 2 ds + C t 0 (t -s) -1 2 ds + C 4 T ≤ C 1 ||∇u 0 || p + C 6 T 1 2 . ( 29
)
Combining ( 27) and (29) we get sup

t∈[0,T ] ||Λu(t)|| W 1,p ≤ C 1 ||u 0 || W 1,p + C 7 T 1 2 . Choosing R = 2 C 1 ||u 0 || W 1,p and T ∈ (0, 1) such that C 1 ||u 0 || W 1,p + C 7 T 1 2 ≤ R,
we obtain that sup

t∈[0,T ] ||Λu(t)|| W 1,p ≤ R.
It follows that Λ maps X R (T ) into itself.

• Step 2. We next show that Λ is a strict contraction for T small enough.

Let u and v be two functions in X R (T ). Using (24) we have

||Λu(t) -Λv(t)|| p ≤ t 0 ∇e (t-s) (δ ∆) [-u ω u + v ω v ] p ds + r t 0 e (t-s)(δ ∆) [ Ẽ(u) -Ẽ(v)] p ds ≤ C 1 δ -1 2 t 0 (t -s) -1 2 ||-u ω u + v ω v || p ds + r C 1 t 0 Ẽ(u) -Ẽ(v) p ds. (30)
Note that, by ( 25) and (28), we have

||u ω u -v ω v || p ≤ ||u ω u -u ω v -v ω v + u ω v || p ≤ ||u|| ∞ ||ω u -ω v || p + ||ω v || ∞ ||u -v|| p ≤ C ||ω u -ω v || p + C ||u -v|| p , (31)
and it follows from Theorem 3.2 and (25) that

||ω u -ω v || W 2,p ≤ C ||∇E(u) -∇E(v)|| p ≤ C ||E ′ (u) ∇u -E ′ (u) ∇v -E ′ (v) ∇v + E ′ (u) ∇v|| p ≤ C ||E ′ || L ∞ (-C 2 ,C 2 ) ||∇u -∇v|| p + C||E ′ (v) -E ′ (u)|| ∞ ||∇v|| p ≤ C ||∇v -∇u|| p + C ||E ′′ || L ∞ (-C 2 ,C 2 ) d X (u, v). ≤ C 8 d X (u, v). (32)
Combining (32) and (31) we obtain

t 0 (t -s) -1 2 || -u ω u + v ω v || p ds ≤ C T 1 2 C 8 d X (u, v) + T 1 2 C 9 d X (u, v) ≤ T 1 2 C 10 d X (u, v). ( 33
)
Since u and v are bounded by (25), we have

r || Ẽ(u) -Ẽ(v)|| p ≤ C ||u -v|| p .
Then, we get

t 0 r || Ẽ(u) -Ẽ(v)|| p (s) ds ≤ C 11 T d X (u, v)
Substituting (33) and the above inequality in (30) we conclude that

||Λu(t) -Λv(t)|| p ≤ C 12 T 1 2 d X (u, v).
Using again (24), we have

||∇Λu(t) -∇Λv(t)|| p ≤ t 0 ∇e (t-s) (δ ∆) [-∇ • (u ω u ) + ∇ • (v ω v )] p ds + r t 0 ∇e (t-s) (δ ∆) ( Ẽ(u) -Ẽ(v)) p ds ≤ δ -1 2 C 1 t 0 (t -s) -1 2 ||-∇ • (u ω u ) + ∇ • (v ω v )|| p ds + r C 1 t 0 ∇ Ẽ(u) -Ẽ(v) p (s) ds. (34)
Since the mapping

W 1,p (Ω) × W 1,p (Ω) -→ W 1,p (Ω) u, v -→ u v
is bilinear and continuous due to p > N , we deduce from (26) and (32) that

δ -1 2 C 1 || -∇ • (u ω u ) + ∇ • (v ω v )|| p ≤ ||u ω u -v ω v || W 1,p ≤ C ||u|| W 1,p ||ω u -ω v || W 1,p + C ||ω v || W 1,p ||u -v|| W 1,p ≤ C d X (u, v) + C ||u -v|| W 1,p ≤ C 13 d X (u, v) Thus, δ -1 2 C 1 t 0 (t -s) -1 2 || -∇ • (u ω u ) + ∇ • (v ω v )|| p ds ≤ T 1 2 C 13 d X (u, v).
On the other hand, due to (25) and the embedding of

W 1,p (Ω) in L ∞ (Ω) ||∇( Ẽ(u) -Ẽ(v))|| p = || Ẽ′ (u) ∇u -Ẽ′ (v) ∇v|| p ≤ C || Ẽ′ (u)|| ∞ ||∇u -∇v|| p + C || Ẽ′ (u) -Ẽ′ (v)|| ∞ ||∇v|| p , ≤ C || Ẽ′ || L ∞ (-C 2 ,C 2 ) ||∇u -∇v|| p + C || Ẽ′′ || L ∞ (-C 2 ,C 2 ) ||u -v|| ∞ ≤ C ||u -v|| W 1,p . Then r C 1 ||∇( Ẽ(u) -Ẽ(v))|| p ≤ C 14 ||u -v|| W 1,p .
Therefore,

||∇Λu(t) -∇Λv(t)|| p ≤ T 1 2 C 13 d X (u, v) + T C 14 d X (u, v) ≤ T 1 2 C 15 d X (u, v). Finally we get d X (Λu(t) -Λv(t)) ≤ T 1 2 C 16 d X (u, v).
Choosing T ∈ (0, 1) such that T 1 2 C 16 < 1 we obtain that Λ is indeed a strict contraction in X R (T ) and thus has a unique fixed point u.

Furthermore, since u ∈ C [0, T ], W 1,p (Ω) and p > N , we have ∇E(u) ∈ C ([0, T ], L p (Ω))
and we infer from Theorem 3.2 that ω u ∈ C [0, T ], W 2,p (Ω) . Combining this property with the fact that W 1,p (Ω) is an algebra, we realize that both ∇ • (u ω u ) and u E(u) belong to C ([0, T ], L p (Ω)). Classical regularity properties of the heat equation then guarantee that u ∈ C (0, T ], W 2,p (Ω) and is a strong solution to (10).

