Yazid Saidi

Bernard Cousin

Resource Sharing Strategies: Which Sharing to Better Protect Primary Shortest Paths?

Keywords: Routing, backup path, local path protection, resource sharing, shortest paths, MPLS, virtual networks

With the widespread use of real-time applications (VoIP, IPTV, Video conference, etc.) in Internet, protection and resource optimization become increasingly desired. Network protection aims to decrease the interruption time of communications by precomputing backup paths capable to receive and route traffics of affected primary paths upon failures. To enhance the utilization of network resources, resource allocation may be optimized by improving data routing and resource sharing: data routing is often optimized by following the shortest paths whereas resource sharing is applied between the backup paths protecting against different failure risks. Two strategies of resource sharing are defined in literature: 1) restrained sharing which limits the resource sharing to the backup paths and 2) global sharing which extends the resource sharing to the primary and backup paths. In this paper, we study the impact of resource sharing strategies on the resource utilization when the primary paths correspond to the shortest ones according to a strictly positive and static metric. With the single failure assumption, we show formally that the resource sharing between primary and backup paths is limited to some few links which cannot form a backup path. Thus, independently of the amount of resources (for instance: bandwidth) that can be shared between the primary and backup paths, the maximum number of backup paths is bounded. In our simulation, we comfort our formal result by showing that the two strategies have close rates of backup path rejection and backup bandwidth utilization.

I. Introduction

Most of today's applications (IPTV, videoconferences, VoIP, etc.) are very sensitive to the disruption of communications and consume more and more resources (such as bandwidth). Hence, protection against failures is becoming very desirable to prevent or reduce the disruption time of communications. However path protection consumes network resources if backup paths are preconfigured and their resource reserved, thus resource optimization is required to improve the network resource utilization.

Network protection [START_REF] Balon | A Scalable and Decentralized Fast-Rerouting Scheme with Efficient Bandwidth Sharing[END_REF], [START_REF] Kodialam | Dynamic Routing of Restorable Bandwidth-Guaranteed Tunnels using Aggregated Network Resource Usage Information[END_REF], [START_REF] Pan | Fast Reroute Extensions to RSVP-TE for LSP Tunnels[END_REF], [START_REF] Saidi | PLR-based Heuristic for Backup Path Computation in MPLS Networks[END_REF], [START_REF] Vasseur | Framework for PCE-based MPLS-TE Fast Reroute Backup Path Computation[END_REF] maintains the communication service continuity by precomputing and generally pre-configuring backup paths capable to reroute traffics of affected primary paths upon a failure. To ensure resource availability 1 after a failure repair, the primary and backup paths should reserve their resources. Whereas the primary paths really use the resources they reserved, the backup paths doesn't consume any resources before a failure occurrence. Hence, under M.-Y. SAIDI is with the L2TI/Institut Galilée, Université Paris 13 Sorbonne Paris Cité, 99 Avenue Jean Baptiste Clément, 93430 Villetaneuse, France, email: saidi@univ-paris13.fr B. COUSIN is with the IRISA/INRIA, Université de Rennes I, France, email: bcousin@irisa.fr 1 In the rest of this document, resource refers to bandwidth.

the single failure assumption, resources can be shared between all the backup paths which protect against different failure risks, since these backup paths cannot be active at the same time. In addition of the resource sharing, the primary paths should follow the shortest paths in order to achieve resource optimization.

For instance, the internet routing protocols (RIP [START_REF] Malkin | RIP Version 2[END_REF] and OSPF [START_REF] Moy | OSPF Version 2[END_REF]) are designed to save resource by allowing the computation of shortest paths. Similarly, for virtual netwok embedding kshortest paths are often chosen to map the virtual links [START_REF] Cheng | Virtual Network Embedding Through Topology-Aware Node Ranking[END_REF] [START_REF] Gang | A Virtual Network Embedding Algorithm based on Mapping Tree[END_REF]. With the arrival of MultiProtocol Label Switching (MPLS) [START_REF] Rosen | Multiprotocol Label Switching Architecture[END_REF] in the few past years, protection [START_REF] Sharma | Framework for Multi-Protocol Label Switching (MPLS)-based Recovery[END_REF] and resource optimization [START_REF] Vasseur | Framework for PCE-based MPLS-TE Fast Reroute Backup Path Computation[END_REF] are provided efficiently.

Firstly, fast recovery and availability of resources are guaranteed with the pre-configuration of local backup paths capable to bypass any failure risk (a failure risk could be a link, a node or a SRLG2 ). Local backup paths ensure fast reaction to failure due to their locality : fast detection and fast rerouting. Two types of backup paths are defined in MPLS for local protection [START_REF] Pan | Fast Reroute Extensions to RSVP-TE for LSP Tunnels[END_REF]: Next HOP backup Label Switched Path (NHOP LSP 3 ) and Next Next HOP backup Label Switched Path (NNHOP LSP). A NHOP LSP (resp. NNHOP LSP) is a backup LSP protecting against link failure (resp. node failure); it is setup between a Label Switched Router (LSR) called Point of Local Repair (PLR) and one LSR called Merge Point (MP). The PLR is the LSR upstream to the failure point. The MP is located between the next-hop (resp. next-next-hop) of the PLR and the destination. The NHOP (resp. NNHOP) backup LSP bypasses the link downstream (resp. the node downstream) to the PLR on the primary LSP. When a link failure (resp. node failure) is detected by an upstream node, this later activates locally all its NHOP and NNHOP (resp. NNHOP) backup LSPs by switching traffic from the affected primary LSPs to their backup LSPs.

Secondly, much resources can be saved thanks to the flexibility in path selection offered by MPLS. Indeed, an appropriate selection of primary and backup paths can increase the bandwidth sharing and thus decrease the bandwidth allocations.

In the last recent years, more attention was given to the virtual networks. For a better use of resources, virtual networks are computed so that they consume less resources. Due to the NP-hardness of the problem of mapping a virtual network to a substrate network (Virtual Network Embedding or VNE), most of the proposed solutions use pre-computed (k-)shortest paths. Like in classical networks, two types of protection could be applied to ensure survivability : global and local. With the global protection, a primary virtual link (which corresponds to a substrate path) is protected by a disjoint backup virtual link con-necting the same extremities [START_REF] Rahman | SVNE: Survivable Virtual Network Embedding Algorithms for Network Virtualization[END_REF] [START_REF] Yu | Virtual Infrastructure Design for Surviving Physical Link Failures[END_REF]. Two virtual links are said disjoint if they don't share any link or internal node in the substrat network. With the local protection [START_REF] Guo | Shared Backup Network Provision for Virtual Network Embedding[END_REF], each link or node belonging to a substrat primary path (i.e. primary virtual link) is protected locally by a backup path which bypasses it.

Two main resources sharing strategies are defined in literature: 1) restrained resource sharing [START_REF] Kodialam | Dynamic Routing of Restorable Bandwidth-Guaranteed Tunnels using Aggregated Network Resource Usage Information[END_REF] and 2) global resource sharing [START_REF] Balon | A Scalable and Decentralized Fast-Rerouting Scheme with Efficient Bandwidth Sharing[END_REF]. With the first strategy, the resource sharing is limited and applied to the backup paths protecting against different failure risks. As these backup paths cannot be active at the same time (due to the single failure assumption), they cannot ask for their resources simultaneously and thus they can share them. With the second strategy, the resource sharing is extended and applied to primary and backup paths. Concretely, since a backup path can bypass several links and/or nodes of a primary path, some resources can be freed on the primary affected paths. Such resources can be reallocated to the backup paths.

In this paper, we study the impact of resources sharing strategies on the resource utilization when the primary paths are the shortest ones. After reviewing works related to the resource sharing in Section II, we introduce and explain in more details the principles of the restrained and global resource sharing strategies. Then, we determine in Section III the formulas computing the amount of sharable and allocated resources with application to the two sharing strategies. In Section IV, we study formally the impact of resource sharing strategies on the resource utilization when the primary paths are the shortest ones. We show that the impact of the primary path resources freed upon a failure is very low and negligible on the protection capability. In Section V, we compare and measure by simulations the gain obtained by global resource sharing instead of restrained resource sharing. Finally, Section VI is dedicated to the conclusions.

II. Related Works

In the last two decades, a great deal of work is addressing network protection to find efficient algorithms and mechanisms providing survivability and optimizing network resource utilization.

In [START_REF] Kamal | 1+n Network Protection for Mesh Networks: Network coding-based protection using p-cycles[END_REF], [START_REF] Kamal | Network Coding-based Protection[END_REF], several network coding-based strategies are described to provide protection in optical and also higher layers. In [START_REF] Cholda | A Survey of Resilience Differentiation Frameworks in Communication Networks[END_REF], an extensive survey of the recovery methods is given. These methods are classified according to different criteria such as the layer in which recovery methods are applied (Physical Layer, Link Layer, Network Layer, etc.), computation and/or establishment moment of the backup paths (before failure for protection and after failure occurrence for the restoration), resource usage (without resource sharing or with resource sharing), scope (global or local protection) and domain (intradomain or inter-domain protection). In MPLS networks, and under different network parameters and constraints, [START_REF] Calle | Protection Performance Components in MPLS Networks[END_REF], [START_REF] Hariyawan | Comparison Analysis Of Recovery Mechanism At MPLS Network[END_REF] propose various comparison metrics, such as the packet loss, rejection probability and restoration time, to evaluate the level of protection.Unfortunately, nether [START_REF] Calle | Protection Performance Components in MPLS Networks[END_REF], [START_REF] Hariyawan | Comparison Analysis Of Recovery Mechanism At MPLS Network[END_REF] nor [START_REF] Cholda | A Survey of Resilience Differentiation Frameworks in Communication Networks[END_REF] consider global sharing in their studies.

For MPLS networks, global and local protection with/without resource sharing can be applied in both intra-domain and interdomain. With the global protection [START_REF] Bhandari | Survivable Networks : Algorithms for Diverse Routing[END_REF], [START_REF] Kodialam | Dynamic Routing of Restorable Bandwidth-Guaranteed Tunnels using Aggregated Network Resource Usage Information[END_REF], two disjoint paths connecting the source and target nodes are computed: one primary path used to transmit traffic before any failure occurrence and backup path that should be activated and used for routing upon any failure affecting the primary path. With the local protection [START_REF] Balon | A Scalable and Decentralized Fast-Rerouting Scheme with Efficient Bandwidth Sharing[END_REF], [START_REF] Mélon | Decentralized Local Backup LSP Calculation with Efficient Bandwidth Sharing[END_REF], [START_REF] Vasseur | Network Recovery: Protection and Restoration of Optical, SONET-SDH, IP, and MPLS[END_REF], for every link and/or node of the primary path, one local backup path (NHOP LSP or NNHOP LSP) bypassing the protected link and/or node is computed. When a failure occurs, the traffic is switched locally at the PLR to the (precomputed) backup paths bypassing the failed risk. In [START_REF] Li | Routing Bandwidth Guaranteed Paths with Local Restoration in Label Switched Networks[END_REF], Li et al. proves that joint resource optimization of primary and local backup path is an NP-hard problem.

Recently, several protection methods were proposed for virtual netwok protection [START_REF] Guo | Shared Backup Network Provision for Virtual Network Embedding[END_REF], [START_REF] Yu | Virtual Infrastructure Design for Surviving Physical Link Failures[END_REF]. Due to the complexity of the survivable virtual newtork embedding (SVNE) problem, this later is generally subdivided into two sub-problems (VNE and protection) which can be solved separately. Whereas on-line protection is applied to protect one path in classical networks, many substrat paths should be protected together to provide protection. In [START_REF] Yu | Virtual Infrastructure Design for Surviving Physical Link Failures[END_REF], the authors proposed to protect each primary substrat path by a disjoint shortest substrat path. To save resources, the backup paths minimize the additional bandwidth. In [START_REF] Guo | Shared Backup Network Provision for Virtual Network Embedding[END_REF], the authors propose to protect locally the substrat links which are used to form the virtual links. In their approach, Guo et al. firstly chose a subset of primary and backup paths before calling a linear procedure that optimizes bandwidth allocation while balancing the load.

