Philippe Laurenc ¸ot
email: laurenco@math.univ-toulouse.fr

GLOBAL BOUNDED AND UNBOUNDED SOLUTIONS TO A CHEMOTAXIS SYSTEM WITH INDIRECT SIGNAL PRODUCTION

Keywords: Mathematics Subject Classification, 35B40 -35M33 -35K10 -35Q92, chemotaxis system -global existence -boundedness -unbounded global solutions

The well-posedness of a chemotaxis system with indirect signal production in a twodimensional domain is shown, all solutions being global unlike the classical Keller-Segel chemotaxis system. Nevertheless, there is a threshold value M c of the mass of the first component which separates two different behaviours: solutions are bounded when the mass is below M c while there are unbounded solutions starting from initial conditions having a mass exceeding M c . This result extends to arbitrary two-dimensional domains a previous result of Tao & Winkler (2017) obtained for radially symmetric solutions to a simplified version of the model in a ball and relies on a different approach involving a Liapunov functional.

Introduction

Chemotaxis models have been derived in [START_REF] Powell | Connecting a chemotactic model for mass attack to a rapid integro-difference emulation strategy[END_REF][START_REF] Strohm | Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data[END_REF][START_REF] White | Spatial invasion of pine beetles into lodgepole forests: a numerical approach[END_REF] to describe the spreading and aggregative behaviour of the mountain pine beetle which has a major impact on the forest industry in North America. These models describe the space and time evolution of the density u of flying beetles, the density v of nesting beetles, and the concentration w of beetle pheromone (chemoattractant). The main behavioural difference between flying and nesting beetles is that the latter do not disperse in space while the former move randomly in space, their motion being biased by high gradients of the pheromone concentration. A very important feature in the model is that the pheromone is produced by the nesting beetles and not by the flying ones, though it influences only the motion of the latter. This is in sharp contrast with the classical Keller-Segel model where the chemical inducing a bias in the motion of the species is produced directly by the species itself. As we shall see below, this feature alters significantly the dynamics, at least in two space dimensions.

A simplified version of the models derived in [START_REF] Powell | Connecting a chemotactic model for mass attack to a rapid integro-difference emulation strategy[END_REF][START_REF] Strohm | Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data[END_REF][START_REF] White | Spatial invasion of pine beetles into lodgepole forests: a numerical approach[END_REF] is considered in [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF] and reads

∂ t u = div (∇u -u∇w) in (0, ∞) × Ω , (1.1a
)

νε∂ t v = u -v in (0, ∞) × Ω , (1.1b) 0 = D∆w -v + v in (0, ∞) × Ω , (1.1c)
supplemented with no-flux boundary conditions for u and w

∇u • n = ∇w • n = 0 on (0, ∞) × ∂Ω , (1.1d)
the normalization condition for w w = 0 in (0, ∞) , (1.1e) and non-negative initial conditions (u in , v in ) for (u, v).

Here Ω is a smooth bounded domain of R 2 , ν, ε, and D are positive parameters, and v in (1.1c) and w in (1.1e) denote the mean value with respect to the space variable of v and w, respectively. Recall that, for z ∈ L 1 (Ω), its mean value is given by

z := 1 |Ω| Ω z(x) dx .
A Jäger-Luckhaus version of the Keller-Segel chemotaxis system [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF] is recovered from (1.1) by setting νε = 0 and reads, since v = u in that case,

∂ t u = div (∇u -u∇w) in (0, ∞) × Ω , (1.2a
)

0 = D∆w -u + u in (0, ∞) × Ω , (1.2b)
supplemented with no-flux boundary conditions for u and w

∇u • n = ∇w • n = 0 on (0, ∞) × ∂Ω , (1.2c)
the normalization condition for w w = 0 in (0, ∞) ,

and a non-negative initial condition u in for u. Observe that, in (1.2), the chemoattractant is produced directly by the species with density u, instead of being produced through another species as in (1.1).

A first consequence of this feature is that all solutions to (1.1) are global [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF]Theorem 1.1]. This global existence property contrasts markedly with the situation for (1.2) for which the following is known: solutions to (1.2) are global when either u in 1 < 4πD or Ω is a ball, u in is radially symmetric, and u in 1 > 8πD [START_REF] Biler | Local and global solvability of some parabolic systems modelling chemotaxis[END_REF][START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF][START_REF] Nagai | Blow-up of radially symmetric solutions to a chemotaxis system[END_REF]. Finite time blowup may take place when either u in 1 > 4πD or Ω is a ball, u in is radially symmetric, and u in 1 > 8πD [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF][START_REF] Nagai | Blow-up of radially symmetric solutions to a chemotaxis system[END_REF][START_REF]Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains[END_REF][START_REF] Senba | Blowup behavior of solutions to the rescaled Jäger-Luckhaus system[END_REF], see also the survey [START_REF]From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I[END_REF]. A striking feature of the dynamics of (1.1), uncovered in [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF], is that, though all solutions to (1.1) are global, a threshold phenomenon occurs in infinite time. More precisely, when Ω is a ball, say Ω = B 1 (0) := {x ∈ R 2 : |x| < 1}, and the initial conditions (u in , v in ) are radially symmetric, so that the solution (u, v, w) of (1.1) is also radially symmetric for all times, it is shown in [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF]Theorems 1.2 & 1.3] that all (radially symmetric) solutions are actually bounded when u in 1 < 8πD while there are unbounded solutions emanating from initial conditions satisfying u in 1 > 8πD, the L ∞ -norm of u(t) growing up as t → ∞ at an exponential rate. The approach of [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF] exploits the classical fact that the specific structure of (1.1) and the radial symmetry of the solutions allow one to reduce (1.1) to a single nonlocal parabolic equation for the cumulative distribution function U defined by

U(t, |x|) := √ |x| 0 σū(t, σ) dσ , (t, x) ∈ (0, ∞) × B 1 (0) ,
where ū(t, |x|) := u(t, x) for (t, x) ∈ (0, ∞) × B 1 (0 This technique is however restricted to (1.1) in a radially symmetric setting and does not allow one to handle more general initial conditions or to consider an arbitrary domain Ω ⊂ R 2 . Neither does it extend to the version of (1.1) involving degradation of the chemoattractant

∂ t u = div (∇u -u∇w) in (0, ∞) × Ω , (1.3a
)

νε∂ t v = u -v in (0, ∞) × Ω , (1.3b) 0 = D∆w -δw + v in (0, ∞) × Ω , (1.3c)
with δ > 0, supplemented with no-flux boundary conditions for u and w and non-negative initial conditions (u in , v in ) for (u, v), nor to its parabolic counterpart

∂ t u = div (∇u -u∇w) in (0, ∞) × Ω , (1.4a
)

νε∂ t v = u -v in (0, ∞) × Ω , (1.4b
)

ν∂ t w = D∆w -δw + v in (0, ∞) × Ω , (1.4c)
supplemented with no-flux boundary conditions for u and w

(∇u -u∇w) • n = ∇w • n = 0 on (0, ∞) × ∂Ω , (1.4d)
and initial conditions

(u, v, w)(0) = (u in , v in , w in ) in Ω . (1.5)
The main contribution of this paper is to show that the systems (1.1), (1.3), and (1.4) share the same infinite time threshold phenomenon uncovered in [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF], without restrictions on the twodimensional domain Ω and the initial data. This requires a different argument and we actually show that (1.1), (1.3), and (1.4) all possess a Liapunov functional, and that the properties of this Liapunov functional provide insight on the boundedness or unboundedness of the solutions.