• Step 3. Thanks to the analysis performed in Steps 1 and 2, the existence and uniqueness of a maximal solution follows by classical argument, see [START_REF] Cazenave | An Introduction to semilinear evolution equations[END_REF] for instance.

• Step 4. Since 0 clearly solves (10), and u 0 ≥ 0, the positivity of u follows from the comparison principle.

Global existence

From now on we choose N = 1, Ω = (-1, 1), p = 2 and we set ϕ = ω u to simplify the notation.

The bistable case:

E(u) = (1 -u)(u -a).
In this case, the system (8) now reads

       ∂ t u = δ ∂ 2 xx u -∂ x (u ϕ) + r u (u -a)(1 -u), x ∈ (-1, 1), t > 0 -ε ∂ 2 xx ϕ + ϕ = (-2u + (a + 1)) ∂ x u, x ∈ (-1, 1), t > 0 ∂ x u(t, ±1) = ϕ(t, ±1) = 0, t > 0, u(0, x) = u 0 (x), x ∈ (-1, 1), (35)
for a some a ∈ (0, 1).

Since E ∈ C 2 (R), Theorem 2.2 ensures that there is a maximal solution u of (35) in C [0, T max ), W 1,p ((-1, 1)) ∩ C (0, T max ), W 2,p (-1, 1) .
To prove Theorem 2.3 we show that, for all T > 0 and t ∈

[0, T ] ∩ [0, T max ), u(t) is bounded in W 1,2 (-1, 1).
We begin the proof by the following lemmas which give some estimates on u and ϕ.

Lemma 5.1. Let the same assumptions as that of Theorem 2.3 hold, and u be the nonnegative maximal solution of (35). Then for all T > 0 there exists C 1 (T ), such that u and ϕ satisfy the following estimates

||u(t)|| 2 ≤ C 1 (T ), for all t ∈ [0, T ] ∩ [0, T max ), ( 36
) t 0 ||∂ x u|| 2 2 dt ≤ C 1 (T ) for all t ∈ [0, T ] ∩ [0, T max ), ( 37
)
and t 0 ||ϕ(t)|| 2 W 1,2 dt ≤ C 1 (T ) for all t ∈ [0, T ] ∩ [0, T max ). ( 38
)
Proof. Multiplying the first equation in (35) by u(t) and integrating it over (-1, 1), we obtain

d dt 1 -1 |u| 2 dx = -2 δ 1 -1 |∂ x u| 2 dx + 2 1 -1 u ϕ ∂ x u dx + 2 r 1 -1 u 2 E(u) dx. ( 39
)
Multiplying now the second equation in (35) by ϕ and integrating it over (-1, 1) we obtain

ε 1 -1 |∂ x ϕ| 2 dx + 1 -1 |ϕ| 2 dx = -2 1 -1 u ϕ ∂ x u dx + (a + 1) 1 -1 ∂ x u ϕ dx. ( 40
)
At this point we notice that the cubic terms on the right hand side of (39) and (40) cancel one with the other, and summing (40) and ( 39) we obtain

d dt ||u|| 2 2 + ε ||∂ x ϕ|| 2 2 + ||ϕ|| 2 2 + 2 δ ||∂ x u|| 2 2 = 2 r 1 -1 u 2 E(u) dx + (a + 1) 1 -1 ∂ x u ϕ dx. (41)
We integrate by parts and use Cauchy-Schwarz inequality to obtain (a + 1)

1 -1 ∂ x u ϕ dx = -(a + 1) 1 -1 u ∂ x ϕ dx ≤ (a + 1) 2 2 ε ||u|| 2 2 + ε 2 ||∂ x ϕ|| 2 2 .
On the other hand, u 2 E(u) ≤ 0 if u / ∈ (a, 1) so that

1 -1 u 2 E(u) dx ≤ 2 (1 -a)
The previous inequalities give that

d dt ||u|| 2 2 + ε 2 ||∂ x ϕ|| 2 2 + ||ϕ|| 2 2 + 2 δ ||∂ x u|| 2 2 ≤ (a + 1) 2 2 ε ||u|| 2 2 + 4 r (1 -a).
Therefore, for all T > 0 there exists C 1 (T ) such that (36), (37) and (38) hold.

Lemma 5.2. Let the same assumptions as that of Theorem 2.3 hold, and u be the nonnegative maximal strong solution of (35). For all T > 0, there is C ∞ (T ) such that

||u(t)|| ∞ ≤ C ∞ (T ), for all t ∈ [0, T ] ∩ [0, T max ). ( 42
)
Proof. The estimates (36) and (37) and the Gagliardo-Nirenberg inequality (20) yield that there exists C 2 (T ) such that

t 0 ||u ∂ x u|| 2 ds ≤ C t 0 ||u|| 1 2 2 ||∂ x u|| 3 2 2 ds ≤ C 2 (T ) for all t ∈ [0, T ] ∩ [0, T max ).
The second equation in (35) and classical elliptic regularity theory ensure that there exists C(T ) such that

t 0 ||ϕ|| W 2,2 ds ≤ C(T ) for all t ∈ [0, T ] ∩ [0, T max ), which gives in particular, since W 2,2 (-1, 1) is embedded in W 1,∞ (-1, 1), t 0 ||∂ x ϕ|| ∞ ds ≤ C 3 (T ) for all t ∈ [0, T ] ∩ [0, T max ). (43)
Now, we multiply the first equation in (35) by q u q-1 where q > 1 and integrate it over (-1, 1) to obtain

d dt 1 -1 |u| q dx = -4 δ (q -1) q 1 -1 |∂ x u q 2 | 2 dx + (q -1) 1 -1 ϕ ∂ x u q dx + r q 1 -1 u q (u -a) (1 -u) dx ≤ -(q -1) 1 -1 ∂ x ϕ u q dx + r q 2 (1 -a)
Using Hölder's inequality , we obtain

d dt ||u|| q q ≤ (q -1) ||∂ x ϕ|| ∞ ||u|| q q + 2 q r. (44)
Introducing

φ(t) = t 0 ||∂ x ϕ(s)|| ∞ ds ≤ C 3 (T ) for all t ∈ [0, T ] ∩ [0, T max ),
the bound being a sequence of (43), we integrate (44) and find ||u(t)|| q q ≤ ||u 0 || q q e (q-1) φ(t) + 2 q r t 0 e (q-1) (-φ(s)+φ(t)) ds ≤ (||u 0 || q q + 2 q r) T e q C(T ) , ||u(t)|| q ≤ (||u 0 || q q + 2 q r) T

1 q e C(T ) for all t ∈ [0, T ] ∩ [0, T max ).