To decrease the rejection rate of protection requests (i.e., to improve the resource utilization), [START_REF] Mélon | Decentralized Local Backup LSP Calculation with Efficient Bandwidth Sharing[END_REF] proposes a global resource sharing strategy for on-line protection. Contrarily to the restrained resource sharing strategy which limits the resource sharing to the backup paths, Mélon et al. suggest to pre-allocate the resources freed by the deactivated (or bypassed) primary path segments upon a failure to the backup paths which will be activated to recover from that failure (see SectionIII). In order to minimize the resource allocation, [START_REF] Balon | A Scalable and Decentralized Fast-Rerouting Scheme with Efficient Bandwidth Sharing[END_REF] proposes a resource sharing-based cost function that measures the amount of extra spare resources required to cross a given link. Obviously, larger are the primary resources freed on a link upon a failure, smaller is the cost of this link for that failure. As this link cost function depends only on the capacities of resource sharing, the backup path and thus the recovery time may be arbitrary long. Indeed, the backup paths optimizing the cost function may include very long paths which induce high transmission delays. In addition, optimizing the resource allocation does not systematically improve the request rejection rate because of resource sharing capabilities of backup path computations.

Although the primary paths often correspond to the shortest ones, none of the described works studies the impact of such primary routing decision on the blocking probability and the bandwidth sharing capabilities. In this paper, we try fill the gap by studying and measuring the impact of an optimal primary routing (according to a strictly positive and static metric) on the performances of the backup path routing.

III. Bandwidth Allocation Model

Before the presentation of the LSP admission control that takes into account bandwidth sharing (section III-A) for path computation, we first give some notations and definitions use-ful to the understanding of our control admission models (section III-B). Furthermore, these definitions enable the description of the context of our study.

A. Notations

Let us consider a directed graph

G = (V, E, -→ w , -→ C = --→ P C + -→ RC)
where V is the set of vertices, E the set of links and -→ w , --→ P C and -→ RC are functions that associate respectively to each link (λ ∈ E) a strictly positive constant weight w(λ), a primary bandwidth capacity expressed in bandwidth units P C λ and a finite protection bandwidth capacity RC λ expressed in bandwidth units. We • the weight w (π) of a primary path π as follows:

w (π) = ∑ λ∈π w(λ). • P r (s, t) r
as the set of primary paths crossing the failure risk r (link, node or a set of links and/or nodes) and interconnecting the source node s to the target node t. All the paths in P r (s, t) r are the shortest ones.

• P λ as the primary bandwidth that should be reserved on each link λ to carry out traffic before failures. It is computed as the cumulated bandwidth of primary paths crossing the risk λ and connecting s to t.

• Bp (s, t) r
as the set of backup paths protecting the primary paths in the set P r (s, t) r .

• δ λ r as the protection cost of risk r on link λ. It corresponds to the cumulated bandwidth of backup paths which will be activated on link λ to cope with the failure of risk r.

• L λ r corresponds to the (primary) bandwidth freed on link λ upon failure of risk r.

• two bandwidth allocation methods on links: bidirectional and unidirectional. With the first method, any two unidirectional links (u → v and v → u) which share the same physical conductor u -v use the same pool for bandwidth allocations. With the second bandwidth allocation method, each unidirectional link has its own autonomous pool that it uses for bandwidth allocations. For the ease of understanding and without loss of generality, we will focus in this paper on the case of unidirectional bandwidth allocations. As the case of bidirectional bandwidth allocations can be treated in the same way, only the results of simulations are given and discussed.

B. Admission control of backup paths

Using local protection mode, regardless of resource sharing strategies, N -1 local backup paths (detours) should be built to protect a path that traverses N nodes. For instance, to fully protect path

p 1 = D → C → F in Figure 1 (a), two backup paths b 1C = D → G → F and b 1F = C → B → E → F are established.
The first backup path interconnects the PLR node D to the merge node F whereas the second backup path connects the PLR node C to the merge node F . Thus, the first backup path is a NNHOP LSP protecting against the failures of PLR's downstream node (C) and link (D -C) whereas the second backup path is a NHOP LSP protecting against the PLR's downstream link (C -F ).

To save resources and accept more path establishment requests, resources like the bandwidth could be shared between paths. For instance, under the practical hypothesis of simple failure that we adopt in this article as in many articles ( [START_REF] Balon | A Scalable and Decentralized Fast-Rerouting Scheme with Efficient Bandwidth Sharing[END_REF], [START_REF] Grover | Cycle-Oriented Distributed Preconfiguration: Ring-like Speed with Mesh-like Capacity for Self-planning Network Restoration[END_REF], [START_REF] Kini | Shared Backup Label Switched Path Restoration[END_REF]), some paths cannot carry traffic at the same time: they can therefore share their bandwidth allocations. For this purpose, two main bandwidth sharing strategies were defined: 1) restrained bandwidth sharing and 2) global bandwidth sharing.

With the first bandwidth sharing strategy, the bandwidth sharing is applied and limited to the backup paths that protect against different failure risks. In Figure 1 ). Thus, after determining the protection cost δ λ r of risk r on link λ which correspond to the cumulative bandwidth of backup paths that should be activated to recover from failure r, we determine the protection bandwidth4 R λ that should be reserved for protection on the (unidirectional) link λ as follows:

R λ = max r δ λ r (1)
The total bandwidth bw (λ) allocated on λ must be always smaller than the capacity C λ of link λ:

bw (λ) = P λ + R λ = P λ + max r δ λ r ≤ C λ (2)
where P λ is the cumulative bandwidth of the primary LSPs traversing link λ.

In addition of the bandwidth sharing between the backup paths, more bandwidth could be saved by reallocating the bandwidth freed by the bypassed part of the primary path affected by the failure [START_REF] Balon | A Scalable and Decentralized Fast-Rerouting Scheme with Efficient Bandwidth Sharing[END_REF]. For instance, to recover from failure of node C in Figure 1 (a), the traffics of the primary paths p 1 and p 2 will be rerouted and switched to the backup paths b 1C and b 2C respectively. As shown in Figure 1 (b), the recovery from the failure of node C frees up bandwidth on some primary links. Typically, when node C fails, the traffic of primary path p 2 (old p 2 in Figure 1 (b)) will be switched onto backup path b 2C (repaired path p 2 in Figure 1 (b)). Thus, some bandwidth will be freed on link D -G after the node failure recovery. In order to save bandwidth, the global bandwidth sharing strategy proposes to reuse the bandwidth freed up 5 after the recovery of failure r by reallocating it to the backup paths that protect against the same failure r. In our example of Figure 1, the bandwidth allocated on link D -G can be shared between the primary path p 2 and the backup path b 1C (see Figure 1

R λ = max r (δ λ r -L λ r , 0) (3)
We deduce the total amount of bandwidth bw (λ) allocated on the link λ as follows:

bw (λ) = P λ + R λ = P λ + max r (δ λ r -L λ r , 0) ≤ C λ (4)
Note that all the parameters (P λ , δ λ r and L λ r ) that are necessary to verify the admission control (i.e. ensure the availability of enough bandwidth before and after any failure) are known by the extremity nodes of link λ since these two nodes know all the paths that traverse them.

Furthermore, to control and specify the amount of resources that should be used for protection and to separate the computation task of primary LSPs from that of backup LSPs, the bandwidth capacity of each link λ can be divided in two separate pools: primary bandwidth pool P C λ and protection bandwidth pool RC λ . The primary bandwidth pool is used to allocate bandwidth for primary LSPs whereas the protection bandwidth pool is used to allocate bandwidth for backup LSPs. With such bandwidth allocation model, link λ can be included in the computation of a new backup path iff the resulted protection bandwidth on link λ remains lower or equal to the protection capacity.

When the restrained resource sharing strategy is applied:

R λ = max r δ λ r ≤ RC λ (5)
When of global resource sharing strategy is applied:

R λ = max r (δ λ r -L λ r , 0) ≤ RC λ (6)
Although it seems that the global resource sharing strategy is more efficient than the restrained resource sharing strategy, the blocking probabilities 6 of the two strategies could be very close, especially when the primary LSPs correspond to the shortest ones in terms of a strictly positive and static7 metric. In the two following sections, we prove formally and by simulations that both the two strategies of resource sharing have close blocking probabilities.

IV. Impact of the Amount of Primary Bandwidth Freed upon Failure on the Blocking Probability

The majority of the well known IGP protocols computes the primary paths as the shortest ones in terms of a static metric (i.e., traffic independent costs). For instance, RIP minimizes the hop number (number of intermediate routers in a path) while OSPF applies the SPF (shortest path first) algorithm to optimize a static metric that depends generally on bandwidth capacities of links. With the advent of MPLS, the IP routing protocols (OSPF-TE and ISIS-TE) are extended to take into account the traffic characteristics in route computations. This leads to the definition of new (semi-)dynamic metric-based routing algorithms which often applies the Dijkstra's shortest path algorithm. For instance, CSPF (constrained shortest path first) prunes links that do not meet the configured constraints from the topology network before applying the SPF algorithm that derives the best available path based on the information in the traffic engineering database. In other words, CSPF always returns a shortest path while the pruned links don't cut all the possible shortest paths between two nodes. In addition of the IGP protocols, we note that for VNE the kshortest paths are often selected to map the virtual links to the primary substrat paths (i.e. primary virtual links).

In this section, we show that when the primary paths follow the shortest paths according to any static metric, the maximum number of backup paths is bounded even if the primary bandwidth freed upon failure is infinite (i.e., the freed bandwidth is very larger than the protection bandwidth). This means that any backup path must cross at least one link which cannot free up bandwidth upon failure of the protected risk (i.e. we cannot build a backup path with only links freeing bandwidth upon the failure of the protected risk). Hence, the maximum number of backup paths which can be built is bounded at least by the capacities of links which cannot free up bandwidth (upon the considered failure). Before detailing the proof of our assertion, let us consider an example. In Figure 2, a network topology of equal-cost links is depicted. Assume that any primary path follows a shortest path and requires a minimum of 1 bandwidth unit. To protect a primary path traversing node D, link D -G then node G (in this order), it is sufficient to determine a backup path that connects the PLR D to node G or any node downstream to G (on the shortest primary path). As the primary paths should follow the shortest paths (dashed arrows in Figure 2 Even if we consider that the freed bandwidth upon failure r is infinite on all the links that are capable to free up bandwidth (example: when the primary capacities are infinite whereas the protection capacities are finite), we show formally in the next paragraphs that the maximum number of backup paths is bounded. Without loss of generality, we assume that any backup path requires a minimum of 1 bandwidth unit and the protection bandwidth is bounded.

Lemma IV.1: Any backup path protecting a primary shortest path (according to a strictly positive and static metric) against a link failure risk must include a link which doesn't free up any bandwidth. Formally: ∀r ∈ E, ∀π ∈ Bp (s, t) r , ∃λ ∈ π : L λ r = 0 Proof: To free up bandwidth on a link λ upon failure of link plr -p 18 , λ must belong to at least one shortest primary Fig. 3. Links forming primary and backup paths path traversing link plr -p 1 in one direction (from plr to p 1 or from p 1 to plr). In addition, link λ must be located on the downstream of link plr -p 1 .

Let us define Down (plr,p1) as a set of nodes located downstream to plr → p 1 (in this direction) on the primary paths traversing plr and p 1 (see Figure 3). Here we prove that Down (plr,p1) ∩ Down (p1,plr) = ∅.