From now on, we shall focus on (1.4)-(1.5) and will return briefly to (1.1) and (1.3) in Section 5. Introducing

L(r) := r ln r -r + 1 ≥ 0 , r ≥ 0 , (1.6)
and

E 0 (u, w) := Ω [L(u) -uw] dx + D 2 ∇w 2 2 + δ 2 w 2 2 , (1.7a) E(u, v, w) := E 0 (u, w) + ε 2 -D∆w + δw -v 2 2 , (1.7b)
the main observation is that E is a Liapunov functional for (1.4)-(1.5), see Lemma 3.2. Interestingly, E 0 is a Liapunov functional for the classical Keller-Segel system [START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF][START_REF] Nagai | Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[END_REF]

∂ t u = div (∇u -u∇w) in (0, ∞) × Ω , (1.8a
)

ν∂ t w = D∆w -δw + u in (0, ∞) × Ω , (1.8b) ∇u • n = ∇w • n = 0 on (0, ∞) × ∂Ω , (1.8c) (u, w)(0) = (u in , w in ) in Ω , (1.8d)
and, as such, its properties have been thoroughly studied [9-11, 13, 17]. In particular, introducing the set

A M := (u, w) ∈ L 1 (Ω) × W 1 2 (Ω) : u ≥ 0 , w ≥ 0 , u 1 = M (1.9) for M ≥ 0, it is known that inf (u,w)∈A M E 0 (u, w) > -∞ if M ∈ [0, 4πD] , inf (u,w)∈A M E 0 (u, w) = -∞ if M > 4πD , (1.10)
a property which has far-reaching consequences on the dynamics of (1.8). Indeed, given M > 0, global existence and blowup of solutions to (1.8) are intimately related to the finiteness or not of the infimum of E 0 on A M [START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF][START_REF] Horstmann | The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results[END_REF][START_REF]On the existence of radially symmetric blow-up solutions for the Keller-Segel model[END_REF][START_REF] Horstmann | Blow-up in a chemotaxis model without symmetry assumptions[END_REF][START_REF] Nagai | Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[END_REF]. As we shall see below, the property (1.10) also plays an important role in the dynamics of (1.4)-(1.5), a feature which is actually not so surprising as there is a close connection between (1.4)-(1.5) and (1.8). Indeed, (1.8) can formally be derived from (1.4)-(1.5) by setting ε = 0. We now describe our results on (1.4)-(1.5) and begin with its well-posedness. For θ ∈ (2/3, 1) and M ≥ 0, we set

I M,θ := (u, v, w) ∈ W 2θ 3,B,+ (Ω) × W 1 3,B,+ (Ω) × W 2 3,B,+ (Ω) : u 1 = M , (1.11)
where

W m r,B (Ω) :=            {z ∈ W m r (Ω) : ∇z • n = 0 on ∂Ω} , 1 + 1/r < m ≤ 2 , W m r (Ω) , -1 + 1/r < m < 1 + 1/r , W -s r/(r-1) (Ω) ′ , -2 + 1/r < s ≤ -1 + 1/r , (1.12)
and

W m r,B,+ (Ω) := z ∈ W m r,B (Ω) : z ≥ 0 in Ω (1.13) for r ∈ (1, ∞), see [3, Section 5].
Theorem 1.1. Let θ ∈ (5/6, 1) and consider initial conditions (u in , v in , w in ) ∈ I M,θ . Then the system (1.4)-(1.5) has a unique non-negative weak solution

(u, v, w) in W 1 3 defined on [0, ∞) satisfying u ∈ C([0, ∞); W 2θ 3,B (Ω)) ∩ C 1 ([0, ∞); W 2θ-2 3,B (Ω)) , v ∈ C 1 ([0, ∞); W 1 3 (Ω)) , w ∈ C([0, ∞); W 2 3,B (Ω)) ∩ C 1 ([0, ∞); L 3 (Ω)) , and
u(t) 1 = M := u in 1 , t ≥ 0 . (1.14)
In particular, (u(t), v(t), w(t)) ∈ I M,θ for all t ≥ 0. Furthermore,

u ∈ C((0, ∞); W 2 3,B (Ω)) ∩ C 1 ((0, ∞); L 3 (Ω))
. Several approaches may be used to deal with the well-posedness of (1.4)-(1.5). Since (1.4b) can be solved explicitly to find v in terms of u, one possibility would be to solve the parabolic system (1.4a)-(1.4c)-(1.4d)-(1.5) with a source term which is nonlocal with respect to time. However, since an abstract theory is developed in [START_REF]Highly degenerate quasilinear parabolic systems[END_REF] to handle systems coupling parabolic equations and ordinary differential equations, we rather follow this route which provides the local well-posedness of (1.4)-(1.5) in W 1 3 (Ω) and may cope as well with nonlinear reaction terms in (1.4b) or (1.4c). The parabolic regularising effects of both (1.4a) and (1.4c) are then used to prove that it is actually a strong solution for positive times. Several estimates are next needed to prove that the solution is global, the positivity of ε being of utmost importance already in Lemma 2.3.

Having established the global well-posedness of (1.4)-(1.5), we next turn to qualitative information on the dynamics of (1.4)-(1.5) which is the main goal of this paper. We first study the boundedness of solutions, thereby extending [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF]Theorem 1.2] to (1.4)-(1.5) in an arbitrary domain Ω ⊂ R 2 . Theorem 1.2. Let θ ∈ (5/6, 1) and consider initial conditions (u in , v in , w in ) ∈ I M,θ . We denote the corresponding solution to (1.4)- (1.5) given by Theorem 1.1 by (u, v, w).