Consequently, by letting q tend to ∞, we see that there exists C ∞ (T ) such that

||u(t)|| ∞ ≤ C ∞ (T ), for all t ∈ [0, T ] ∩ [0, T max ).
Lemma 5.3. Let the same assumptions as that of Theorem 2.3 hold, and u be the nonnegative maximal strong solution of (35). For all T > 0, there is C 4 (T ) such that

||∂ x u(t)|| 2 ≤ C 4 (T ) for all t ∈ [0, T ] ∩ [0, T max ). ( 45
)
Proof. We multiply the first equation in ( 35) by (-∂ 2 xx u) and integrate it over (-1, 1) to obtain

1 2 d dt 1 -1 |∂ x u| 2 dx = -δ 1 -1 |∂ 2 xx u| 2 dx + 1 -1 ∂ x (u ϕ) ∂ 2 xx u dx + r 1 -1 u (1 -u) (u -a) (-∂ 2 xx u) dx = -δ 1 -1 |∂ 2 xx u| 2 dx + 1 -1 (u ∂ x ϕ + ∂ x u ϕ) ∂ 2 xx u dx + r 1 -1 u (-u 3 + (a + 1)u 2 -a) ∂ 2 xx u dx.
Using Cauchy-Schwarz inequality and Lemma 5.2 we obtain,

1 2 d dt 1 -1 |∂ x u| 2 dx ≤ -δ 1 -1 |∂ 2 xx u| 2 dx + δ 3 1 -1 |∂ 2 xx u| 2 dx + C ||u|| 2 ∞ 1 -1 |∂ x ϕ| 2 dx + δ 3 1 -1 |∂ 2 xx u| 2 dx + C ||ϕ|| 2 ∞ 1 -1 |∂ x u| 2 dx + δ 3 1 -1 |∂ 2 xx u| 2 dx + C ||(u (-u 3 + (a + 1)u 2 -a))|| 2 ∞ ≤ C(T ) 1 -1 |∂ x ϕ| 2 dx + C ||ϕ(t)|| 2 ∞ 1 -1 |∂ x u| 2 dx + C(T ). ( 46
)
Using (38) and Sobolev embedding theorem we obtain the following estimate

t 0 ||ϕ|| 2 ∞ ds ≤ C(T ) for all t ∈ [0, T ] ∩ [0, T max ). ( 47
)
Since ( 38) and ( 47) hold, then it follows from (46) after integration that

||∂ x u(t)|| 2 2 ≤ C 4 (T ), for all t ∈ [0, T ] ∩ [0, T max ).
It remains to prove Theorem 2.3.

Proof of Theorem 2.3. For all T > 0, Lemma 5.3 and the estimate (36) ensure that

||u(t)|| W 1,2 ≤ C(T ), for all t ∈ [0, T ] ∩ [0, T max ),
which guarantees that u cannot explode in W 1,2 (-1, 1) in finite time and thus that T max = ∞.

The monostable case: E(u) = 1u

For this choice of E, system (8) now reads

       ∂ t u = δ ∂ 2 xx u -∂ x (u ϕ) + r u (1 -u) x ∈ (-1, 1), t > 0 -ε ∂ 2 xx ϕ + ϕ = -∂ x u, x ∈ (-1, 1), t > 0 ∂ x u(t, ±1) = ϕ(t, ±1) = 0 t > 0, u(0, x) = u 0 (x) x ∈ (-1, 1), (48)
Since E ∈ C 2 (R), Theorem 2.2 ensures that there is a maximal solution u of (48) in

C [0, T max ), W 1,p (-1, 1) ∩ C (0, T max ), W 2,p (-1, 1) .
In contrast to the previous case, it does not seem to be possible to begin the global existence proof with an L ∞ (L 2 ) estimate on u. Nevertheless, there is still a cancellation between the two equations which actually gives us an L ∞ (L log L) bound on u and a L 2 bound on ∂ x √ u. Integrating (48) over (0, t) × (-1, 1) and using the nonnegativity of u, we first observe that,

||u(t)|| 1 ≤ ||u 0 || 1 + 2 r t, for all t ∈ [0, T max ). ( 49
)
To prove Theorem 2.4 we need to prove the following lemmas:

Lemma 5.4. Let the same assumptions as that of Theorem 2.4 hold, and let u be the maximal solution of (48). Then for all T > 0, there exists a constant C 1 (T ) such that

t 0 ||∂ x √ u|| 2 2 ds ≤ C 1 (T ), for all t ∈ [0, T ] ∩ [0, T max ), ( 50
)
and t 0 ||ϕ|| 2 W 1,2 ds ≤ C 1 (T ), for all t ∈ [0, T ] ∩ [0, T max ). ( 51
)
Proof. The proof goes as follows. On the one hand, we multiply the first equation in ( 48) by (log u + 1) and integrate it over (-1, 1). Since u (1 -u) log u ≤ 0 and u (1

-u) ≤ 1, d dt 1 -1 u log u dx = - 1 -1 (δ ∂ x u -u ϕ) ( 1 u ∂ x u) dx + r 1 -1 u (1 -u) (log u + 1) dx ≤ - 1 -1 δ u (∂ x u) 2 dx + 1 -1 ϕ ∂ x u dx + 2 r. (52)
On the other hand, we multiply the second equation in (48) by ϕ and integrate it over (-1, 1) to obtain

ε 1 -1 |∂ x ϕ| 2 dx + 1 -1 |ϕ| 2 dx = - 1 -1 ∂ x u ϕ dx. (53)
Adding ( 53) and (52) yields

d dt 1 -1 u log u dx + ε ||∂ x ϕ|| 2 2 + ||ϕ|| 2 2 ≤ -4 δ 1 -1 |∂ x √ u| 2 dx + 2 r. (54)
Finally, (50) and (51) are obtained by a time integration of (54).