Assume that there is a node b i so that b i ∈ Down (plr,p1) . This means that:

w(plr, b i ) < w(p 1 , b i ) (7)
Where w(plr, b i ) is the weight of any shortest path connecting node plr to node b i and w(plr, b i ) is the weight of any shortest path connecting node p 1 to node b i . Actually, assume that b i ∈ Down (p1,plr) . This means that:

w(p 1 , b i ) < w(plr, b i ) (8)
From formulas ( 7) and ( 8), we conclude that b i ∈ Down (plr,p1) ∩ Down (p1,plr) leads to the following contradiction: w(plr, b i ) < w(p 1 , b i ) < w(plr, b i ). As a result, we deduce that: Down (plr,p1) ∩ Down (p1,plr) = ∅

As the PLR node belongs to Down (p1,plr) whereas the merge node is in Down (plr,p1) , no backup path connecting the PLR to the merge point could be established. Indeed, any link connecting a node in Down (p1,plr) to a node in Down (plr,p1) cannot free up bandwidth (by definition, the extremity nodes of a link freeing bandwidth are in the same set Down (plr,p1) or Down (p1,plr) ).

2 Lemma IV.2: Any backup path protecting a primary shortest path (according to a strictly positive and static metric) against a node failure risk must include a link which doesn't free up any bandwidth. Formally: ∀r ∈ V, ∀π ∈ Bp (s, t) r , ∃λ ∈ π : L λ r = 0 Proof: We prove the validity of lemma IV.2 by contradiction. In other words, if such a backup path exists, it must be shorter than the primary path it protects. Assume that there is one backup path b = plr → b 1 → .. → b j (see Figure3) composed of only links freeing up bandwidth after the failure of node p 1 (downstream to the PLR node plr). The backup path b protects a primary shortest sub-path p = plr → p 1 → .. → p i → b j according to the strictly positive and static metric -→ w (see Figure 3). Let us prove by induction on the k th backup nodes that:

∀k ≤ j, ∃s ∈ P aths (plr,b k ) , ∀π ∈ P aths (p1,b k ) : w(π) ≥ w(p 1 plr → s → b k ) (9)
free up bandwidth upon a failure of link p f -p i , we conclude that Lemma IV.1 is valid for any backup path plr → .. → p j j≥i protecting against the failure risk p f -p i .

where p 1 plr is any shortest path from p 1 to plr. k = 1 To free up bandwidth upon failure of node p 1 (see Figure 3), link plr → b 1 must belong to at least one shortest primary path traversing node p 1 . In addition, link plr → b 1 must be located on the downstream of node p 1 . This implies that: ∃s = plr → b 1 ∈ P aths (plr,b1) so that: p 1 plr → s → b 1 is a shortest path This means that formula ( 9) is valid for k = 1.

Step 1 < k ≤ j Assume that formula ( 9) is valid for n = 1, k -1 and prove that it is valid for n = k. To free up bandwidth upon failure of node p 1 , link b k-1 → b k must belong to at least one shortest primary path traversing node p 1 . In addition, link b k-1 → b k must be located on the downstream of node p 1 . This implies that:

∃s ∈ P aths (p1,b k-1 ) , ∀π ∈ P aths (p1,b k ) : w(π) ≥ w(p 1 → s → b k-1 → b k ) As for n = k -1, we have: ∃ s ′ ∈ P aths (plr,b k-1 ) , ∀π ′ ∈ P aths (p1,b k-1 ) : w(π ′ ) ≥ w(p 1 plr → s ′ → b k-1 ), we deduce that (for π ′ = p 1 → s → b k-1 ): w(π) ≥ w(p 1 → s → b k-1 → b k ) = w(p 1 → s → b k-1 ) + w(b k-1 → b k ) ≥ w(p 1 plr → s ′ → b k-1 ) + w(b k-1 → b k ) = w(p 1 plr → s ′ → b k-1 → b k ) Thus, path p 1 plr → s ′′ → b k (with s ′′ = s ′ → b k-1 and s ′′ ∈ P aths (plr,b k )
) is also a shortest path according to the metric -→ w . In other words, formula ( 9) is verified.

To prove the correctness of Lemma IV.2, we show now that formula (9) contradicts the shortness of the primary path plr → p 1 → p i → b j .

We recall that the primary path plr → p 1 → .. → p i → b j corresponds to a shortest path. This implies that: ∀π ∈ P aths (p1,bj ) : w(π) ≥ w(plr → p 1 → .. → p i → b j ). Thus, for any segment path s ∈ P aths (plr,bj ) , we have:

w(plr → s → b j ) ≥ w(plr → p 1 → .. → p i → b j ).
On the other hand, formula [START_REF] Gang | A Virtual Network Embedding Algorithm based on Mapping Tree[END_REF] implies for k = j that: ∃s ′ ∈ P aths (plr,bj ) ∀π ′ ∈ P aths (p1,bj ) :

w(π ′ ) ≥ w(p 1 plr → s ′ → b j ) = w(p 1 plr) + w(plr → s ′ → b j ) ≥ w(p 1 plr) + w(plr → p 1 → .. → p i → b j ) = w(p 1 plr → p 1 )+w(p 1 → .. → p i → b j ) > w(p 1 → .. → p i → b j ).
This leads to a contradiction since for π ′ = p 1 → .. → p i → b j (π ′ ∈ P aths (p1,bj ) ), we obtain:

w(π ′ ) = w(p 1 → .. → p i → b j ) > w(p 1 → .. → p i → b j ).
Thus, formula (9) cannot be verified. In other words, any backup path protecting against a node failure risk must utilize at least one link which cannot free up bandwidth upon that node failure. 2 Proposition IV.3: Every backup path (NNHOP LSP or NHOP LSP) should traverse a link that cannot free up bandwidth after the failure of a protected risk.

Proof: As both the NHOP and NNHOP paths should protect against link failures 9 , we conclude from Lemma IV.1 that every backup path should traverse a link that don't free up bandwidth after a failure of a protected link.

2 Theorem IV.4: The number of NHOP and NNHOP backup LSPs that can be build in a network

G = (V, E, -→ w , --→ P C, -→ RC) is
9 This is due to the difficulty to distinguish link and node failures.

bounded if |E| and (RC λ ) ∀λ∈E are bounded (by constants).

Proof: For the proof, we first show that for any link, the number of backup paths protecting against its failure is bounded. From Lemma IV.1, we know that any backup path protecting against any link failure risk r l should traverse at least one link λ that cannot free up bandwidth after the failure r l . From formula (6), we have:

δ λ r l -L λ r l = δ λ r l ≤ R λ = max r (δ λ r -L λ r , 0) ≤ RC λ
As the protection cost δ λ r l corresponds to the number of (1 bandwidth unit) backup paths protecting against the failure r l and traversing link λ, we deduce that this number of backup paths is bounded by RC λ . Because the number of links freeing up some bandwidth after the failure of r l is lower than |E| (it is always equal to 0 if we apply the restrained bandwidth sharing strategy), we conclude that the number of backup paths protecting against the link failure risk r l is bounded by

∑ λ∈E RC λ ≤ |E| × max λ∈E (RC λ ).
Similarly, we deduce that the maximum number of backup paths that can be built in the network is bounded by |E| 2 × max λ∈E (RC λ ) since the number of distinct link failure risks is lower or equal to |E|.

2 Interpretation:

• With both the global and restrained bandwidth sharing strategies, the number of backup paths is bounded when the protection capacities (or the link capacities) are bounded and lower than given constants. As any backup path should traverse at least one link that don't free up bandwidth, the use of the global bandwidth sharing strategy instead of the restrained bandwidth sharing strategy could not avoid network redimensioning over the long term.

• When a great amount of traffic is not protected (for instance, best-effort traffic does not require protection), the freed bandwidth on some links upon failure could be high. Even in this case, the maximum number of backup paths is bounded specifically by the capacity of links that cannot free up bandwidth.

Whereas the maximum number of backup paths depends on all the network links with the restrained bandwidth sharing strategy, this number depends more on the links that cannot free up bandwidth with the global bandwidth sharing strategy. In the next section, we compare by simulations these two bandwidth sharing strategies to quantify the gain in performances due to the exploitation of the freed bandwidth.

V. Simulation and Performance Analysis

A. Simulation model

In order to quantify the gain in performances due to the use of the global bandwidth sharing instead of the restrained bandwidth sharing, we compared the two bandwidth sharing strategies by simulations. In our experiments, we used two well known topologies of network: Long Haul and Cost 239. The first network topology, depicted in Figure 4 (a), is composed of 28 nodes and 45 bidirectional links. The protection capacities are equal to 600 units in each direction for the bold links and 200 units for the light links. This network topology is relatively wide and presents a mean connectivity degree of 3.21. The second network topology, depicted in Figure 4 (b), is composed of 11 nodes and 26 bidirectional links. It is small and strongly connected since its mean connectivity degree is equal to 4.73. All To take into account the two possible models of bandwidth allocation (unidirectional bandwidth allocation and bidirectional bandwidth allocation), we considered two test scenarios: unidirectional allocation-based scenario (UAS) and bidirectional allocation-based scenario (BAS). In the first test scenario, the unidirectional bandwidth allocation method is applied for bandwidth allocation. It means that two protection pools are associated to each bidirectional link in Figure 4. In the second test scenario, the bidirectional bandwidth allocation method is applied for bandwidth allocation. It means that only one protection pool is associated to each bidirectional link in Figure 4. Thus, the protection capacities of bold links are equal to 1200 units (600 × 2) whereas they are equal to 400 units (200 × 2) on the light links.

In our simulations, we generated sequentially 1000 demands of protected path establishment asking for bandwidth quantities uniformly distributed between 1 and 10 units. Each demand is composed of one primary path establishment request that is always satisfied (i.e., we assumed that the primary pool capacities of links are sufficient to satisfy all the requests of primary path establishment) and several requests of backup path establishment allowing the protection of the built primary path. The source and target nodes of each primary path are selected uniformly among the set of network nodes. For the computation of primary paths, we applied the shortest path first (SPF) algorithm that optimizes the number of hops whereas we used the constrained shortest path first (CSPF) algorithm for the computation of backup paths. With the restrained resource sharing strategy, a request of backup path establishment is satisfied iff equation (5) is verified. With the global resource sharing strategy, equation ( 6) must be verified to establish the requested backup path.

Two criteria are selected to compare the global and restrained resource sharing strategies: rate of backup path rejection (BPR) and rate of protection bandwidth utilization (PBwU). The first criterion BPR is computed for different network loads. It determines the ratio of backup paths that are rejected because of the lack of protection bandwidth on the network links. It corresponds to the (instantaneous) ratio between the number of backup path requests that are rejected and the total number of backup paths required to protect entirely the last 50 primary paths. Formally, it is determined as follows: BP R = #rejected protection requests

#protection requests

The second criterion PBwU determines and measures the efficiency of bandwidth sharing. It corresponds to the ratio between the sum of all the protection costs and the amount of the bandwidth allocated in the network for the protection. Formally, it is computed as follows:

P BwU = ∑ (λ, r)∈E×(V ∪E) δ λ r / ∑ λ∈E R λ .
For each test scenario (UAS and BAS) and at each establishment of 50 primary paths, the two metrics BPR and PBwU are computed for the two compared strategies. We note that our results correspond to mean values over 1000 experiments.

B. Results and Analysis

Figure 5 shows the evolution of the instantaneous rate of backup path rejection (BPR) as a function of the number of primary paths setup in the network for the unidirectional and bidirectional bandwidth allocations.

As expected, we observe in Figure 5 that the bidirectional bandwidth allocation method is slightly better that the unidirectional bandwidth allocation method. Indeed, Figure 5 clearly shows that the global and restrained bandwidth sharing strategies using the bidirectional bandwidth allocation method have respectively smaller rejection rates than the global and restrained bandwidth sharing strategies using the unidirectional bandwidth allocation method. These observations can be explained by the distribution of the protection costs on links (especially on opposite links) which is heterogeneous [START_REF] Saidi | PLR-based Heuristic for Backup Path Computation in MPLS Networks[END_REF].