(a) Assume further that M = u in 1 ∈ (0, 4πD). Then

sup t≥0 u(t) ∞ + v(t) ∞ + w(t) W 1 ∞ < ∞ . (1.15)
(b) Assume further that Ω is the ball B r (0) of radius r > 0 centered at x = 0, (u in , v in , w in ) are radially symmetric, and M = u in 1 ∈ (0, 8πD). Then (1.15) also holds true.

The proof of Theorem 1.2 (a) relies on the properties of the Liapunov functional E (defined in (1.7b)) which are derived from the properties of E 0 (defined in (1.7a)) [START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF][START_REF] Nagai | Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[END_REF]. The main observation is that, when (u, w) belongs to the set A M defined in (1.9) and M ∈ (0, 4πD), it follows from the Moser-Trudinger inequality [START_REF] Chang | Conformal deformation of metrics on S 2[END_REF] that

E 0 (u, w) ≥ 4πD -M 8π ∇w 2 2 + δ 2 w 2 2 -b
for some constant b ≥ 0 which depends only on Ω, D, δ, M, and w 1 . Therefore, an upper bound on E 0 (u, w) entails an estimate on the W 1 2 -norm of w which, in turn, allows us to derive further estimates and deduce the boundedness of the solution. Similar arguments are used to prove Theorem 1.2 (b).

The final result deals with the unboundedness of some solutions provided u in 1 is large enough. Theorem 1.3. Let θ ∈ (5/6, 1).

(a) Given M ∈ (4πD, ∞) \ 4πDN, there are non-negative functions (u in , v in , w in ) ∈ I M,θ for which the corresponding solution (u, v, w) to (1.4)- (1.5) given by Theorem 1.1 satisfies

lim sup t→∞ u(t) ∞ = ∞ . (1.16) (b)
Assume further that Ω is the ball B r (0) of radius r > 0 centered at x = 0. Given M ∈ (8πD, ∞), there are non-negative radially symmetric functions (u in , v in , w in ) ∈ I M,θ for which the corresponding solution (u, v, w) to (1.4)-(1.5) given by Theorem 1.1 satisfies (1.16).

On the one hand, Theorem 1.3 extends [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF]Theorem 1.3] to (1.4)-(1.5) in an arbitrary domain Ω ⊂ R 2 and, when Ω is a ball and the initial data are radially symmetric, applying the same approach to (1.1) is likely to provide an alternative proof of [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF]Theorem 1.3]. It however only gives the unboundedness of the solution, while [START_REF]Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF]Theorem 1.3] guarantees an exponential growth of the L ∞ -norm of u. On the other hand, Theorem 1.3 is also a consequence of the availability of a Liapunov functional and its properties, the latter being established in [START_REF] Horstmann | The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results[END_REF][START_REF]On the existence of radially symmetric blow-up solutions for the Keller-Segel model[END_REF][START_REF] Horstmann | Blow-up in a chemotaxis model without symmetry assumptions[END_REF]. Roughly speaking, the proof of Theorem 1.3 involves three steps and proceeds along the lines of [START_REF] Horstmann | Blow-up in a chemotaxis model without symmetry assumptions[END_REF]. We first show that, given a solution (u, v, w) to (1.4)-(1.5) such that u ∈ L ∞ ((0, ∞) × Ω), then there are a sequence (t k ) k≥1 , t k → ∞, of positive real numbers and a stationary solution (u * , v * , w * ) to (1.4) such that

E(u * , v * , w * ) ≤ lim inf k→∞ E(u(t k ), v(t k ), w(t k )) .
(1.17)

It next follows from [13, Lemma 3.5] that there is µ M ≥ 0 depending only on the parameters in (1.4) and M such that

E(u * , v * , w * ) ≥ -µ M .
(1.18) It now readily follows from (1.17), (1.18), and the fact that E is a Liapunov functional that, if E(u in , v in , w in ) < -µ M , then (u, v, w) cannot be bounded. The last step is to check that such initial conditions exist, but this is again a consequence of the analysis performed in [START_REF] Horstmann | The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results[END_REF][START_REF]On the existence of radially symmetric blow-up solutions for the Keller-Segel model[END_REF][START_REF] Horstmann | Blow-up in a chemotaxis model without symmetry assumptions[END_REF].

Global existence

2.1. Local existence. We begin with the local existence of solutions to (1.4)-(1.5) which is a consequence of the well-posedness theory developped in [START_REF]Highly degenerate quasilinear parabolic systems[END_REF] for partially diffusive systems.

Proposition 2.1. Given non-negative initial conditions (u in , v in , w in ) in W 1 3 (Ω; R 3 ), the system (1.4)-(1.5) has a unique non-negative weak solution (u, v, w) in W 1 3 defined on a maximal time interval [0, T m ), T m ∈ (0, ∞], satisfying (u, w) ∈ C [0, T m ); W 1 3 (Ω; R 2 ) ∩ C 1 [0, T m ); W 1 3/2 (Ω; R 2 ) ′ , v ∈ C 1 ([0, T m ); W 1 3 (Ω)) , and
u(t) 1 = M := u in 1 , t ∈ [0, T m ) . (2.1) Moreover, if there is T > 0 such that (u, v, w) ∈ BUC [0, T ] ∩ [0, T m ); W 1 3 (Ω; R 3 ) , (2.2)
then T m ≥ T .

Proof. In order to cast (1.4)-(1.5) in a form suitable to apply the abstract theory developed in [START_REF]Highly degenerate quasilinear parabolic systems[END_REF], we set

u 1 = u, u 2 = w, u 3 = v, U := (u 1 , u 2 , u 3 ), U 1 := (u 1 , u 2 )
, and U 2 := (u 3 ), and define matrixvalued functions (a

1 j,k ) 1≤j,k≤2 , (a 2 j,k ) 1≤j,k≤2 in M 2 (R) and vector-valued functions (a 1 j ) 1≤j≤2 , (a 2 j ) 1≤j≤2 in R 2 by a 1 1,1 (U) = a 1 2,2 (U) := A(U) , a 1 1,2 (U) = a 1 2,1 (U) := 0 , U ∈ R 3 , a 2 j,k (U) = 0 , a 1 j (U) = a 2 j (U) := 0 , 1 ≤ j, k ≤ 2 , U ∈ R 3 , the 2 × 2-matrix A(U) being given by A(U) := 1 -u 1 0 D/ν For V = (v 1 , v 2 , v 3 ) ∈ R 3 , we define the operators A 1 (V ), A 2 (V ) and the boundary operators B 1 (V ), B 2 (V ) by A r (V )U r := - 2 j,k=1 ∂ j a r j,k (V )∂ k U r + 2 j=1 a r j (V )∂ j U r , r = 1, 2 , B r (V )U r := 2 j,k=1 a r j,k (V )n j ∂ k U r , r = 1, 2 ,
as well as the source terms