Lemma 5.5. Let the same assumptions as that of Theorem 2.4 hold, and let u be the maximal solution of (48). Then for all T > 0, there exists a constant C 2 (T ) such that

||u(t)|| 2 ≤ C 2 (T ), for all t ∈ [0, T ] ∩ [0, T max ), ( 55
)
and t 0 ||∂ x u|| 2 2 ds ≤ C 2 (T ), for all t ∈ [0, T ] ∩ [0, T max ). ( 56
) Proof. A simple computation shows that, since u 2 (1 -u) ≤ 1, d dt 1 -1 u 2 dx ≤ -2 δ 1 -1 |∂ x u| 2 dx + 2 1 -1 u ϕ ∂ x u dx + 4 r. ( 57
)
Using Cauchy-Schwarz inequality, Gagliardo-Nirenberg inequality (20), Young inequality and (49) we obtain that for all T > 0,

2 1 -1 u ϕ ∂ x u dx = - 1 -1 u 2 ∂ x ϕ dx ≤ ||u|| 2 4 ||∂ x ϕ|| 2 ≤ C ||u|| 1 2 W 1,2 ||u|| 1 2 1 2 ||∂ x ϕ|| 2 ≤ C(T ) ||∂ x ϕ|| 2 ||u|| W 1,2 ≤ δ ||u|| 2 W 1,2 + C(T ) ||∂ x ϕ|| 2 2
We substitute the previous inequality in (57) to obtain

d dt ||u|| 2 2 ≤ -2 δ ||∂ x u|| 2 2 + δ ||u|| 2 W 1,2 + C(T ) ||∂ x ϕ|| 2 2 + 4 r. ( 58
)
Integrating (58) in time, and using (51) yield that there exists C 3 (T ) such that (55) and (56) hold.

Now we are in a position to show the global existence of solution to (48).

Proof of Theorem 2.4 (global existence).

By elliptic regularity, and the continuous embedding of W 2,2 (-1, 1) in W 1,∞ (-1, 1), we have

||∂ x ϕ(t)|| ∞ ≤ C ||ϕ(t)|| W 2,2 ≤ C ||E ′ (u) ∂ x u|| 2 ≤ C ||∂ x u|| 2 ,
which together with (56), implies that

t 0 ||∂ x ϕ(s)|| 2 ∞ ds ≤ C t 0 ||∂ x u(s)|| 2 2 ds ≤ C(T ), for all t ∈ [0, T ] ∩ [0, T max ).
Thanks to this estimate, we now argue as in the proof of Lemma 5.2 and Lemma 5.

3 to get that ||u(t)|| ∞ + ||∂ x u(t)|| 2 ≤ C(T ), for all t ∈ [0, T ] ∩ [0, T max ).
Thus, the maximal solution u of (48) cannot explode in finite time.

To complete the proof of Theorem 2.4, it remains to prove the asymptotic behaviour of u when t → ∞. We note that we have the following lemma which controls the L 1 (-1, 1) norm of u. For f ∈ L 1 (-1, 1), we set

< f >= 1 2 1 -1 f (x) dx
Lemma 5.6. Let the same assumptions as that of Theorem 2.4 hold, and let u be the nonnegative global solution of (48). For r > 0, there exists a constant C 0 > 0 such that

0 ≤< u(t) >≤ C 0 , t ∈ (0, ∞), (59)
and if r = 0

< u(t) >=< u 0 >= 1 2 ||u 0 || 1 , t ∈ (0, ∞). ( 60
)
Proof. We note that if r = 0, d dt < u >= 0, so that

< u(t) >= 1 2 ||u(t)|| 1 = 1 2 ||u 0 || 1 . If r > 0, d dt < u(t) > = r < u(t) > -r < u 2 (t) > ≤ r < u(t) > -r < u(t) > 2 ,
whence < u(t) >≤ max {1, < u 0 >}.

Next we turn to the existence of a Liapunov functional for (48) which is the cornerstone of our analysis. Lemma 5.7. Let the same assumptions as of that Theorem 2.4 hold, and let u be the nonnegative global solution of (48). There exists a constant C 1 such that

r ∞ 0 1 -1 u (u -1) log u dx dt + ∞ 0 ||∂ x √ u|| 2 2 + ε ||∂ x ϕ|| 2 2 + ||ϕ|| 2 2 dt ≤ C 1 , ( 61
)
and ∞ 0 ||ϕ|| 2 ∞ ds ≤ C 1 . ( 62
)
Proof. Let us define the following functional L

L(u) = 1 -1 (u log u -u + 1) dx ≥ 0,
and show that it is a Liapunov functional. Indeed

d dt L(u) = -δ 1 -1 |∂ x u| 2 u dx + 1 -1 ϕ ∂ x u dx + r 1 -1 u log u (1 -u) dx. ( 63
)
Combining ( 63) and ( 53) we obtain that

d dt L(u) = -4 δ 1 -1 |∂ x √ u| 2 dx -ε 1 -1 |∂ x ϕ| 2 dx - 1 -1 |ϕ| 2 dx + r 1 -1 u log u (1 -u) dx. = -D(u, ϕ) ≤ 0, ( 64
) since u log u (1 -u) ≤ 0. Then for all t ≥ 0 L(u(t)) + t 0 D(u(s), ϕ(s)) ds ≤ L(u 0 ).
Since u 0 and u are nonnegative , we have

L(u 0 ) ≤ 1 -1 u 2 0 + 1 dx ≤ ||u 0 || 2 2 + 2,
and

L(u(t)) ≥ - 2 e - 1 -1 u 0 dx, so that t 0 D(u(s), ϕ(s)) ds ≤ 1 + ||u 0 || 2 2 + 2 e + 2 < u 0 >, t ≥ 0. ( 65
)
Therefore, (65) yields there exists

C 1 > 0 such that ∞ 0 D(u, ϕ) dt ≤ C 1 . (66)
From (66), we see that (61) holds true. In addition, inequality (61) together with Sobolev's embedding theorem give (62).