Figure 6 depicts the difference in cumulated rates of backup path rejection for the unidirectional and bidirectional bandwidth allocations. It shows that the difference is small and even imperceptible sometimes. For instance, in Longhaul network topology, the difference in instantaneous rates of backup path rejection does not exceed 8 %, even for high loads of traffic (a large number of primary paths) where the instantaneous rejection rate of backup path is high and inefficient (see Figure 5 (a)). For usual instantaneous rejection rates that should be lower than 10 %, the difference between the compared strategies is often not Obviously, the difference in the backup path rejection is directly related to the amount and distribution of the freed bandwidth on links. Since the freed bandwidth is statically high on the links close to PLRs and generally low on the links located far from PLRs, the difference in backup path rejection rates of the compared strategies is slightly higher in COST 239 network topology than in Longhaul network topology. Indeed, the links are closer to the PLRs in COST 239 since it is more homogeneous and it has a larger connectivity degree than Longhaul.

In addition of the previous observations, we note that even for high freed bandwidth values, the rates of backup path rejection increase with the augmentation of the traffic load and they converge to the saturation state where all the new protection requests are rejected. This corroborates our theoretical results which announces the existence of an upper bound for the number of backup paths that can be established in the network even with unlimited resources.

With regards to the second metric (bandwidth sharing utilization), Figure 7 shows that both the global and restrained band-width sharing strategies have similar bandwidth utilization rates for small and usual backup path rejection rates. For instance, the difference in bandwidth sharing utilization for the compared strategies is very small in Longhaul network (see Figure 7 (a)) when the number of primary paths is lower than 1000 (all the backup path rejection rates are lower than 0.3) whereas the difference is imperceptible in COST 239 network (see Figure 7 (b)) when the number of primary paths is lower than 3000 (all the backup path rejection rates are lower than the usual value 0.15). For high traffic loads, Figure 7 shows that the global bandwidth sharing strategy is better than the restrained bandwidth sharing strategy. This is essentially due to the amount of freed bandwidth which increases with the augmentation of the rejection rate of backup paths. Indeed, whereas the protection bandwidth is completely independent of the freed bandwidth variation when the restrained bandwidth sharing strategy is applied, it decreases with the augmentation of the freed bandwidth when we apply the global bandwidth sharing strategy.

To summarize, these simulations show that the difference in performances between the global and restrained bandwidth sharing strategies is almost imperceptible for low traffic loads where the rejection rate of backup path is small and usual. For high traffic loads where the rejection rate of backup paths is high, the global bandwidth sharing strategy is slightly better than the restrained bandwidth sharing strategy. In addition to the prece- dent remarks, our simulations comfort our theoretical results (see Therem IV.4) and show clearly that the number of backup paths is always bounded even if the freed bandwidth is unlimited on the links that are capable to free up bandwidth.

VI. Conclusion

In this article, we presented two known strategies of resource (bandwidth) sharing: restrained bandwidth sharing and global bandwidth sharing. The first strategy limits the bandwidth sharing to the backup paths that protect against different failure risks whereas the second strategy extends the bandwidth sharing to the primary and backup paths that could not be active at the same time.

To measure the gain obtained by the extension of the bandwidth sharing to the primary and backup paths, we firstly showed theoretically that the bandwidth sharing between the primary and backup paths can never be applied on some backup links when the primary paths correspond to the shortest ones (according to a static metric). Thus, the maximum number of backup paths is limited and bounded by the protection capacities of links. Secondly, to quantify the improvement due the bandwidth sharing between the primary and backup paths, we showed by simulations that the gain in performances (backup paths rejection and bandwidth utilization) is often imperceptible, particularly for low traffic loads where the rejection rate of backup paths is small and usual. For high traffic loads where the rejection rates are high, the global bandwidth sharing strategy is slightly better than the restrained bandwidth sharing strategy, especially in strongly connected networks.

As a result, the global bandwidth sharing strategy cannot be a long term solution for supporting bandwidth-intensive applications especially since the global bandwidth sharing strategy induces an overcost. Indeed, in return of the slight performance improvements the global bandwidth sharing allows, we note the complication of path computation and the necessity to maintain larger information. For instance, additional computations should be done with the global bandwidth sharing strategy to determine the amount of freed bandwidth after each establishment or liberation of a primary path.

  (a), the backup LSP b 2C (A → B → E → F → G) protecting the primary LSP p 2 (A → C → D → G) against failures of node C and link A -C can share its resource allocation (for instance, on links B -E and E -F ) with the backup LSP b 1F (C → B → E → F ) which protects the primary path p 1 (D → C → F ) against the failure of link C -F . Indeed, LSPs b 2C and b 1F cannot be active at the same time since they protect against disjoint sets of failure risks (failure of link A -C for b 2C and, link C -F for b 1F

Fig. 1 .

 1 Fig. 1. MPLS protection and bandwidth sharing

Fig. 2 .

 2 Fig. 2. Links that are able to free up bandwidth upon a failure of link D -G

  ), only nodes F and G can be located on the downstream of G. In addition, the shortest primary paths traversing node D, link D-G and node G (in this order) can only free up bandwidth on links G -F and G -H located on the downstream of the failed link D -G. In a same way, we deduce that any backup path protecting a primary path traversing node G, link G -D and node D (in this order) should connect the PLR G to node D or any node downstream to D on a shortest path (i.e. any node in {A, B, C, E}). The links that are able to free up bandwidth on a primary path traversing node G, link G -D and node D (in this order), upon a failure of link G -D are: D -C, C -A, D -B, B -A, B -E (in the two directions) and D -E (links associated to bold arrows in Figure 2). From the precedent remarks, we conclude that any backup path protecting link D -G should use link H -E or link F -C. Since these two last links will not free up bandwidth after the failure of link G -D, we deduce that the number of backup paths protecting link G-D is bounded by the capacities of links H -E and F -C (the protection costs δ H-E G-D and δ F -C G-D increase with the establishment of backup paths protecting against the failure of link G -D).

Fig. 4 .

 4 Fig. 4. Network topologies

Fig. 5 .Fig. 6 .

 56 Fig. 5. Evolution of the mean rate of backup path rejection

Fig. 7 .

 7 Fig. 7. Evolution of the mean rate of protection bandwidth utilization

A SRLG[START_REF] Kompella | Routing Extensions in Support of Generalized Multi-Protocol Label Switching (GMPLS)[END_REF],[START_REF] Saidi | Using Shared Risk Link Groups to Enhance Backup Path Computation[END_REF] corresponds to a set of logical links that share a common physical component (optical fiber, crossconnect, etc.) whose single failure may impact all links in the set.

A LSP is a path through an MPLS network.

Protection bandwidth corresponds to the minimum amount of bandwidth that should be reserved for backup paths, to ensure the availability of enough bandwidth after any single failure.

[START_REF] Vasseur | Framework for PCE-based MPLS-TE Fast Reroute Backup Path Computation[END_REF] Only the primary links located between the extremity nodes of the activated backup path frees up bandwidth.

The blocking probability corresponds to the probability that a request of path establishment will be rejected due to the lack of network resources (bandwidth).

We recall that a metric is said to be static if its values on links do not change.

Even if we consider that the PLR plr is not adjacent the failed link p f -p i (case of the global protection), we can easily prove the correctness of Lemma IV.1. Indeed, if Lemma IV.1 is valid for any backup path p f → .. → p j j≥i protecting against the failure risk p f -p i , it implies that it is valid for any backup path p f → .. → plr → .. → p j j≥i protecting against the failure risk p f -p i . As all the links of the sub-path p f → .. → plr can


Jie Jiang
email: jiang@apm.ac.cn

Philippe Laurenc ¸ot
email: philippe.laurencot@univ-smb.fr

GLOBAL BOUNDED CLASSICAL SOLUTIONS TO A PARABOLIC-ELLIPTIC CHEMOTAXIS MODEL WITH LOCAL SENSING AND ASYMPTOTICALLY UNBOUNDED MOTILITY

Keywords: Mathematics Subject Classification, 35K51 -35K55 -35A01, chemotaxis -global existence -boundedness -comparison

Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of L ∞ -estimates on the second component of the system and is achieved by various comparison arguments.

Introduction

Let Ω be a smooth bounded domain of R N , N ≥ 1, and consider the initial-boundary value problem

∂ t u = ∆ uγ(v) , (t, x) ∈ (0, ∞) × Ω, (1.1a)
0 = ∆v -v + u, (t, x) ∈ (0, ∞) × Ω, (1.1b)
∇ uγ(v) • n = ∇v • n = 0, (t, x) ∈ (0, ∞) × ∂Ω, (1.1c)

u(0) = u in , x ∈ Ω, (1.1d)
where n denotes the outward unit normal vector field to ∂Ω.

The system (1.1) is a classical parabolic-elliptic simplification of the well-known fully parabolic system of partial differential equations named after Keller & Segel, which is originally proposed in their seminal work [START_REF] Keller | Model for chemotaxis[END_REF] to model the chemotaxis phenomenon due to a local sensing mechanism:

∂ t u = ∆(uγ(v)), (t, x) ∈ (0, ∞) × Ω, (1.2a)
τ ∂ t v = ∆v -v + u, (t, x) ∈ (0, ∞) × Ω, (1.2b) ∇ uγ(v) • n = ∇v • n = 0, (t, x) ∈ (0, ∞) × ∂Ω, (1.2c) (u, τ v)(0) = (u in , τ v in ), x ∈ Ω, (1.2d)
Here, u and v represent the cell density and the chemical concentration, respectively. The cellular motility γ is a positive function on (0, ∞) and its dependence on v accounts for the influence of the chemical signal on the motion of cells which is better seen after expanding ∆(uγ(v)) = div γ(v)∇u + uγ ′ (v)∇v) :

the diffusion of cells in space is governed by the values of γ(v), while the (chemotactic) bias on their motion induced by the signal is attractive when γ ′ (v) < 0 and repulsive when γ ′ (v) > 0. More recently, an extended model involving a third component n accounting for the nutrient level has been introduced in [START_REF] Liu | Sequential establishment of stripe patterns in an expanding cell population[END_REF] and reads:

∂ t u = ∆(uγ(v)) + θuf (n), (t, x) ∈ (0, ∞) × Ω, (1.3a
)

τ ∂ t v = ∆v -v + u, (t, x) ∈ (0, ∞) × Ω, (1.3b)
∂ t n = ∆n -θuf (n), (t, x) ∈ (t, x) ∈ (0, ∞) × Ω, (1.3c)
also supplemented with no-flux boundary conditions and initial conditions. In [START_REF] Liu | Sequential establishment of stripe patterns in an expanding cell population[END_REF], the motility γ is assumed to be a positive non-increasing function, which reflects a repressive effect of the signal on cellular motility. Formation of spatially periodic patterns are shown by numerical simulations and experimental analysis in a growing bacteria population merely under this motility control. We note that setting θ = 0 in (1.3) cancels the coupling between u and n and we then recover the system (1.2) for (u, v).