S 1 (V ) := 0 v 3 -v 2 ν , S 2 (V ) := v 1 -v 3 νε .
With this notation, the system (1.4)-(1.5) reads

∂ t U 1 + A 1 (U)U 1 + A 2 (U)U 2 = S 1 (U) in (0, ∞) × Ω , (2.3a)
∂ t U 2 = S 2 (U) in (0, ∞) × Ω , (2.3b) B 1 (U)U 1 + B 2 (U)U 2 = 0 on (0, ∞) × Ω , (2.3c)
U(0) = U in in Ω , (2.3d)
where

U in := (u in , w in , v in ) ∈ W 1 3 (Ω; R 3 ). Since A(U) has two positive eigenvalues (1, D/ν) for all U ∈ R 3 , it follows from the structure of (a 1 j,k ) 1≤j,k≤2 and [2, Remarks 4.1 (a)-(iii)] that (A 1 , B 1
) is normally elliptic and, since B 2 = 0, the condition (6.1) in [START_REF]Highly degenerate quasilinear parabolic systems[END_REF] is satisfied. We then infer from [2, Theorem 6.4] that there is a unique weak solution U = (U 1 , U 2 ) to (2.3) defined on a maximal time interval [0, T m ), T m ∈ (0, ∞], which satisfies

U 1 ∈ C [0, T m ); W 1 3 (Ω; R 2 ) ∩ C 1 [0, T m ); W 1 3/2 (Ω; R 2 ) ′ , U 2 ∈ C 1 [0, T m ); W 1 3 (Ω)
. Setting (u, w, v) := U, we have thus constructed a unique weak solution (u, v, w) to (1.4)-(1.5) endowed with the properties listed in Proposition 2.1, except for the non-negativity of (u, v, w) and (2.1). To prove the former, we first notice that (1.4a)-(1.4d) and the comparison principle entail that u ≥ 0 in [0, T m ) × Ω. Owing to (1.4b), this in turn implies that v ≥ 0 in [0, T m ) × Ω. This last property, along with (1.4c)-(1.4d) and a further use of the comparison principle, ensures that w ≥ 0 in [0, T m ) × Ω. We finally integrate (1.4a) over Ω and deduce (2.1) from the no-flux boundary conditions (1.4d) and the non-negativity of u, thereby completing the proof.

We next derive additional regularity properties of u and w with the help of parabolic regularity results.

Corollary 2.2. Consider non-negative initial conditions (u in , v in , w in ) in W 1 3 (Ω; R 3 ) and denote the corresponding weak solution in W 1 3 to (1.4)-(1.5) by (u, v, w), defined up to its maximal time

T m ∈ (0, T m ], see Proposition 2.1. Assume further that u in ∈ W 2θ 3,B (Ω) and w in ∈ W 2 3,B (Ω) (2.4)
for some θ ∈ (5/6, 1). Then

u ∈ C([0, T m ); W 2θ 3,B (Ω)) ∩ C 1 ([0, T m ); W 2θ-2 3,B (Ω)) , (2.5
)

u ∈ C((0, T m ); W 2 3,B (Ω)) ∩ C 1 ((0, T m ); L 3 (Ω)) , (2.6
)

w ∈ C([0, T m ); W 2 3,B (Ω)) ∩ C 1 ([0, T m ); L 3 (Ω)) .
(2.7)

Proof. Let β ∈ (0, 1). Since v ∈ C 1 ([0, T m ); W 1 3 (Ω)
) and -D∆ + δ id generates an analytic semigroup in L 3 (Ω), we infer from (1.4c), (2.4), and [3, Theorem 10.1] (with ρ = β, E 0 = L 3 (Ω), and

E 1 = W 2 3,B (Ω)) that w ∈ C([0, T m ); W 2 3,B (Ω)) ∩ C 1 ([0, T m ); L 3 (Ω)) , (2.8)
w ∈ C β ((0, T m ); W 2 3,B (Ω)) ∩ C 1+β ((0, T m ); L 3 (Ω)) . (2.9) Next, since u ∈ C([0, T m ); W 1 3,B (Ω)) ∩ C 1 ([0, T m ); W -1 3,B (
Ω)) by (1.12) and Proposition 2.1 and

W -1 3,B (Ω), W 1 3,B (Ω) θ • = W 2θ-1 3,B (Ω) (up to equivalent norms) by [3, Theorem 7.2], an interpolation argument guarantees that u ∈ C 1-θ ([0, T m ); W 2θ-1 3,B (Ω)) .
Owing to (2.8), a similar argument implies that w

∈ C 1-θ ([0, T m ); W 2θ 3,B (Ω)). Consequently, recalling that W 2θ-1 3,B (Ω) is an algebra as 2θ -1 > 2/3, we conclude that u∇w ∈ C 1-θ ([0, T m ); W 2θ-1 3,B (Ω; R 2 )). Thus, div(u∇w) ∈ C 1-θ ([0, T m ); W 2θ-2 3,B (Ω)) . ( 2
(with ρ = 1 -θ, E 0 = W 2θ-2
3,B (Ω), and

E 1 = W 2θ 3,B (Ω)) that u ∈ C([0, T m ); W 2θ 3,B (Ω)) ∩ C 1 ([0, T m ); W 2θ-2 3,B (Ω)) . (2.11)
Consider next α ∈ (5/3 -θ, 1). Using once more [3, Theorem 7.2] which guarantees that

W 2θ-2 3,B (Ω), W 2θ 3,B (Ω) α • = W 2(α+θ)-2 3,B (Ω)
(up to equivalent norms) and an interpolation argument, we deduce from (2.11) that

u ∈ C 1-α ([0, T m ); W 2(α+θ)-2 3,B (Ω)) ; Hence, since 2(α + θ) -2 > 4/3 > 1, u ∈ C 1-α ([0, T m ); W 1 3,B (Ω)) . (2.12)
Combining (2.9) (with β = 1 -α) and (2.12) with the fact that W 1 3,B (Ω) is an algebra entail that u∇w belongs to C 1-α ((0, T m ); W 1 3,B (Ω; R 2 )) and thus div(u∇w) ∈ C 1-α ((0, T m ); L 3 (Ω)) .