In the following lemma we show that {u(t) : t ≥ 0} is bounded in W 1,2 (-1, 1).

Lemma 5.8. Let the same assumptions as that of Theorem 2.4 hold, and let u be the nonnegative global solution of (48). Then u belongs to L ∞ (0, ∞); W 1,2 (-1, 1) .

Proof. First,

d dt ||u-< u > || 2 2 = d dt ||u|| 2 2 - 1 -1 ∂ t u dx 1 -1 u dx = d dt ||u|| 2 2 -4 r < u > 2 +4 r < u > < u 2 > ≤ d dt ||u|| 2 2 + 2 r C 0 ||u|| 2 2 . ( 67
)
Multiplying the first equation in (48) by 2 u, integrating it over (-1, 1), and using the Cauchy-Schwarz inequality and the fact that u ≥ 0 we obtain

d dt ||u|| 2 2 ≤ -2 δ 1 -1 |∂ x u| 2 dx - 1 -1 u 2 ∂ x ϕ dx + 2 r 1 -1 u 2 (1 -u) dx ≤ -2 δ ||∂ x u|| 2 2 + ||u|| 2 4 ||∂ x ϕ|| 2 + 2 r. (68)
Gagliardo-Nirenberg inequality (20) together with the Poincaré inequality ( 19) and ( 59) give

||u|| 2 4 ≤ C ||u|| W 1,2 ||u|| 1 ≤ C ( ||∂ x u|| 2 + ||u|| 1 ) ≤ C (||∂ x u|| 2 + 1) . (69)
Thus

||u|| 2 4 ||∂ x ϕ|| 2 ≤ C (||∂ x u|| 2 + 1) ||∂ x ϕ|| 2 . (70)
Substituting (70), ( 69) and ( 68) into (67), and using Young and Hölder inequalities to obtain

d dt ||u-< u > || 2 2 ≤ -2 δ ||∂ x u|| 2 2 + C (||∂ x u|| 2 + 1) ||∂ x ϕ|| 2 + 2r + 2r C 0 ||u|| 2 3 1 ||u|| 4 3 4 ≤ -2 δ ||∂ x u|| 2 2 + C ||∂ x ϕ|| 2 2 + δ ||∂ x u|| 2 2 + C. Using Poincaré's inequality we get d dt ||u(t)-< u(t) > || 2 2 + α ||u(t)-< u(t) > || 2 2 ≤ C ||∂ x ϕ(t)|| 2 2 + C,
for some α > 0 independent of t. Integrating this differential inequality gives

e α t ||u(t)-< u(t) > || 2 2 ≤ ||u 0 -< u 0 > || 2 2 + t 0 e α s C ||∂ x ϕ(s)|| 2 2 + C ds ||u(t)-< u(t) > || 2 2 ≤ C e -α t + C t 0 e α (s-t) (||∂ x ϕ(s)|| 2 2 + 1) ds
Since e -α t ≤ 1, and e α (s-t) ≤ 1 as s ≤ t we obtain

||u(t)-< u(t) > || 2 2 ≤ C + C t 0 ||∂ x ϕ(s)|| 2 2 ds + 1 α ,
Using (61) we end up with

||u(t)-< u(t) > || 2 2 ≤ C, (71)
where C is independent of t. Therefore, u belongs to L ∞ ((0, ∞); L 2 (-1, 1)).

It remains to show that ∂ x u is in L ∞ ((0, ∞); L 2 (-1, 1)). We multiply the first equation in (48) by -∂ 2 xx u and integrate it over (-1, 1). Since u ≥ 0 we use Cauchy-Schwarz and Young inequalities and (71) to obtain

1 2 d dt ||∂ x u|| 2 2 ≤ -δ ||∂ 2 xx u|| 2 2 + 1 -1 ∂ 2 xx u (∂ x u ϕ + ∂ x ϕ u) -r 1 -1 u (1 -u) ∂ 2 xx u dx ≤ -δ ||∂ 2 xx u|| 2 2 - 1 2 1 -1 ∂ x ϕ |∂ x u| 2 dx + δ 8 ||∂ 2 xx u|| 2 2 + C 1 -1 |u| 2 |∂ x ϕ| 2 dx -r 1 -1 u ∂ 2 xx u dx -2 r 1 -1 u |∂ x u| 2 dx ≤ -δ ||∂ 2 xx u|| 2 2 + 1 2 ||∂ x ϕ|| 2 ||∂ x u|| 2 4 + δ 8 ||∂ 2 xx u|| 2 2 + C ||u|| 2 ∞ ||∂ x ϕ|| 2 2 + C + δ 8 ||∂ 2 xx u|| 2 2 . (72)
Since ∂ x u ∈ W 1,2 0 (-1, 1), using the Gagliardo-Nirenberg inequality (20) and the classical Poincaré inequality (19), we obtain

||∂ x u|| 4 ≤ C ||∂ 2 xx u|| 1 2 2 ||∂ x u|| 1 2 1 . (73)
Then, we substitute (73) into (72), and by Young inequality, the Sobolev embedding, ( 59) and (71), we obtain

1 2 d dt ||∂ x u|| 2 2 ≤ - 3 δ 4 ||∂ 2 xx u|| 2 2 + C ||∂ x ϕ|| 2 ||∂ 2 xx u|| 2 ||∂ x u|| 1 + C ||u|| 2 ∞ ||∂ x ϕ|| 2 2 + C ≤ - δ 2 ||∂ 2 xx u|| 2 2 + C ||∂ x ϕ|| 2 2 ||∂ x u|| 2 1 + C ||u|| 2 W 1,2 ||∂ x ϕ|| 2 2 + C ≤ - δ 2 ||∂ 2 xx u|| 2 2 + C ||∂ x u|| 2 2 ||∂ x ϕ|| 2 2 + C (||∂ x ϕ|| 2 2 + 1). (74)
Since ∂ x u ∈ W 1,2 0 (-1, 1), we use once more the classical Poincaré inequality to obtain

d dt ||∂ x u|| 2 2 + β||∂ x u|| 2 2 ≤ C ||∂ x u|| 2 2 ||∂ x ϕ|| 2 2 + C (||∂ x ϕ|| 2 2 + 1),
for some β > 0 independent of t.