The mathematical analysis of the Keller-Segel-type system (1.2) and its variants (such as (1.1) or (1.3)) has attracted a lot of interest in recent years. Besides the quasilinear structure of the cell's equation (1.2a) coupling in a nonlinear way the dynamics of the cells and the signal, the cells' equation (1.2a) features a (signal-dependent) degeneracy when γ vanishes, which is likely to occur at infinity (γ(s) = s -k , k > 0, or γ(s) = e -χs , χ > 0, for instance) or at zero (γ(s) = s k , k > 0, for instance). In that case, equation (1.2a) is a second-order degenerate quasilinear parabolic equation which, unlike the celebrated porous medium equation ∂ t z -∆z m = 0, m > 1, does not have a variational structure in general. Nevertheless, energy methods turn out to be adequate in some special cases which we describe now. When γ ∈ W 1,∞ (0, ∞) ranges in a compact subinterval of (0, ∞), thereby excluding the aforementioned possible degeneracy, and Ω is convex, the existence of a global bounded solution to (1.2) is shown in space dimension N = 2 in [START_REF] Tao | Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system[END_REF], along with the existence of global weak solutions in higher space dimensions N ≥ 3. When the motility γ is the specific algebraically decaying function γ(s) = c 0 s -k with c 0 > 0, it is proved in [START_REF] Yoon | Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion[END_REF] that global bounded classical solutions whatever the value of k > 0, provided that c 0 is sufficiently small. The smallness condition on c 0 is subsequently relaxed in [START_REF] Ahn | Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing[END_REF][START_REF] Wang | On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: a paradigm for global boundedness and steady states[END_REF] when τ = 0 and k < 2 N -2 . When γ(s) = 1/(c + s k ), c ≥ 0, global weak solutions to (1.2) are constructed in [START_REF] Desvillettes | A logarithmic chemotaxis model featuring global existence and aggregation[END_REF] when N ∈ {1, 2, 3} and k ∈ (0, k N ) with k 1 = ∞, k 2 = 2, and k 3 = 4/3. When the motility function γ is a negative exponential, γ(s) = e -χs with χ > 0, global existence of weak solutions is established in [START_REF] Burger | Delayed blow-up for chemotaxis models with local sensing[END_REF] in arbitrary space dimensions N ≥ 1. Moreover, when N = 2, there is a threshold value of u in 1 , which is exactly the same as that of the minimal Keller-Segel system according to [START_REF] Jin | Critical mass on the Keller-Segel system with signal-dependent motility[END_REF], separating two different dynamics of (1.2): global classical solutions to (1.2) exist when u in 1 lies beyond the threshold value, while there are initial conditions with u in 1 above the threshold value for which the corresponding classical solution to (1.2) becomes unbounded in finite or infinite time.

In a series of works [8-14, 16-18, 22, 24, 25, 28], a different approach is developed and proved to be efficient in generic cases. The main idea is to exploit the specific structure of (1.2a) to derive a quasilinear degenerate parabolic equation involving a non-local term for an auxiliary unknown function. A systematic argument based on comparison techniques, monotonicity tricks, iteration procedures as well as applications of abstract semi-group theory gives rise to advanced theories on well-posedness and qualitative behavior of the solutions.

Let us summarize the results obtained in the literature for the initial-boundary value problem (1.2) or its variant (1.3) (also supplemented with no-flux boundary conditions) by this alternative method. When τ > 0, it is proved in [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with densitysuppressed motilities[END_REF][START_REF]Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions[END_REF][START_REF] Jiang | Global existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption[END_REF] that there exists a unique global classical solution to (1.2) for a large class of non-negative initial conditions under a rather general assumption on γ:

γ ∈ C 3 ((0, ∞)), γ > 0 on (0, ∞), (1.4
) The monotonicity (1.5) of γ is actually not needed, as shown recently in [START_REF] Xiao | Global existence and uniform boundedness in a fully parabolic Keller-Segel system with non-monotonic signal-dependent motility[END_REF], where the existence of global classical solutions to (1.2) is established under the only assumptions (1.4) and (1.6). In the same vein, global weak solutions are constructed in [START_REF] Li | Global existence of weak solutions to a signal-dependent Keller-Segel model for local sensing chemotaxis, Nonlinear Anal[END_REF] when γ satisfies (1.4) and sup s>0 γ(s) < 1/τ , and in [START_REF] Desvillettes | Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing, Nonlinear Anal[END_REF] when γ satisfies (1.4) and decays at most algebraically at infinity. All these existence results share the property that no finite time blowup occurs in (1.2), a feature which significantly differs from the minimal Keller-Segel system or the logarithmic Keller-Segel system which both involve a linear cellular diffusion and are well-known to trigger finite time singularities. Furthermore, the boundedness of global solutions to (1.2) is shown to be closely related to the decay property of γ at infinity. Specifically, for γ satisfying (1.4), (1.5), and vanishing at infinity lim s→∞ γ(s) = 0, (1.7) it is proved that exponential decay is critical in space dimension N = 2 in the following sense: there exist unbounded solutions to (1.2) when γ(s) = e -χs for some χ > 0 and the L 1 -norm of u in is sufficiently large [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with densitysuppressed motilities[END_REF][START_REF] Jin | Critical mass on the Keller-Segel system with signal-dependent motility[END_REF], whereas all solutions to (1.2) are bounded if γ decays slower than a negative exponential function; that is, γ satisfies lim inf

γ ′ ≤ 0 on (0, ∞), (1.5
s→∞ e χs γ(s) > 0 for all χ > 0, typical examples including γ(s) = s -k 1 log -k 2 (1 + s) with (k 1 , k 2 ) ∈ (0, ∞) × [0, ∞), or γ(s) = e -s α with 0 < α < 1.
When N ≥ 3 and γ(s) ∼ s -k , boundedness of global solutions holds true for k < N/(N -2) [START_REF] Fujie | Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions[END_REF][START_REF] Jiang | Global existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption[END_REF], which in particular improves previous results for the specific case τ = 0 and γ(s) = s -k by substantially enlarging the admissible range of k. Recently, the above boundedness results have been improved in [START_REF] Xiao | Global existence and uniform boundedness in a fully parabolic Keller-Segel system with non-monotonic signal-dependent motility[END_REF] by removing the monotonicity requirement (1.5) and replacing the asymptotically vanishing property (1.7) by the asymptotically smallness assumption (1.6). Moreover, if γ simply satisfies (1.4) and is bounded by positive constants from above and below, so that (1.2a) is uniformly parabolic, existence of uniform-in-time bounded solution is also proved in [START_REF] Xiao | Global existence and uniform boundedness in a fully parabolic Keller-Segel system with non-monotonic signal-dependent motility[END_REF] in any bounded domain Ω in arbitrary space dimension N ≥ 1, thus improving the corresponding one in [START_REF] Tao | Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system[END_REF] obtained in two-dimensional convex domains.

As for the initial-boundary value problem (1.1), which is recovered from (1.2) by setting formally τ = 0, the existence and boundedness/unboundedness results available for (1.2) are still true under the same assumptions by setting 1/0 = ∞ in (1.6). Indeed, global existence of classical solutions is proved in our previous work [START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF] for motility functions γ satisfying (1.4) and such that γ ∈ L ∞ (s, ∞) for any s > 0. Note that, thanks to the continuity (1.4) of γ, the latter is obviously equivalent to lim sup s→∞ γ(s) < ∞.

(1.8)

According to the above discussion, the studies of (1.1) and (1.2) performed up to now require the motility γ to be a bounded function on [s, ∞) for any s > 0, as implied by, either (1.6), or the assumption that γ ranges in a compact subinterval of (0, ∞). This raises the natural question of the dynamics of (1.1) and (1.2) when γ becomes unbounded near infinity, and the main purpose of the present contribution is a detailed analysis of the dynamics of (1.1) in that case.

We first show that the initial-boundary value problem (1.1) has a unique global bounded classical solution when γ is, either non-decreasing, or asymptotically unbounded.

Theorem 1.1. Assume that γ satisfies (1.4), as well as, either γ ′ ≥ 0 in (0, ∞), or

lim sup s→∞ γ(s) = ∞.
(1.9)

Suppose that u in ∈ W 1,∞ + (Ω) with m = u in 1 > 0, where W 1,∞ + (Ω) := z ∈ W 1,∞ (Ω) : z ≥ 0
in Ω and we use the short notation • p for the norm

• L p (Ω) with p ∈ [1, ∞].
Then there is a unique non-negative global classical solution

u ∈ C [0, ∞) × Ω ∩ C 1,2 (0, ∞) × Ω , v ∈ C 1,2 (0, ∞) × Ω ,
to (1.1) which satisfies the conservation of matter

u(t) 1 = m = u in 1 , t ≥ 0, (1.10)
and is uniformly bounded; that is,

sup t∈(0,∞) ( u(t) ∞ + v(t) ∞ ) < ∞. (1.11)
Combining the outcome of Theorem 1.1 with [17, Theorem 1.1] provides the existence of a unique non-negative classical solution to (1.1) under the sole assumption (1.4) on γ, as stated now for future reference.

Corollary 1.2. Assume that γ satisfies (1.4) and that

u in ∈ W 1,∞ + (Ω) with m = u in 1 > 0.
Then there is a unique non-negative global classical solution to (1.1) which satisfies the conservation of matter (1.10).

Another consequence of Theorem 1.1 is the boundedness of the non-negative global classical solution to (1.1) given by Corollary 1.2 in the chemorepulsive case γ ′ ≥ 0. The dynamics is more complex in the chemoattractive case γ ′ ≤ 0 as the solution to (1.1) may blow up in infinite time according to [START_REF] Fujie | Global existence for a kinetic model of pattern formation with density-suppressed motilities[END_REF].

We next supplement Theorem 1.1 with explicit L ∞ -bounds on u or v when γ is endowed with additional properties. First, define the (unbounded) linear operator A on L 2 (Ω) by

D(A) := z ∈ H 2 (Ω) : ∇z • n = 0 on ∂Ω , A[z] := z -∆z, z ∈ D(A).
(1.12)

Then v = A -1 [u] according to (1.1b) and (1.1c) and v in := A -1 [u in ] ∈ W 3,p (Ω) for any 1 ≤ p < ∞
by standard regularity theory of elliptic equations. In particular, v in is bounded and positive in Ω, the latter following from Lemma 2.2 below while the former is ensured by Sobolev embeddings. Now, we show that explicit upper and lower bounds on v are available under a local monotonicity assumption on γ, implying the existence of an invariant region for the second component v.

Theorem 1.3. Assume that γ satisfies (1.4) and that u in ∈ W 1,∞ + (Ω) with m = u in 1 > 0. Let (u, v) be the corresponding non-negative classical solution to (1.1) given by Corollary 1.2. If γ ′ ≥ 0 on min Ω {v in }, max Ω {v in } , (1.13)
then the solution to (1.1) satisfies the uniform bound (1.11). More precisely,

min Ω {v in } ≤ v(t, x) ≤ max Ω {v in }, (t, x) ∈ [0, ∞) × Ω.
In the same vein, we have the following explicit upper bounds on u.

Theorem 1.4. Assume that γ satisfies (1.4), along with (1.13), and that

u in ∈ W 1,∞ + (Ω) with m = u in 1 > 0. Let (u, v) be the corresponding non-negative global classical solution to (1.1) provided by Corollary 1.2. Then u(t) ∞ ≤ u in γ v in ∞ γ(minΩ{v in }) ≤ u in ∞ γ maxΩ{v in } γ(minΩ{v in }) , t ≥ 0 . Assume in addition that γ ′′ ≤ 0 in minΩ{v in }, maxΩ{v in } . Then u(t) ∞ ≤ u in ∞ for t ≥ 0.
We finally deal with the large time behaviour of solutions to (1.1) when γ is non-decreasing and report that there is no pattern formation in that case, as intuitively expected in the chemorepulsive regime.

Theorem 1.5. Assume that γ satisfies (1.4) and (1.13). Consider u in ∈ W 1,∞ + (Ω) with m = u in 1 > 0 and denote the corresponding non-negative global classical solution to (1.1) provided by Corollary 1.2 by (u, v). Then

lim t→∞ u(t) - m |Ω| ∞ + v(t) - m |Ω| W 1,∞ = 0.
A similar stability result is proved in [1, Theorem 1.3] when γ(s) = s -k for some k ∈ (0, 1]. Their proof relies on the availability of a Liapunov functional for (1.1) when γ(s) = s -k for some k ∈ (0, 1] which turns out to be also a Liapunov functional when γ is non-decreasing. The proof of of Theorem 1.5 is thus the same as that of [START_REF] Ahn | Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing[END_REF]Theorem 1.3], to which we refer.