(2.13)

Owing to (1.4a) and (2.13), we are again in a position to apply [3, Theorem 10.1] (with

ρ = 1 -α, E 0 = L 3 (Ω), E 1 = W 2 3,B (Ω)) on any subinterval [t 0 , T m ) of (0, T m ) to conclude that u ∈ C 1-α ((t 0 , T m ); W 2 3,B (Ω)) ∩ C 2-α ((t 0 , T m ); L 3 (Ω)) . Since t 0 ∈ (0, T m ) is arbitrary, the proof of Corollary 2.2 is complete. 2.2. Estimates in W 1 3 (Ω). Let θ ∈ (5/6, 1
) and consider initial conditions (u in , v in , w in ) ∈ I M,θ . We denote the corresponding weak solution in W 1,3 to (1.4)-(1.5) by (u, v, w), defined up to its maximal time

T m ∈ (0, ∞], see Proposition 2.1.
The aim of this section is to show that (2.2) holds true for any T > 0. We first recall that, according to (2.1),

u(t) 1 = M = u in 1 , t ∈ [0, T m ) . (2.14)
Throughout this section, C and (C i ) i≥1 are positive constants depending only on Ω, ν, ε, D, δ, θ, u in , v in , and w in . Dependence upon additional parameters will be indicated explicitly.

Lemma 2.3. Let T > 0. There is C 1 (T ) > 0 such that L(u(t)) 1 + v(t) 2 + w(t) W 1 2 + ∂ t w(t) 2 ≤ C 1 (T ) , t ∈ [0, T ] ∩ [0, T m ) , t 0 ∇ √ u(s) 2 2 + ∇∂ t w(s) 2 2 ds ≤ C 1 (T ) , t ∈ [0, T ] ∩ [0, T m ) ,
the function L being defined in (1.6).

Proof. By (1.4a) and (1.4d),

d dt L(u) 1 = - Ω |∇u| 2 u dx + Ω ∇u • ∇w dx = -4 ∇ √ u 2 2 - Ω u∆w dx . (2.15) Next, by (1.4b), ν Ω u∂ t w dx = ν 2 ε Ω ∂ t v∂ t w dx + ν Ω v∂ t w dx .
On the one hand, we infer from (1.4c) and (1.4d) that

Ω v∂ t w dx = ν ∂ t w 2 2 + 1 2 d dt D ∇w 2 2 + δ w 2 2 .
On the other hand, differentiating (1.4c) with respect to time gives

Ω ∂ t v∂ t w dx = D ∇∂ t w 2 2 + δ ∂ t w 2 2 + ν 2 d dt ∂ t w 2 2 .
Gathering the previous three identities leads us to

ν Ω u∂ t w dx = ν 2 d dt D ∇w 2 2 + δ w 2 2 + ν 2 ε ∂ t w 2 2 + ν 2 (1 + δε) ∂ t w 2 2 + ν 2 εD ∇∂ t w 2 2 . ( 2
(u) 1 + ν 2 D ∇w 2 2 + δ w 2 2 + ν 2 ε ∂ t w 2 2 + 4D ∇ √ u 2 2 + ν 2 (1 + δε) ∂ t w 2 2 + ν 2 εD ∇∂ t w 2 2 = Ω u (ν∂ t w -D∆w) dx = Ω u(v -δw) dx .
(2.17)

Finally, we deduce from (1.4b) that νε 2

d dt v 2 2 + v 2 2 = Ω uv dx . (2.18)
We next recall the Gagliardo-Nirenberg inequality

z 4 ≤ c 0 ∇z 1/2 2 z 1/2 2 + z 2 , z ∈ W 1 2 (Ω) , (2.19)
where c 0 is a positive constant depending only on Ω. Combining (2.14) and (2.19) gives

u 2 = √ u 2 4 ≤ 2c 2 0 ∇ √ u 2 √ u 2 + √ u 2 2 ≤ 2c 2 0 √ M ∇ √ u 2 + M .
Consequently, by Hölder's and Young's inequalities,

Ω uv dx ≤ u 2 v 2 ≤ 2c 2 0 √ M v 2 ∇ √ u 2 + 2c 2 0 M v 2 ≤ 2D ∇ √ u 2 2 + C(1 + v 2 2
) . We then infer from (2.17), (2.18), the non-negativity of u and w, and the previous inequality that

d dt D L(u) 1 + ν 2 ε v 2 2 + D ∇w 2 2 + δ w 2 2 + ν 2 ε ∂ t w 2 2 + 2D ∇ √ u 2 2 + v 2 2 + ν 2 (1 + δε) ∂ t w 2 2 + ν 2 εD ∇∂ t w 2 2 ≤ C(1 + v 2 2
) . We finally apply Gronwall's lemma and use the positivity of the parameters to complete the proof.

An immediate consequence of Lemma 2.3 and (1.4c) is the following improved estimate on w.

Corollary 2.4. Let T > 0. There is C 2 (T ) > 0 such that

w(t) W 2 2 ≤ C 2 (T ) , t ∈ [0, T ] ∩ [0, T m ) . Proof. By (1.4c), for t ∈ [0, T ] ∩ [0, T m ), D ∆w(t) 2 ≤ ν ∂ t w(t) 2 + δ w(t) 2 + v(t) 2 ≤ (ν + δ + 1)C 1 (T ) ,
the last estimate being a direct consequence of Lemma 2.3.

Thanks to Corollary 2.4, we are in a position to obtain additional estimates on u.

Lemma 2.5. Let T > 0 and r > 0. There are C 3 (T, r) > 0 such that

u(t) r+1 ≤ C 3 (T, r) , t ∈ [0, T ] ∩ [0, T m ) .
Proof. Let T > 0, r > 0, and t ∈ [0, T ] ∩ [0, T m ). By (1.4a) and (1.4d),

1 r + 1 d dt u r+1 r+1 = - Ω u r-1 |∇u| 2 dx + r r + 1 Ω ∇ u r+1 • ∇w dx = - 4r (r + 1) 2 ∇ u (r+1)/2 2 2 - r r + 1 Ω u r+1 ∆w dx .
Owing to the Gagliardo-Nirenberg inequality (2.19), Hölder's inequality, and Corollary 2.4, we further obtain

d dt u r+1 r+1 + 4r r + 1 ∇ u (r+1)/2 2 2 ≤ r ∆w 2 u (r+1)/2 2 4 ≤ 2rc 2 0 C 2 (T ) ∇ u (r+1)/2 2 u (r+1)/2 2 + u (r+1)/2 2 2 ≤ rC(T ) ∇ u (r+1)/2 2 u (r+1)/2 r+1 + u r+1 r+1 .
We now use Young's inequality to obtain

d dt u r+1 r+1 + 2r r + 1 ∇ u (r+1)/2 2 2 ≤ r(1 + r)C(T ) u r+1 r+1 ,
from which Lemma 2.5 follows after integration with respect to time.