Define

φ(t) = t 0 ||∂ x ϕ(s)|| 2 ds, t ≥ 0,
and notice that , since ||∂ x ϕ|| 2 2 belongs to L 1 (0, ∞) by (61),

0 ≤ φ(t) ≤ φ ∞ = ∞ 0 ||∂ x ϕ(s)|| 2 2 ds.
Integrating the previous differential inequality we find

||∂ x u(t)|| 2 2 ≤ ||∂ x u 0 || 2 2 e Cφ(t)-β t + C t 0
(1 + φ ′ (s)) e β(s-t)+Cφ(t)-Cφ(s) ds

≤ ||∂ x u 0 || 2 2 e C φ∞ + C e C φ∞ t 0 [e β(s-t) + φ ′ (s)] ds ≤ ||∂ x u 0 || 2 2 e C φ∞ + C e C φ∞ 1 β + φ ∞ .
Therefore, ∂ x u belongs to L ∞ (0, ∞), L 2 (-1, 1) , and Lemma 5.8 is proved.

Lemma 5.9. Let the same assumptions as that of Theorem 2.4 hold, and let u be the nonnegative global solution of (48). There is

C 2 such that ∞ 0 ||∂ t u|| 2 2 dt ≤ C 2 . ( 75
)
Proof. We multiply the first equation in (48) by ∂ t u and integrate it over (-1, 1) to obtain

1 -1 (∂ t u) 2 dx = δ 1 -1 ∂ 2 xx u ∂ t u dx - 1 -1 ∂ x (u ϕ) ∂ t u dx + r 1 -1 u (1 -u) ∂ t u dx = - δ 2 d dt ||∂ x u|| 2 2 - 1 -1 (∂ x u ϕ + u ∂ x ϕ) ∂ t u dx + r 1 -1 (u -u 2 ) ∂ t u dx.
Using Young and Cauchy-Schwarz inequalities we obtain

||∂ t u|| 2 2 ≤ - δ 2 d dt ||∂ x u|| 2 2 + 1 -1 |∂ x u| 2 |ϕ| 2 dx + 1 4 ||∂ t u|| 2 2 + 1 -1 |u| 2 |∂ x ϕ| 2 dx + 1 4 ||∂ t u|| 2 2 + r d dt 1 -1 u 2 2 - u 3 3 dx, which gives d dt δ 2 ||∂ x u|| 2 2 + 1 -1 F (u) dx + 1 2 ||∂ t u|| 2 2 ≤ ||∂ x u|| 2 2 ||ϕ|| 2 ∞ + ||u|| 2 ∞ ||∂ x ϕ|| 2 2 ,
where

F (u) = r -u 2 2 + u 3 3 ≥ -r 6 .
Next we integrate the above inequality in time, and use (62), (61) and Lemma 5.8 to obtain

- r 3 + 1 2 t 0 ||∂ t u|| 2 2 ds ≤ C + C t 0 ||ϕ|| 2 ∞ + ||∂ x ϕ|| 2 2 ds ≤ C
for t ≥ 0 where C is independent of t. We have thus proved (75).

To end the proof of Theorem 2.4, our aim now is to look at the large time behaviour of the solution.

Proof of Theorem 2.4, (large time behaviour). In this proof, we follow [START_REF] Langlais | Stabilization of solutions of nonlinear and degenerate evolution equations[END_REF]. By Lemma 5.8, the family {u(t), t ≥ 0} is bounded in W 1,2 (-1, 1). Since the embedding of W 1,2 (-1, 1) in L 2 (-1, 1) is compact then, there are a sequence of positive time (t n ), such that t n → ∞, and z ∈ L 2 (-1, 1) such that z = lim n→∞ u(t n ) in L 2 (-1, 1) and a.e. in (-1, 1).

Consider

U n (s, x) = u(t n + s, x), x ∈ (-1, 1), -1 < s < 1, n > 0, and Φ n (s, x) = ϕ(t n + s, x), -1 < s < 1.

We first prove that

U n -→ z as n → ∞, in C [-1, 1]; L 2 (-1, 1) . ( 76
)
Indeed for each s ∈ (-1, 1)

1 -1 |u(t n + s, x) -u(t n , x)| 2 dx ≤ 1 -1 tn+1 tn-1 |∂ t u| 2 dt dx. Hence sup s∈[-1,1] ||U n (s) -u(t n )|| 2 ≤ 2 1 -1 ∞ tn-1 |∂ t u| 2 dt dx 1 2
.

The right hand side goes to zero as n → ∞ by Lemma 5.9. Letting n → ∞ in the above inequality gives (76).