The cornerstone of the proof of our existence and boundedness results is the derivation of an upper bound for the second component v, which is achieved by a new delicate comparison argument developed in the current contribution. Let us illustrate the difficulty to be faced and sketch the main idea in the simple case where γ satisfies (1.4) and γ ′ ≥ 0 on (0, ∞). To begin with, we derive from (1.1a)-(1.1b) the following key identity

∂ t v -γ(v)∆v + vγ(v) = A -1 [uγ(v)] , (t, x) ∈ (0, ∞) × Ω , (1.14)
recalling that A is defined in (1.12). This identity is uncovered in [START_REF] Fujie | Global existence for a kinetic model of pattern formation with density-suppressed motilities[END_REF] and has since then been used efficiently to investigate the global existence and boundedness of classical solutions to (1.1) and its variants, see [START_REF] Desvillettes | Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing, Nonlinear Anal[END_REF][START_REF] Fujie | A short remark on comparison estimates in a chemotaxis system with local sensing[END_REF][START_REF] Fujie | Boundedness of classical solutions to a degenerate Keller-Segel type model with signaldependent motilities[END_REF][START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with densitysuppressed motilities[END_REF][START_REF] Fujie | Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions[END_REF][START_REF]Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions[END_REF][START_REF] Jiang | Boundedness and exponential stabilization in a parabolic-elliptic Keller-Segel model with signaldependent motilities for local sensing chemotaxis[END_REF][START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF][START_REF] Jiang | Global existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption[END_REF][START_REF] Li | Global existence of weak solutions to a signal-dependent Keller-Segel model for local sensing chemotaxis, Nonlinear Anal[END_REF][START_REF] Lyu | Global classical solutions for a class of reaction-diffusion system with densitysuppressed motility[END_REF][START_REF]Logistic damping effect in chemotaxis models with density-suppressed motility[END_REF][START_REF] Xiao | Global existence and uniform boundedness in a fully parabolic Keller-Segel system with non-monotonic signal-dependent motility[END_REF], where an upper bound on γ is essentially used to control the non-local source term on the right-hand side of (1.14). Indeed, supposing that γ(v) is bounded from above on [0, T ] by some possibly time-dependent upper bound γ * (T ) > 0, an application of the elliptic comparison principle yields an upper control of the non-local term by a term growing linearly as a function of v which reads

A -1 [uγ(v)] ≤ γ * (T )A -1 [u] = γ * (T )v , (t, x) ∈ [0, T ] × Ω .
We also refer to [START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF] for a more tricky derivation of a sublinear control of this term when γ is non-increasing on (0, ∞). Then a systematic argument based on comparison techniques and Moser-Alikakos iterations is developed in [START_REF] Fujie | Global existence for a kinetic model of pattern formation with density-suppressed motilities[END_REF][START_REF] Fujie | Boundedness of classical solutions to a degenerate Keller-Segel type model with signaldependent motilities[END_REF][START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF][START_REF] Xiao | Global existence and uniform boundedness in a fully parabolic Keller-Segel system with non-monotonic signal-dependent motility[END_REF] to derive estimate for v in L ∞ (0, T ) × Ω . In contrast, when γ is non-decreasing and unbounded at infinity, the non-local term features in principle a superlinear growth with respect to v. Indeed, since γ(v) ≤ γ( v ∞ ), it follows that

A -1 [uγ(v)] ≤ γ( v ∞ )v , (t, x) ∈ (0, ∞) × Ω ,
and thus

∂ t v -γ(v)∆v + vγ(v) ≤ γ( v ∞ )v , (t, x) ∈ (0, ∞) × Ω .
(1.15) The previous methods fail in this case due to the superlinear dependence of the right-hand side on v ∞ .

To overcome this difficulty, we develop a novel approach to derive time-independent upper bounds on v solely relying on comparison techniques. Specifically, introducing the solution V to the ordinary differential equation

dV dt + V γ(V ) = γ( v ∞ )V , t ≥ 0 , V (0) = v in ∞ , (1.16
)

recalling that v in = A -1 [u in ],
an immediate consequence of (1.16) is that V is a supersolution to (1.15) and we deduce from the parabolic comparison principle that v(t, x) ≤ V (t) for (t, x) ∈ [0, ∞) × Ω. In particular, v(t) ∞ ≤ V (t) for t ≥ 0 and the monotonicity of γ, the non-negativity of V , and (1.16)

entail that dV /dt = (γ( v ∞ ) -γ(V ))V ≤ 0. As a result, we have v(t) ∞ ≤ V (t) ≤ v in ∞ for t ≥ 0,
as stated in Theorem 1.3. The lower bound in Theorem 1.3 is derived with a similar argument.

In the general case where γ satisfies (1.4) but need not be monotone, the main idea is to split γ as a sum of its increasing and decreasing parts. A more delicate argument combining the above technique and the monotonicity trick developed in our previous work [START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF] is carried out to derive the time-independent upper bound for v; see Section 3.2. Once the upper bound for v is obtained, we accomplish the proof according to Proposition 2.3.

The remainder of this paper is organized as follows. In Section 2, we recall the local existence result and establish a blowup criterion, which ensures that global existence of classical solutions to (1.1) is a direct, though far from straightforward, consequence of the boundedness of v. We then derive the uniform-in-time boundedness of v by the new comparison technique and hence prove Theorem 1.1 in Section 3. In addition, we discuss the continuous dependence of classical solutions of (1.1) on initial conditions. We next derive explicit L ∞ -estimates for v and u in Section 4 and Section 5, respectively, under additional conditions on γ. Finally, we study the stabilization of solutions to (1.1) and prove Theorem 1.5 in Section 6.

Preliminaries

We first state the local well-posedness of (1.1), which can be proved as in [

u ∈ C [0, T max ) × Ω ∩ C 1,2 (0, T max ) × Ω , v ∈ C 1,2 (0, T max ) × Ω ,
defined on a maximal time interval [0, T max ) with T max ∈ (0, ∞] which satisfies

u(t) 1 = m = u in 1 , t ∈ [0, T max ).
(2.1)

In addition, if T max < ∞, then lim t→Tmax u(t) ∞ = ∞. (2.2)
We next recall the following lemma [15, Lemma 2.1].

Lemma 2.2.

There is ω * > 0 depending only on Ω such that, for any f ∈ L 1 (Ω) satisfying f ≥ 0 a.e. in Ω and f 1 = m , there holds

A -1 [f ] ≥ mω *
in Ω , recalling that the elliptic operator A is defined by (1.12).

We now fix γ satisfying (1.4) and u in ∈ W 1,∞ + (Ω) with m = u in 1 > 0 and denote the corresponding classical solution to (1.1) defined on [0, T max ) provided by Proposition 2.1 by (u, v). We then infer from (1.1) that v solves

∂ t v -γ(v)∆v + vγ(v) = A -1 [uγ(v)], (t, x) ∈ (0, T max ) × Ω, (2.3a) ∇v • n = 0, (t, x) ∈ (0, T max ) × ∂Ω, (2.3b) v(0) = v in , x ∈ Ω, (2.3c)
where

v in = A -1 u in . (2.4)
Observe that the properties of u in , the elliptic comparison principle, Lemma 2.2, (2.1), and (2.4)

imply that v in ∈ L ∞ (Ω) with v in ∞ ≤ u in ∞ and v(t, x) ≥ v * := mω * > 0, (t, x) ∈ [0, T max ) × Ω. (2.5)
We conclude this section by showing that an L ∞ -estimate on v on (0, T ) for some T > 0 guarantees that T max ≥ T . Proposition 2.3. Under the assumption of Proposition 2.1, if there is T > 0 such that

V(T ) := sup [0,T ]∩[0,Tmax) v(t) ∞ < ∞ , (2.6)
then T max ≥ T and U(T

) := sup [0,T ] u(t) ∞ < ∞ . (2.7)
In addition, if (2.6) holds true for all T > 0 and there is

V ∞ > 0 such that V(T ) ≤ V ∞ for all T > 0, then there is U ∞ > 0 such that U(T ) ≤ U ∞ for all T > 0.
Proof. We outline the proof here since it relies on a well-established argument already described in previous works [START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF][START_REF] Jiang | Global existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption[END_REF][START_REF] Xiao | Global existence and uniform boundedness in a fully parabolic Keller-Segel system with non-monotonic signal-dependent motility[END_REF], see also [START_REF] Fujie | Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions[END_REF][START_REF]Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions[END_REF]. First, arguing as in [17, Lemma 3.2], we derive a Hölder estimate for v in C α [0, T ] × Ω , with a possible dependence upon T of both the estimate and the exponent α ∈ (0, 1). Let us mention that, as already noticed in [START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF], no monotonicity property of γ is needed here and the proof only requires positive upper and lower bounds on γ(v).

The second step is the derivation of estimates for v in L ∞ (0, T ), W 2,r (Ω) for r > 1, still possibly depending on T , as well as on r. The just established Hölder continuity of v enables us to regard γ(v)∆ as a generator of a parabolic evolution operator on L r (Ω) for r ∈ (1, ∞). Then estimates in L ∞ (0, T ), W 2,r (Ω) of v follows from (2.3) and applications of the abstract theory for non-autonomous parabolic equations developed in [START_REF] Amann | Dynamic theory of quasilinear parabolic systems. III: Global existence[END_REF][START_REF]Dynamic theory of quasilinear parabolic equations. II: Reaction-diffusion systems[END_REF][START_REF]Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems[END_REF][START_REF]Linear and quasilinear parabolic problems[END_REF]; see [17, Sect. 3.3 and Sect. 4.2] for detailed proofs.

Finally, we finish the proof of Proposition 2.3 by establish estimates for u in L ∞ (0, T ), L r (Ω) , r ∈ [1, ∞], by a bootstrap argument as done in [START_REF] Ahn | Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing[END_REF].

Global existence and boundedness

The purpose of this section is two-fold: we show the global existence and uniform-in-time boundedness of classical solutions to (1.1), as well as their continuous dependence on initial data.

According to Proposition 2.3, it is sufficient to prove that the L ∞ -norm of v is bounded on [0, T max ) by a time-independent bound to obtain Theorem 1.1. In order to illustrate the approach we use and avoid some technicalities, we first provide a proof under the additional assumption that γ is nondecreasing on (0, ∞), see Section 3.1. The proof in the general case is then given in Section 3.2.

3.1. L ∞ -estimate on v: the non-decreasing case. Lemma 3.1. Assume γ satisfy (1.4) and γ ′ ≥ 0 on (0, ∞). Then

v(t) ∞ ≤ v in ∞ , t ∈ [0, T max ). (3.1)
Proof. Owing to (1.1b), the monotonicity of γ, and the non-negativity of u, there holds

uγ(v) ≤ γ( v ∞ )A[v] in (0, T max ) × Ω.
Consequently, the elliptic comparison implies that

A -1 [uγ(v)] ≤ γ( v ∞ )v in (0, T max ) × Ω,
and it readily follows from (2.3) that v satisfies

∂ t v -γ(v)∆v + vγ(v) ≤ γ( v ∞ )v , (t, x) ∈ (0, T max ) × Ω , (3.2a) ∇v • n = 0, (t, x) ∈ (0, T max ) × ∂Ω , (3.2b) v(0) = v in , x ∈ Ω . (3.2c)
Introducing the solution V to the ordinary differential equation

dV dt + V γ(V ) = γ( v ∞ )V , t ∈ (0, T max ) , (3.3a)
V (0) = v in ∞ , (3.3b)
we infer from (3.2), (3.3), and the parabolic comparison principle that v(t, x) ≤ V (t), (t, x) ∈ (0, T max ) × Ω.

(3.4)

In particular, for t ∈ (0, T max ), there holds v(t) ∞ ≤ V (t) and we deduce from (3.3a), the nonnegativity of V , and the monotonicity of γ that dV dt

(t) + V (t)γ(V (t)) ≤ γ(V (t))V (t), t ∈ (0, T max ).
Hence, dV /dt ≤ 0 in (0, T max ), from which (3.1) follows after using (3.3b) and (3.4).