We are now in a position to apply [22, Lemma A.1] and derive an L ∞ -estimate on u.

Lemma 2.6. Let T > 0. There is C 4 (T ) > 0 such that

u(t) ∞ + v(t) ∞ + ∂ t v(t) ∞ ≤ C 4 (T ) , t ∈ [0, T ] ∩ [0, T m ) .
Proof. Let T > 0. Owing to Corollary 2.4 and Lemma 2.5, we infer from [22, Lemma A.1] that

u(t) ∞ ≤ C(T ) , t ∈ [0, T ] ∩ [0, T m ) .
Next, (1.4b) and the non-negativity of u and v readily imply that, for t ∈ [0, T ] ∩ [0, T m ) and x ∈ Ω,

0 ≤ v(t, x) ≤ v in ∞ e -t/νε + 1 νε t 0 u(s) ∞ e -(t-s)/νε ds ≤ v in ∞ + sup s∈[0,t] { u(s) ∞ } .
We finally use (1.4b) to complete the proof.

We now exploit the properties of the heat semigroup and (1.4c) to derive additional estimates on w.

Corollary 2.7. Let T > 0. There is C 5 (T ) > 0 such that

w(t) W 1 ∞ ≤ C 5 (T ) , t ∈ [0, T ] ∩ [0, T m ) .
Proof. Let T > 0 and t ∈ [0, T ] ∩ [0, T m ). Since W

+ C t 0 e -δ(t-s)/(2ν) (t -s) -11/12 v(s) 3 ds ≤ C w in W 2 3 + CC 4 (T ) t 0 (t -s) -11/12 ds ≤ C(T ) .
We complete the proof by observing that w(t) ∞ ≤ C(T ) due to Corollary 2.4 and the continuous embedding of

W 2 2 (Ω) in L ∞ (Ω).
Thanks to the just established W 1 ∞ -estimate on w, we may derive a W 1 2 -estimate on u.

Lemma 2.8. Let T > 0. There is C 6 (T ) > 0 such that

∇u(t) 2 + t 0 ∂ t u(s) 2 2 ds ≤ C 6 (T ) , t ∈ [0, T ] ∩ [0, T m ) .
Proof. Let T > 0 and t ∈ [0, T ] ∩ [0, T m ). It follows from (1.4a) and Hölder's and Young's inequalities that

∂ t u 2 2 + 1 2 d dt ∇u 2 2 = - Ω ∂ t u div(u∇w) dx ≤ ∂ t u 2 u ∞ ∆w 2 + ∂ t u 2 ∇u 2 ∇w ∞ ≤ 1 2 ∂ t u 2 2 + u 2 ∞ ∆w 2 2 + ∇u 2 2 ∇w 2 ∞ .
Using Corollary 2.4, Lemma 2.6, and Corollary 2.7, we further obtain

∂ t u 2 2 + d dt ∇u 2 2 ≤ 2C 4 (T ) 2 C 2 (T ) 2 + 2C 5 (T ) 2 ∇u 2 2 ,
from which Lemma 2.8 readily follows.

As for w in Corollary 2.7, the properties of the heat semigroup and (1.4a) provide additional estimates on u. Corollary 2.9. Let T > 0. There is C 7 (T ) > 0 such that

u(t) W 1 3 + v(t) W 1 3 ≤ C 7 (T ) , t ∈ [0, T ] ∩ [0, T m ) .
Proof. Let T > 0 and t ∈ [0, T ] ∩ [0, T m ). We infer from (1.4a), Duhamel's formula, properties of the heat semigroup, see [1, Proposition 12.5] and [4, Theorem V.2.1.3], and Hölder's inequality that

∇u(t) 3 ≤ C ∇u in 3 + C t 0 (t -s) -2/3 div(u∇w)(s) 2 ds ≤ C + C t 0 (t -s) -2/3 ∇w(s) ∞ ∇u(s) 2 ds + C t 0 (t -s) -2/3 u(s) ∞ ∆w(s) 2 ds .
Owing to Corollary 2.4, Lemma 2.6, Corollary 2.7, and Lemma 2.8, we further obtain

∇u(t) 3 ≤ C + CC 5 (T )C 6 (T ) t 0 (t -s) -2/3 ds + CC 2 (T )C 4 (T ) t 0 (t -s) -2/3 ds ≤ C(T ) . (2.20)
As Summarizing the outcome of the above analysis, we have shown that, for any T > 0, there is

∇v(t, x) = ∇v in (x)e -t/
C 8 (T ) > 0 such that u(t) W 1 3 + v(t) W 1 3 + w(t) W 1 3 ≤ C 8 (T ) , t ∈ [0, T ] ∩ [0, T m ) , (2.21)
see Corollary 2.7 and Corollary 2.9. According to (2.2), excluding finite time blowup requires uniformly continuous estimates with respect to time which we derive now. In fact, we shall establish Hölder estimates with respect to time.

Lemma 2.10. Let T > 0. There is C 9 (T ) > 0 such that, for

t 1 ∈ [0, T ] ∩ [0, T m ) and t 2 ∈ [0, T ] ∩ [0, T m ), u(t 2 ) -u(t 1 ) W 1 3 + v(t 2 ) -v(t 1 ) W 1 3 + w(t 2 ) -w(t 1 ) W 1 3 ≤ C 9 (T )|t 2 -t 1 | α , where α := (3θ -2)/6θ. Proof. Let T > 0, t 1 ∈ [0, T ] ∩ [0, T m ), and t 2 ∈ [0, T ] ∩ [0, T m ), t 2 > t 1 . It first follows from (1.4b) and (2.21) that ∂ t v(t) W 1 3 ≤ 1 νε u(t) W 1 3 + v(t) W 1 3 ≤ C(T ) , t ∈ [0, T ] ∩ [0, T m ) .
Consequently,

v(t 2 ) -v(t 1 ) W 1 3 ≤ t 2 t 1 ∂ t v(s) W 1 3 ds ≤ C(T )(t 2 -t 1 ) . (2.22)
Next, by Hölder's inequality,

w(t 2 ) -w(t 1 ) W 1 3 ≤ C w(t 2 ) -w(t 1 ) 1/3 W 1 ∞ w(t 2 ) -w(t 1 ) 2/3 W 1 2 ≤ C w(t 2 ) W 1 ∞ + w(t 1 ) W 1 ∞ 1/3 t 2 t 1 ∂ t w(s) W 1 2 ds 2/3
.