Next, using the definition of D(u, ϕ) which is given in (64) we obtain that

1 -1 r ||U n log U n (U n -1)|| 1 + ||∂ x U n || 2 2 + ||Φ n || 2 2 + ε ||∂ x Φ n || 2 2 ds ≤ tn+1 tn-1 r ||u(s) log u(s) (u(s) -1)|| 1 + ||∂ x u(s)|| 2 2 + ||ϕ(s)|| 2 2 + ε ||∂ x ϕ(s)|| 2 2 ds ≤ 2 ∞ tn-1 D(u, ϕ) ds. (77)
The right-hand side of (77) goes to zero as n → ∞ by (66), so that

Φ n -→ 0 as n → ∞, in L 2 (-1, 1); W 1,2 (-1, 1) .
In addition, using Cauchy-Schwarz inequality, (59) and ( 77) we obtain

1 -1 ||∂ x U n (s)|| 2 1 ds = 4 1 -1 1 -1 U n (s) |∂ x U n (s)| dx 2 ds ≤ 4 1 -1 ||U n (s)|| 1 ||∂ x U n (s)|| 2 2 ds ≤ C ∞ tn-1 D(u, ϕ) ds.
Since the right-hand side goes to zero as n → ∞ by (66), then we have

∂ x U n -→ 0 as n → ∞, in L 2 (-1, 1); L 1 (-1, 1) . ( 78
)
Since the limit in the sense of distribution is unique, (76) and (78) yield that

∂ x z = 0. ( 79
)
If r = 0, (79) together with (60) and (76) give that z =< u 0 >. We have thus shown that < u 0 > is the only cluster point of {u(t), t ≥ 0}. Since {u(t), t ≥ 0} is relatively compact in L 2 (-1, 1) thanks to its boundedness in W 1,2 (-1, 1) (see Lemma 5.8), we conclude that u(t) converges to < u 0 > in L 2 (-1, 1) as t -→ ∞.

If r > 0, by (77) we have

1 -1 ||U n log U n (U n -1)|| 1 ds -→ 0, as n → ∞, (80) Since
(U n ) is bounded in L ∞ ((-1, 1) × (-1, 1
)) thanks to the boundness of {u(t), t ≥ 0} in W 1,2 (-1, 1) and the embedding of W 1,2 (-1, 1) in L ∞ (-1, 1), we infer from (76), ( 79), (80) that z log z (z -1) = 0, that is z = 0 or z = 1. Therefore 0 and 1 are the only two cluster points of {u(t), t ≥ 0} as t → ∞. Since the ω-limit set of u is a compact connected subset of L 2 (-1, 1), see [START_REF] Cazenave | An Introduction to semilinear evolution equations[END_REF]Theorem 9.1.8] for instance, we conclude that u(t) converges either to 0 or to 1 in L 2 (-1, 1) as t -→ ∞.

6 Limiting behaviour as ε → 0

When E(u) = 1 -u, letting ε → 0 in (48) formally leads to (9) which is well-posed since E ′ < 0 and δ > 0. The purpose of this section is to justify rigorously this fact and prove Theorem 2.6. Let T > 0, δ > 0, r ≥ 0, ε > 0 and a nonnegative initial condition u 0 ∈ W 1,2 (-1, 1). We discuss the limit as ε → 0 of the unique solution u ε of

   ∂ t u ε = δ ∂ 2 xx u ε -∂ x (u ε ϕ ε ) + r u ε (1 -u ε ) in (0, T ) × (-1, 1), u ε (0, x) = u 0 (x) in (-1, 1), ∂ x u ε (t, ±1) = 0 on (0, T ), (81)
given by Theorem 2.4, where ϕ ε is the unique solution of

-ε ∂ 2 xx ϕ ε + ϕ ε = -∂ x u ε in (0, T ) × (-1, 1), ϕ ε (t, ±1)
= 0 on (0, T ).

Estimates

Lemma 6.1. There is C 1 (T ) independent of ε such that

T 0 δ ||∂ x √ u ε || 2 2 + ε ||∂ x ϕ ε || 2 2 + ||ϕ ε || 2 2 dt ≤ C 1 (T ), (82)
Proof. By (54) see (the proof of Lemma 5.4), we have

d dt 1 -1 u ε log u ε dx + ε ||∂ x ϕ ε || 2 2 + ||ϕ ε || 2 2 ≤ -4 δ 1 -1 |∂ x √ u ε | 2 dx + 2 r,
from which (82) follows by a time integration.

Using Gagliardo-Nirenberg inequality (20) we obtain the following estimate: Lemma 6.2. For 2 ≤ p ≤ 6, there exists C 2 (T, p) independent of ε such that

T 0 ||u ε || p 2 p 2 dt ≤ C 2 (T, p). (83)
Proof. For t ∈ (0, T ), thanks to (82), we can use Gagliardo-Nirenberg inequality (20) on √ u ε and we obtain for all p ∈ [2, ∞)

|| u ε (t)|| p ≤ C || u ε (t)|| θ W 1,2 || u ε (t)|| 1-θ 2 , (84)
where

θ = p -2 2 p . Therefore || u ε (t)|| p p ≤ C || u ε (t)|| p-2 2 W 1,2 || u ε (t)|| p+2 2 2 ||u ε (t)|| p 2 p 2 ≤ C || u ε (t)|| p-2 2 W 1,2 ||u ε (t)|| p+2 4 1 . Since ||u ε (t)|| 1 ≤ ||u 0 || 1 + 2 r T, and p-2 2
≤ 2 for 2 ≤ p ≤ 6 , the estimate (83) follows from (82) and the previous inequalities.

Lemma 6.3. There is C 3 (T ) independent of ε such that T 0 ||∂ x u ε || 3 2 3 2 dt ≤ C 3 (T ). (85)
Proof. Hölder and Young inequalities together with (82) and ( 83) with p = 6 yield

T 0 ||∂ x u ε || 3 2 3 2 dt ≤ 2 3 2 T 0 || √ u ε ∂ x √ u ε || 3 2 3 2 dt ≤ 2 3 2 T 0 ||u ε || 3 4 3 ||∂ x √ u ε || 3 2 2 dt ≤ C T 0 ||u ε || 3 3 + ||∂ x √ u ε || 2 2 dt ≤ C 3 (T ),
which gives the result.

Lemma 6.4. There is C 4 (T ) independent of ε such that

T 0 ||∂ t u ε || (W 2, 3
2 ) ′ dt ≤ C 4 (T ).