3.2. L ∞ -estimate on v: the general case. When γ is not monotone but becomes unbounded near infinity, the previous argument no longer works and requires to be suitably adapted. To this end, we begin with the following auxiliary result.

Lemma 3.2. Under the assumptions (1.4) and (1.9), there is

s * ≥ v in ∞ such that γ(s * ) = max s∈[v * ,s * ] {γ(s)}, (3.5)
recalling that the lower bound v * on v is defined in (2.5).

We may obviously choose s * = v in ∞ when γ is non-decreasing.

Proof. Let j ≥ 1 be a positive integer and set

M j := max s∈ v * ,j v in ∞ {γ(s)}, s j := sup s ∈ v * , j v in ∞ : γ(s) = M j , so that γ(s j ) = M j = max s∈ v * ,s j {γ(s)}.
Then (M j ) j≥1 and (s j ) j≥1 are non-decreasing sequences of positive real numbers and the unboundedness (1.9) of γ at infinity guarantees that

lim j→∞ M j = ∞ and lim j→∞ s j = ∞.
Consequently,

j 0 := inf{j ≥ 1 : s j ≥ v in ∞ } < ∞. Setting s * := s j 0 , it is clear that s * ≥ v in ∞ with γ(s * ) = γ(s j 0 ) = M js 0 = max s∈[v * ,s j 0 ] {γ(s)} = max s∈[v * ,s * ] {γ(s)},
and the proof is complete.

We next define

γ ′ i (s) := 0, s ∈ [v * , s * ), (γ ′ (s)) + = max γ ′ (s), 0 , s ≥ s * , (3.6)
and

γ ′ d (s) := 0, s ∈ [v * , s * ), -(γ ′ (s)) -= min γ ′ (s), 0 , s ≥ s * , (3.7)
with γ i (s * ) = γ d (s * ) = 0 and notice that

γ i ≥ 0 ≥ γ d on [v * , ∞) (3.8) and γ(s) = γ(s * ) + γ i (s) + γ d (s), s ∈ [s * , ∞). (3.9) In addition, γ i (s) = γ d (s) = 0, s ∈ [v * , s * ]. (3.10) We also set Γ d (s) := s s * γ d (σ)dσ, s ∈ [v * , ∞), (3.11)
and deduce from (3.8) and (3.10) that

Γ d (s) = 0, s ∈ [v * , s * ], 0 ≥Γ d (s) ≥ (s -s * )γ d (s) ≥ sγ d (s), s ≥ s * . (3.12)
With this notation, we are in a position to derive an upper bound on uγ(v).

Lemma 3.3. There holds

uγ(v) ≤ u γ(s * ) + γ i ( v ∞ ) + A[Γ d (v)] in (0, T max ) × Ω.
Proof. Let t ∈ (0, T max ). We first consider the case where v(t) ∞ ≥ s * . Then, for x ∈ Ω, either v(t, x) ≥ s * and it follows from (1.1b), (3.9), the monotonicity of γ i and γ d , and the non-negativity of u that

u(t, x)γ(v(t, x)) =u(t, x)γ(s * ) + u(t, x)γ i (v(t, x)) + u(t, x)γ d (v(t, x)) ≤u(t, x)γ(s * ) + u(t, x)γ i ( v(t) ∞ ) + γ d (v(t, x))(v -∆v)(t, x) =u(t, x)γ(s * ) + u(t, x)γ i ( v(t) ∞ ) -div(γ d (v)∇v)(t, x) + v(t, x)γ d (v(t, x)) + γ ′ d (v(t, x))|∇v(t, x)| 2 ≤u(t, x)γ(s * ) + u(t, x)γ i ( v(t) ∞ ) -∆Γ d (v)(t, x) + v(t, x)γ d (v(t, x)).
We next use (3.12) to arrive at

u(t, x)γ(v(t, x)) ≤ u(t, x) γ(s * ) + γ i ( v(t) ∞ ) + A[Γ d (v)](t, x).
Or v * ≤ v(t, x) < s * , and there is r > 0 such v * ≤ v(t, y) < s * for all y ∈ B r (x) ⊂ Ω by the continuity of v and (2.5). Therefore, Γ d (v(t)) ≡ 0 in B r (x) by (3.12) and

A[Γ d (v)](t, x) = 0. Consequently, recalling (3.5), u(t, x)γ(v(t, x)) ≤ u(t, x)γ(s * ) ≤ u(t, x) γ(s * ) + γ i ( v(t) ∞ ) + A[Γ d (v)](t, x).
We next argue as above to conclude that, when v(t) ∞ < s * , the inequality

u(t, x)γ(v(t, x)) ≤ u(t, x)γ(s * ) ≤ u(t, x) γ(s * ) + γ i ( v(t) ∞ ) + A[Γ d (v)](t, x)
also holds true for all x ∈ Ω. This completes the proof.

After this preparation, we are ready to establish an L ∞ -estimate on v. Proposition 3.4. Recalling that s * is defined in Lemma 3.2, there holds

v(t) ∞ ≤ s * , t ∈ [0, T max ). (3.13)
Proof. It readily follows from Lemma 3.3 and the elliptic comparison principle that

A -1 [uγ(v)] ≤ γ(s * ) + γ i ( v ∞ ) v + Γ d (v) in (0, T max ) × Ω.
Consequently, recalling (2.3), we realize that v satisfies

∂ t v -γ(v)∆v + vγ(v) ≤ γ(s * ) + γ i ( v ∞ ) v + Γ d (v), (t, x) ∈ (0, T max ) × Ω, (3.14a) ∇v • n = 0, (t, x) ∈ (0, T max ) × ∂Ω, (3.14b)
v(0) = v in , x ∈ Ω. (3.14c)
We next introduce the solution V to the following ordinary differential equation

dV dt + V γ(V ) = γ(s * ) + γ i ( v(t) ∞ ) V + Γ d (V ), t ∈ (0, T max ), (3.15a)
V (0) = s * , (3.15b)
noticing that the non-positivity (3.12) of Γ d indeed guarantees that V is well-defined on [0, T max ). Since s * ≥ v in ∞ by Lemma 3.2, we deduce from (3.14), (3.15), and the parabolic comparison principle that

v(t) ∞ ≤ V (t), t ∈ (0, T max ). (3.16)
Let T ∈ (0, T max ). On the one hand, the continuity of v, the positivity of γ, and the nonpositivity (3.12) of Γ d imply that

V (t) ≤ V T := s * exp T γ(s * ) + sup τ ∈[0,T ] {γ i ( v(τ ) ∞ )} , t ∈ [0, T ]. (3.17)
On the other hand, owing to the monotonicity of γ i , it follows from (3.15a) and (3.17) that

dV dt + V γ(V ) ≤ γ(s * ) + γ i (V ) V + Γ d (V ), t ∈ (0, T max ). (3.18)
Now, set G(s

) := Γ d (s) -sγ d (s) for s ≥ v * and notice that G(s) = 0 for s ∈ [v * , s * ] by (3.10) and (3.12
). We infer from (3.9) and (3.18) that

d(V -s * ) + dt ≤ γ(s * ) + γ i (V ) -γ(V ) V sign + (V -s * ) + Γ d (V )sign + (V -s * ) = γ(s * ) + γ i (V ) -γ(s * ) -γ i (V ) -γ d (V ) V sign + (V -s * ) + Γ d (V )sign + (V -s * ) = G(V ) -G(s * ) sign + (V -s * ) = G(V ) -G(s * ) V -s * (V -s * ) + . Now, since G ′ (s) = -sγ ′ d (s) ≥ 0 for s ≥ s * , we deduce from (3.17) that 0 ≤ G(V ) -G(s * ) V -s * (V -s * ) + ≤ (V -s * ) + sup s∈[s * ,V T ] {G ′ (s)}, t ∈ [0, T ],
and we conclude that 3.3. Continuous dependence. An interesting corollary of the previous analysis is the continuous dependence of the solutions to (1.1) with respect to the initial condition u in , a property which is likely to follow as well from the proof of [START_REF] Ahn | Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing[END_REF]Lemma 3.1]. This property is needed later in the proof of Theorem 1.3. We recall that, according to Corollary 1.2, the problem (1.1) is well-posed in a classical sense under the sole assumption (1.4) on γ.

V (t) ≤ s * for t ∈ [0, T ]. As T is arbitrary in (0, T max ),
Proposition 3.5. Assume that γ satisfies (1.4) and consider a sequence (u in j ) j≥1 in W 1,∞ + (Ω) and

u in ∈ W 1,∞ + (Ω) such that u in 1 > 0 and lim j→∞ u in j -u in ∞ = 0. (3.19)
For j ≥ 1, let (u j , v j ) be the classical solution to (1.1) with initial condition u in j and denote the classical solution to (1.1) with initial condition u in by (u, v), their existence being guaranteed by Corollary 1.2. Then, for any T > 0,

lim j→∞ sup t∈[0,T ] (u j -u)(t) ∞ + (v j -v)(t) ∞ = 0. (3.20) Proof. Setting v in j := A -1 [u in j ], j ≥ 1, and v in := A -1 [u in ],
it readily follows from (3.19) that we may assume without loss of generality that there are m 0 > 0 and M 0 > 0 such that

u in j 1 ≥ m 0 and v in j ∞ ≤ M 0 , j ≥ 1. (3.21)
We then argue as in Section 3.2 (with m 0 instead of m and M 0 instead of v in ∞ ) to establish that there is s 0 ≥ M 0 such that

m 0 ω * ≤ v j (t, x) ≤ s 0 , (t, x) ∈ [0, ∞) × Ω, j ≥ 1, (3.22)
the lower bound in (3.22) being a consequence of Lemma 2.2 and (3.21). Owing to (3.22), we next proceed as in [START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF]Proposition 4.6] to show that there is α ∈ (0, 1/2) such that

(v j ) j≥1 is bounded in BUC 2α [0, ∞), C 2α ( Ω) . (3.23)
In addition, arguing as in [START_REF] Jiang | Global existence and uniform boundedness in a chemotaxis model with signaldependent motility[END_REF]Proposition 4.7], we obtain that, for any p ∈ (N, ∞), there is

C 0 (p) > 0 such that v j (t) W 2,p ≤ C 0 (p), t ≥ 0, j ≥ 1.
(3.24) Fix p > 2N/(1-2α). Since C 2α ( Ω) is continuously embedded in L p (Ω) and interpolation and Sobolev inequalities guarantee that there are C 1 (p) > 0 and C 2 (p) > 0 such that

z C 1+α ≤ C 1 (p) z W 3/2,p ≤ C 2 (p) z 3/4 W 2,p z 1/4 p , z ∈ W 2,p (Ω),
we infer from (3.24) and the above inequality that, for j ≥ 1 and (t,

s) ∈ [0, ∞) 2 , v j (t) -v j (s) C 1+α ≤ C 2 (p) v j (t) 3/4 W 2,p + v j (s) 3/4 W 2,p v j (t) -v j (s) 1/4 p ≤ 2C 0 (p) 3/4 C 2 (p)|Ω| 1/4p v j (t) -v j (s) 1/4 C 2α .
Hence, owing to (3.23), we conclude that

(v j ) j≥1 is bounded in BUC α/2 [0, ∞), C 1+α ( Ω) . (3.25a)
It then follows from (3.22), (3.25a), and parabolic regularity for linear equations [21, Chapter V, Theorem 1.1] applied to (1.1a) considered as a linear equation for u j that there is α 1 ∈ (0, 1) such that (u j ) j≥1 is bounded in C α 1 /2,α 1 [0, T ] × Ω for any T > 0.

(3.25b)

We now use elliptic regularity theory to deduce from (1.1b) and (3.25b) that

(v j ) j≥1 is bounded in C α 1 /2,2+α 1 [0, T ] × Ω for any T > 0, (3.25c)
which, together with parabolic regularity applied to (1.1a), see [21, Chapter IV], gives that (u j ) j≥1 is bounded in C (2+α 1 )/2,2+α 1 (0, T ) × Ω for any T > 0.