We then deduce from Lemma 2.3, Corollary 2.7, and Hölder's inequality that

w(t 2 ) -w(t 1 ) W 1 3 ≤ C(2C 5 (T )) 1/3 (t 2 -t 1 ) 1/3 t 2 t 1 ∂ t w(s) 2 W 1 2 ds 1/3 ≤ C(T ) C 1 (T ) 2 (t 2 -t 1 ) + C 1 (T ) 1/3 (t 2 -t 1 ) 1/3 ≤ C(T )(t 2 -t 1 ) 1/3 . (2.23)
Finally, as in the proof of Corollary 2.9, we infer from (1.4a), Duhamel's formula, properties of the heat semigroup [4, Theorem V.2.1.3], and Hölder's inequality that, for t ∈ [0, T ] ∩ [0, T m ),

u(t) W 2θ 2 ≤ C u in W 2θ 2 + C t 0 (t -s) -θ div(u∇w)(s) 2 ds ≤ C + C t 0 (t -s) -θ ∇u(s) 2 ∇w(s) ∞ ds + C t 0 (t -s) -θ u(s) ∞ ∆w(s) 2 ds .
Recalling that θ ∈ (5/6, 1), it follows from Corollary 2.4, Lemma 2.6, Corollary 2.7, and Lemma 2.8 that

u(t) W 2θ 2 ≤ C + CC 6 (T )C 5 (T ) t 0 (t -s) -θ ds + CC 4 (T )C 2 (T ) t 0 (t -s) -θ ds ≤ C(T ) . ( 2
u(t 2 ) -u(t 1 ) W 1 3 ≤ C u(t 2 ) -u(t 1 ) W 4/3 2 ≤ C u(t 2 ) -u(t 1 ) 2/3θ W 2θ 2 u(t 2 ) -u(t 1 ) (3θ-2)/3θ 2 ≤ u(t 2 ) W 2θ 2 + u(t 1 ) W 2θ 2 2/3θ t 2 t 1 ∂ t u(s) 2 ds (3θ-2)/3θ ≤ C(T )(t 2 -t 1 ) α t 2 t 1 ∂ t u(s) 2 2 ds α .
Combining the previous inequality with Lemma 2.8 and recalling (2.22) and (2.23) complete the proof, after noticing that 0 < α < 1/3 < 1.

Proof of Theorem 1.1. The well-posedness in W 1 3 on some maximal time interval [0, T m ) is provided by Proposition 2.1. According to Lemma 2.10, the condition (2.2) of Proposition 2.1 is satisfied for any T > 0, and we thus conclude that T m = ∞.

Bounded solutions

Let θ ∈ (5/6, 1). Consider initial conditions (u in , v in , w in ) ∈ I M,θ and denote the corresponding solution to (1.4)- (1.5) given in Theorem 1.1 by (u, v, w). We begin with the evolution of the L 1 -norms of u, v, and w. Lemma 3.1. For t ≥ 0,

u(t) 1 = M = u in 1 , (3.1) v(t) 1 ≤ v in 1 + M , (3.2)
w(t) 1 ≤ w in 1 + M δ + b 0 ( v in 1 + M) , (3.3)
where b 0 := ε/|δε -1| if δε = 1 and b 0 := 1/δe when δε = 1.

Proof. The identity (3.1) is nothing but (2.1), while it readily follows from (1.4b) and (3.1) that

v(t) 1 = v in 1 e -t/νε + M 1 -e -t/νε , t ≥ 0 . (3.4)
The upper bound (3.2) is then an immediate consequence of (3.4). Finally, by (1.4c) and (1.4d),

ν d dt w 1 + δ w 1 = v 1 , t ≥ 0 ,
which allows us to compute explicitly the time evolution of the L 1 -norm of w and find, for t ≥ 0,

w(t) 1 = M δ + w in 1 - M δ e -δt/ν + ε δε -1 v in 1 -M e -t/νε -e -δt/ν (3.5a)
when δε = 1 and

w(t) 1 = M δ + w in 1 - M δ e -δt/ν + v in 1 -M ν te -δt/ν (3.5b)
when δε = 1. Since ze -z ≤ 1/e for z ≥ 0, the upper bound (3.3) readily follows from (3.5).

3.1.

A Liapunov functional. We next turn to the availability of a Liapunov function and recall that E 0 and E are defined by

E 0 (u, w) := Ω [L(u) -uw] dx + D 2 ∇w 2 2 + δ 2 w 2 2 , E(u, v, w) := E 0 (u, w) + ε 2 -D∆w + δw -v 2 2 ,
see (1.7), the function L being given by (1.6).

Lemma 3.2. For t ≥ 0, there holds

d dt E(u(t), v(t), w(t)) + D(u(t), v(t), w(t)) = 0 , t ≥ 0 ,
where

D(u, v, w) := Ω u|∇(ln u -w)| 2 dx + 1 + δε ν -D∆w + δw -v 2 2 + εD ν ∇(-D∆w + δw -v) 2 2 . (3.6) Proof. By (1.4), d dt E 0 (u, w) = Ω (ln u -w)∂ t u dx - Ω u∂ t w dx + Ω (-D∆w + δw) ∂ t w dx = - Ω u|∇(ln u -w)| 2 dx - Ω (εν∂ t v + v)∂ t w dx + Ω (v -ν∂ t w) ∂ t w dx = - Ω u|∇(ln u -w)| 2 dx -ν ∂ t w 2 2 -εν Ω ∂ t v∂ t w dx .
Owing to (1.4c),

Ω ∂ t v∂ t w dx = Ω ν∂ 2 t w -D∆∂ t w + δ∂ t w ∂ t w dx = ν 2 d dt ∂ t w 2 2 + D ∇∂ t w 2 2 + δ ∂ t w 2 2 .
Combining the previous two identities and using (1.4c) to replace ∂ t w complete the proof.

Throughout the remainder of Section 3, b and (b i ) i≥1 are positive constants depending only on Ω, ν, ε, D, δ, θ, u in , v in , and w in . Dependence upon additional parameters will be indicated explicitly.

Time-independent estimates.

As in [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles[END_REF][START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF][START_REF] Nagai | Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[END_REF], we exploit the structure of the Liapunov functional E and first show that it is bounded from below as soon as M is suitably small. To this end, we first recall the following consequence of the Moser-Trudinger inequality [8, Proposition 2.3], see [START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF]Corollary 2.6]. Proposition 3.3. There is K 0 > 0 depending only on Ω such that, for z ∈ W 1 2 (Ω),

Ω e |z| dx ≤ K 0 exp ∇z 2 2 8π + z 1 |Ω| .
We now derive upper and lower bounds on E(u, v, w).