Proof. Consider ψ ∈ W 2, 3 2 (-1, 1) and t ∈ (0, T ), we have

1 -1 ∂ t u ε ψ dx = 1 -1 [∂ x (δ ∂ x u ε -u ε ϕ ε ) + r u ε E(u ε )] ψ dx = 1 -1 (-δ ∂ x u ε ∂ x ψ + u ε ϕ ε ∂ x ψ + r u ε (1 -u ε ) ψ) dx ≤ δ ||∂ x ψ|| ∞ ||∂ x u ε || 1 + ||∂ x ψ|| ∞ ||u ε || 2 ||ϕ ε || 2 + r ||u ε (1 -u ε )|| 1 ||ψ|| ∞ .
Using the embedding of W 2, 3 2 (-1, 1) in W By Lemma 6.1, Lemma 6.3 and Lemma 6.2 with p = 4 the right-hand side of the above inequality is bounded independently of ε and the proof of Lemma 6.4 is complete.

Convergence

In this section we discuss the limit of u ε as ε → 0. For that purpose, we study the compactness properties of (u ε , ϕ ε ).

Proof of Theorem 2.6. Thanks to Lemma 6.2 and Lemma 6.3, (u ε ) ε is bounded in L 2 ) ′ (-1, 1), it follows from [9, Corollary 4] that (u ε ) ε is relatively compact in L 3 2 ((0, T ); C([-1, 1])). Therefore, there are a sequence (ε j ) of positive real numbers, ε j → 0, and u ∈ L ∞ dt → 0, when ε j → 0. In particular, we have u ε j -→ u, in L 2 ((0, T ) × (-1, 1)) .

(88)

Observe that the nonnegativity of u follows easily from that of (u ε j ) by (88).

Owing to Lemma 6.1, we may also assume that ϕ ε j ⇀ ϕ in L 2 ((0, T ) × (-1, 1)) as ε j → 0, (89) (-δ ∂ x u ε j + u ε j ϕ ε j ) ∂ x ψ + r u ε j E(u ε j ) ψ dxdt (91) and

ε j ∂ x ϕ ε j → 0 in
ε j T 0 1 -1 ∂ x ϕ ε j ∂ x ψ dxdt + T 0 1 -1 ϕ ε j ψ dxdt = - T 0 1 -1 ∂ x u ε j ψ dxdt. (92)
Owing to (86), ( 89) and (90), it is straightforward to pass to the limit as ε j → 0 in (92) and find for all test functions ψ. Therefore, u is a weak solution of (15), and classical regularity results ensure that u is actually a classical solution of (15). Since it is unique and the only possible cluster point of (u ε ) ε in L 2 ((0, T ) × (-1, 1)), we conclude that the whole family (u ε ) ε converges to u in L 2 ((0, T ) × (-1, 1)) as ε → 0.

3 2 (

 2 (0, T ); W 1, 3 2 (-1, 1)) while (∂ t u ε ) ε is bounded in L 1 ((0, T ); (W 2,3 2 ) ′ (-1, 1)) by Lemma 6.4. Since W 1, 3 2 (-1, 1) is compactly embedded in C([-1, 1]) and C([-1, 1]) is continuously embedded in (W 2,

3

 3

u ε j → u in L 3 2 0 ||u ε j -u|| 3 2 dt ≤ T 0 ||u ε j -u|| 3 2 1 ||u ε j -u|| 3 2 ∞ dt ≤ T 0 ||u ε j -u|| 3 2

 3002202 ((0, T ); C[-1, 1]) and a.e. in (0, T ) × (-1, 1). (87)Since (u ε ) ε is bounded in L ∞ (0, T ); L 1 (-1, 1) by (59), it follows from (87) T

1 - 1 ∂ t u ε j ψ dxdt = T 0 1 - 1

 1111 L 2 ((0, T ) × (-1, 1)) as ε j → 0. (90) It remains to identify the equations solved by the limit u of (uε j ). Let ψ ∈ C 2 ([0, T ] × [-1, 1]) with ψ(T ) = 0. Since T 0

1 ∂ 1 - 1 ∂ 1 - 1 u ∂ t ψ dxdt - 1 - 1 u 0 1 - 1 ∂ 1 - 1 ∂ 1 - 1 u 1 - 1 uFrom 1 u 1 - 1 u 0 ∂ t u, ψ dt = T 0 1 - 1 (

 1111111011111111111011 x u ψ dxdt,which gives that ϕ = -∂ x u.(93)Next, by (87) and (86) we see thatT 0 t u ε j ψ dxdt -→ -T 0 (x) ψ(0, x) dx, as ε j → 0,andT 0 x u ε j ∂ x ψ dxdt -→ T 0 x u ∂ x ψ dxdt as ε j → 0.From (88), (89) and (93) we see thatT 0 ε j ϕ ε j ∂ x ψ dxdt -→ -T 0 ∂ x u ∂ x ψ dxdt, as ε j → 0. ε j E(u ε j ) ψ dxdt -→ r T 0 E(u) ψ dxdt, as ε j → 0.Thus we conclude that u satisfiesT (-δ ∂ x u -u ∂ x u) ∂ x ψ + r u E(u) ψ) dx dt,

  1,∞ (-1, 1), and Young's inequality, we end up with∂ t u ε ψ dx ≤ δ ||∂ x u ε || 1 + ||u ε || 2 ||ϕ ε || 2 + r ||u ε || 1 + r ||u ε || 2 2 ||ψ|| ) ′ ≤ C ||∂ x u ε || 3 2 + ||u ε || 2 2 + ||ϕ ε || 2 2 + 1 .Integrating the above inequality over (0, T ) and using Young's inequality we obtain+ ||u ε || 2 2 + ||ϕ ε || 2 2 + 1 dt.

	1
	-1				W 2, 3 2
		≤ C ||∂ x u ε || 3 2	+ ||u ε || 2 2 + ||ϕ ε || 2 2 + 1 ||ψ|| W 2, 3 2 ,
	and a duality argument gives
		||∂ t u ε (t)|| (W 2, 3 2 T	T		3
	0	||∂ t u ε (t)|| (W 2, 3 2 ) ′ dt ≤ C(T )	0	||∂ x u ε ||	2 3 2

Acknowledgment

I thank Philippe Laurençot for his helpful advices and comments during this work.