(3.25d)

We now infer from (3.25) and the Arzelà-Ascoli theorem that there are a subsequence of (u j , v j ) j≥1 (not relabeled) and

ũ ∈ C [0, ∞) × Ω ∩ C 1,2 (0, ∞) × Ω , ṽ ∈ C [0, ∞), C 1 ( Ω) such that lim j→∞ sup t∈[0,T ] { (u j -ũ)(t) ∞ + (v j -ṽ)(t) C 1 } = 0 for any T > 0. (3.26)
Since (u j , v j ) solves (1.1) with initial condition u in j , we combine (3.25) with the convergence (3.26) to conclude that (ũ, ṽ) is a non-negative global classical solution to (1.1) with initial condition u in ; that is, (ũ, ṽ) = (u, v) by the uniqueness of classical solutions to (1.1), see Proposition 2.1, so that it is the whole sequence (u j , v j ) j≥1 which actually converges to (u, v) and satisfies (3.26). We have thus completed the proof.

Explicit upper and lower bounds on v

The main building block of the proof of Theorem 1.3 is the following lemma, which is somewhat a generalisation of Lemma 3.1.

γ ′ ≥ 0 on [a, b], (4.1)
a < min Ω {v in } ≤ max Ω {v in } < b, (4.2)
where

v in = A -1 [u in ]. Then a ≤ v(t, x) ≤ b, (t, x) ∈ [0, ∞) × Ω.
Proof. For t ≥ 0, we put

(µ)v = A -1 [γ(µ)u] ≤ A -1 [γ(v)u] ≤ A -1 [γ(M)u] = γ(M)v in [0, τ ) × Ω. ( 4.4)
It follows from (2.3) and (4.4) that v satisfies

γ(M)v ≥ ∂ t v -γ(v)∆v + vγ(v) ≥ γ(µ)v, (t, x) ∈ (0, τ ) × Ω, (4.5a) ∇v • n = 0, (t, x) ∈ (0, τ ) × ∂Ω, (4.5b)
v(0) = v in , x ∈ Ω. ( 4.5c)
The parabolic comparison principle then entails that

V u (t) ≥ v(t, x) ≥ V l (t), (t, x) ∈ [0, τ ) × Ω, (4.6)
where V l and V u are the solutions to the ordinary differential equations

dV l dt + V l γ(V l ) = γ(µ)V l , t ∈ (0, τ ), (4.7a) V l (0) = µ(0), (4.7b)
and

dV u dt + V u γ(V u ) = γ(M)V u , t ∈ (0, τ ), (4.8a) V u (0) = M(0), (4.8b) respectively. Since b > V u (0) ≥ V l (0) > a by (4.2), τ * := inf {t ∈ (0, τ ) : V u (t) > b or V l (t) < a} ∈ (0, τ ],
and we infer from (4.1), (4.3), and (4.6) that Hence, owing to (4.2)

γ(V l (t)) ≤ γ(µ(t)) ≤ γ(v(t, x)) ≤ γ(M(t)) ≤ γ(V u (t)), (t, x) ∈ [0, τ * ) × Ω. ( 4
V l (t) ≥ V l (0) > a and V u (t) ≤ V u (0) < b, t ∈ [0, τ * ),
so that τ * = τ . Gathering (4.6) and the above inequalities, we conclude that

a < V l (t) ≤ v(t, x) ≤ V u (t) < b, (t, x) ∈ [0, τ ) × Ω.
Consequently, τ = ∞ and the proof of Lemma 4.1 is complete. Let us thus assume that u in is not constant, so that v in is also not constant. We then pick ∈ minΩ{v in }, maxΩ{v in } and define

u in j := j -1 j u in + 1 j ν, v in j := A -1 [u in j ] = j -1 j v in + 1 j ν, j ≥ 1.
Clearly, for each j ≥ 1,

min Ω {v in } < j -1 j min Ω {v in } + 1 j ν = min Ω {v in j }, max Ω {v in } > j -1 j max Ω {v in } + 1 j ν = max Ω {v in j },
and we infer from Lemma 4.1 that the global classical solution (u j , v j ) to (1.1) with initial condition u in j satisfies min Ω {v in } ≤ v j (t, x) ≤ max Ω {v in }, (t, x) ∈ [0, ∞) × Ω.

We then pass to the limit j → ∞ in the above inequality with the help of Proposition 3.5 to complete the proof.

Explicit L ∞ -estimates on u

This section is devoted to the proof of Theorem 1.4 and we begin with the case of non-increasing and concave motility function γ. In this case, the evolution equation (2.3a) for v is not used and the proof relies on direct computations involving (1.1a) and (1.1b) and comparison arguments.

U(0) = u in ∞ , (5.2b)
it readily follows from (5.1), (5.2), and the parabolic comparison principle that u(t, x) ≤ U(t), (t, x) ∈ [0, ∞) × Ω.

(5.3)

In particular, u(t) ∞ ≤ U(t) for all t ≥ 0 by (5.3) and we infer from (5.2a) and the non-negativity of Y that dU/dt ≤ 0 in (0, ∞). Therefore, U(t) ≤ U(0) for t ≥ 0 which, together with (5.2b) and (5.3), completes the proof.

We next turn to general non-decreasing motility functions γ and use the comparison principle applied to the parabolic equation satisfied by the auxiliary function uγ(v).

Proof of Theorem 1.4 (Part 2). We assume here that γ satisfies (1.4) and (1.13). Introducing ϕ := uγ(v), we infer from (1.1a) and (2.3a) that

∂ t ϕ = γ(v)∂ t u + uγ ′ (v)∂ t v = γ(v)∆ϕ + uγ ′ (v) A -1 [ϕ] -ϕ = γ(v)∆ϕ + γ ′ γ (v)ϕ A -1 [ϕ] -ϕ . Since A -1 [ϕ] ≤ ϕ ∞ in [0, ∞) × Ω
by the elliptic comparison principle, it follows from the monotonicity of γ, the non-negativity of ϕ, and Theorem 1.

3 that γ ′ γ (v)ϕ A -1 [ϕ] -ϕ ≤ γ ′ γ (v)ϕ ( ϕ ∞ -ϕ) ≤ Y γ ′ (v) ∞ ϕ ϕ ∞ -ϕ ,
with 1/Y := γ(minΩ{v in }) > 0. Therefore, ϕ satisfies

∂ t ϕ -γ(v)∆ϕ ≤ Y γ ′ (v) ∞ ϕ ϕ ∞ -ϕ in (0, ∞) × Ω, (5.4a)
∇ϕ • n = 0 on (0, ∞) × ∂Ω, (5.4b) ϕ(0) = u in γ v in in Ω.

(5.4c)

1 ,Proposition 2 . 1 .

 121 Lemma 3.1]. Assume that γ satisfies (1.4) and thatu in ∈ W 1,∞ + (Ω) with m = u in 1 > 0.Then the initial-boundary value problem (1.1) has a unique non-negative classical solution

3 .

 3 we have shown (3.13) in view of (3.16). Proof of Theorem 1.1. Theorem 1.1 now readily follows from Proposition 3.4 and Proposition 2.Proof of Corollary 1.2. Either γ satisfies (1.9) and Corollary 1.2 is an immediate consequence of Theorem 1.1. Or γ belongs to L ∞ (v * , ∞) and Corollary 1.2 follows at once from [17, Theorem 1.1].

Lemma 4 . 1 .

 41 Assume that γ satisfies (1.4) and letu in ∈ W 1,∞ + (Ω) with m = u in 1 > 0.We denote the corresponding global classical solution to (1.1) by (u, v), see Corollary 1.2. Assume further that there are 0 < a < b such that

v

  * ≤ µ(t) := min x∈ Ω {v(t, x)} ≤ M(t) := max x∈ Ω {v(t, x)}, which are well-defined and positive according to elliptic regularity and Lemma 2.2. Owing to the continuity of v, it follows from (4.2) that τ := inf {t > 0 : M(t) > b or µ(t) < a} ∈ (0, ∞]. In particular, a ≤ µ(t) ≤ M(t) ≤ b, t ∈ [0, τ ). (4.3) By (4.1), (4.3), and the non-negativity of u, uγ(µ) ≤ uγ(v) ≤ uγ(M) in [0, τ ) × Ω, and the elliptic comparison principle and (1.1b) imply that γ

Proof of Theorem 1 . 3 .

 13 If u in ≡ m/|Ω|, then (u, v) = (m/|Ω|, m/|Ω|) and Theorem 1.3 is obvious.

Proof of Theorem 1 . 4 (

 14 Part 1). Let us thus consider a function γ satisfying (1.4) and being nondecreasing and concave on minΩ{v in }, maxΩ{v in } . By (1.1a) and (1.1b),∂ t u = div γ(v)∇u + uγ ′ (v)∇v = γ(v)∆u + 2γ ′ (v)∇v • ∇u + uγ ′′ (v)|∇v| 2 + uγ ′ (v)∆v = γ(v)∆u + 2γ ′ (v)∇v • ∇u + uγ ′′ (v)|∇v| 2 + uγ ′ (v)(v -u). Since v = A -1 [u] and u ≤ u ∞ in Ω, the elliptic comparison principle implies that v(t, x) ≤ u(t) ∞ , (t, x) ∈ [0, ∞) × Ω.Consequently, recalling that v ranges in minΩ{v in }, maxΩ{v in } by Theorem 1.3, it follows from the monotonicity and concavity of γ and the non-negativity of u thatuγ ′′ (v)|∇v| 2 + uγ ′ (v)(v -u) ≤ uγ ′ (v) u ∞ -u ≤ Y u u ∞ -uwith Y := γ ′ (minΩ{v in }) ≥ 0 and we conclude that∂ t u -γ(v)∆u -2γ ′ (v)∇v • ∇u ≤ Y u u ∞ -u in (0, ∞) × Ω, (5.1a) ∇u • n = 0 on (0, ∞) × ∂Ω,(5.1b)u(0) = u in in Ω. (5.1c) Introducing the solution U ∈ C 1 ([0, ∞)) to the ordinary differential equation dU dt = Y U u ∞ -U , t > 0, (5.2a)

Acknowledgments Jiang is supported by National Natural Science Foundation of China (NSFC) under grants No. 12271505 & No. 12071084, the Training Program of Interdisciplinary Cooperation of Innovation Academy for Precision Measurement Science and Technology, CAS (No. S21S3202), and by Knowledge Innovation Program of Wuhan-Basic Research (No. 2022010801010135).

Let ψ ∈ C 1 ([0, ∞)) be the solution to the ordinary differential equation dψ dt = Y γ ′ (v) ∞ ψ ϕ ∞ -ψ , t > 0, (5.5a)

(5.5b)

Owing to (5.4), (5.5), and the parabolic comparison principle,

In particular, for t > 0, ϕ(t) ∞ ≤ ψ(t) and we deduce from (5.5a) that dψ/dt ≤ 0 on (0, ∞).

Combining the just established time monotonicity of ψ with (5.6) gives

Now, on the one hand, the monotonicity of γ and Theorem 1.3 imply that

On the other hand, the elliptic comparison principle and the definition

∞ and we use once more the monotonicity of γ to obtain

Gathering (5.7), (5.8), and (5.9) completes the proof.

Large time behaviour for non-decreasing γ

Proof of of Theorem 1.5. As already pointed out in the introduction, the monotonicity of γ implies the existence of a Liapunov function. Indeed, it readily follows from (1.1) that 1 2

and the monotonicity and positivity of γ, along with Theorem 1.3, guarantee that the right-hand side of the above identity is non-positive. We then proceed as in the proof of [START_REF] Ahn | Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing[END_REF]Theorem 1.3] to complete that of Theorem 1.5.