Lemma 3.4.

There is b 1 > 0 such that, for t ≥ 0,

E(u(t), v(t), w(t)) ≤ E(u in , v in , w in ) , (3.7)
E(u(t), v(t), w(t)) ≥ 4πD -M 8π ∇w(t) 2 2 + δ 2 w(t) 2 2 + ε 2 -D∆w(t) + δw(t) -v(t) 2 2 -b 1 . (3.8)
Proof. We first argue as in the proof of [9, Lemma 4.5], see also [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles[END_REF]Theorem 2 (iv)] and [START_REF] Nagai | Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[END_REF]Lemma 3.4], to obtain that

E 0 (u, w) ≥ D 2 ∇w 2 2 + δ 2 w 2 2 -M ln( e w 1 ) + M ln M -M + |Ω| . (3.9)
It then follows from Proposition 3.3 that

E 0 (u, w) ≥ 4πD -M 8π ∇w 2 2 + δ 2 w 2 2 -M ln K 0 - M |Ω| w 1 -1 .
Combining the previous inequality with (3.3) gives

E 0 (u, w) ≥ 4πD -M 8π ∇w 2 2 + δ 2 w 2 2 -b 1 . Consequently, E(u, v, w) ≥ 4πD -M 8π ∇w 2 2 + δ 2 w 2 2 + ε 2 -D∆w + δw -v 2 2 -b 1
, and we have shown the expected lower bound. The upper bound is a straightforward consequence of Lemma 3.2 and the non-negativity of D.

We are now in a position to deduce a first set of time-independent estimates, provided M ∈ (0, 4πD). Lemma 3.5. Assume that M ∈ (0, 4πD). There is b 2 > 0 such that, for t ≥ 0,

u(t) ln u(t) 1 + w(t) W 1 2 + ∂ t w(t) 2 + ∞ 0 ∂ t w(s) 2 W 1 2 ds ≤ b 2 .
Proof. Let t ≥ 0. It readily follows from (1.4c), (3.7), and (3.8) that

min 4πD -M 8π , δ 2 , ν 2 ε 2 w(t) 2 W 1 2 + ∂ t w(t) 2 2 ≤ b 1 + E(u in , v in , w in ) .
Hence, since M ∈ (0, 4πD),

w(t) W 1 2 + ∂ t w(t) 2 ≤ b , t ≥ 0 . (3.10)
Next, thanks to the non-negativity and convexity of the function L defined in (1.6), it follows from (1.7), (3.7), and Young's inequality that

L(u(t)) 1 = Ω L(u(t, x)) dx ≤ E 0 (u(t), w(t)) + Ω u(t, x)w(t, x) dx ≤ E(u(t), v(t), w(t)) + 1 2 Ω [L(u(t, x)) + L * (2w(t, x))] dx ≤ E(u in , v in , w in ) + 1 2 L(u(t)) 1 + 1 2 Ω L * (2w(t, x)) dx , where L * is the convex conjugate of L; that is, L * (z) := e z -1, z ∈ R. Since Ω L * (2w(t, x)) dx ≤ Ω e 2w(t,x) dx ≤ K 0 exp ∇w(t) 2 2 2π + 2 |Ω| w(t) 1 ≤ b by (3.
3), (3.10), and Proposition 3.3, we end up with

L(u(t)) 1 ≤ 2E(u in , v in , w in ) + b , t ≥ 0 . (3.11)
An immediate consequence of (3.11) and the elementary inequality z| ln z| ≤ L(z) + z + 1 for z ≥ 0 is an L ln L-estimate for u:

u(t) ln u(t) 1 ≤ b , t ≥ 0 . (3.12)
We finally infer from (1.4c), (3.7), (3.8), and Lemma 3.2 that

ν min {(1 + δε), εD} t 0 ∂ t w(s) 2 W 1 2 ds ≤ t 0 ν(1 + δε) ∂ t w(s) 2 2 + νεD ∇∂ t w(s) 2 2 ds ≤ t 0 D(u(s), v(s), w(s)) ds ≤ E(u in , v in , w in ) -E(u(t), v(t), w(t)) ≤ E(u in , v in , w in ) + b 1 ,
which, together with (3.10) and (3.12), complete the proof of Lemma 3.5.

Introducing Y := u 2 2 + νε v 3 3 , we sum up (3.14) and (3.15) and use (3.1), (3.13), and Lemma 3.5 to obtain, for η > 0,

dY dt + 3 2 ∇u 2 2 + 2 3 v 3 3 ≤ 1 + 2 3D 3/2 u 3 3 + b ∂ t w 2 u 2 2 ≤ 1 + 2 3D 3/2 ηb 2 u 2 W 1 2 + κ η M + b ∂ t w 2 u 2 2 ≤ b η ∇u 2 2 + η u 2 2 + ∂ t w 2 u 2 2 . By Poincaré's inequality, there is K 1 > 0 depending only on Ω such that z - 1 |Ω| Ω z(x) dx 2 ≤ K 1 ∇z 2 , z ∈ W 1 2 (Ω) ,
which implies that

z 2 2 ≤ 2K 2 1 ∇z 2 2 + 2 |Ω| z 2 1 , z ∈ W 1 2 (Ω) . (3.16)
Consequently, it follows from (3.1) and (3.16) that

dY dt + 3 2 ∇u 2 2 + 2 3 v 3 3 ≤ b η ∇u 2 2 + 2ηK 2 1 ∇u 2 2 + 2η M 2 |Ω| + ∂ t w 2 u 2 2 . Choosing η = 1/[2b(1 + 2K 2 1 )], we end up with dY dt + ∇u 2 2 + 2 3 v 3 3 ≤ b 1 + ∂ t w 2 u 2 2 .
Using (3.1) and (3.16) again, we deduce that

dY dt + 1 2K 2 1 u 2 2 - 2M 2 |Ω| + 2 3 v 3 3 ≤ b 1 + ∂ t w 2 u 2 2 .
Therefore, there are b 4 > 0 and b

5 > 0 such that dY dt + b 4 Y ≤ b 5 1 + ∂ t w 2 u 2 2 ≤ b 5 (1 + ∂ t w 2 Y ) . Equivalently, d dt Y (t) exp b 4 t -b 5 t 0 ∂ t w(s) 2 ds ≤ b 5 exp b 4 t -b 5 t 0 ∂ t w(s) 2 ds ,
and, after integration with respect to time, Combining (3.17) and (3.18) gives

Y (t) ≤ Y (0) exp b 5 t 0 ∂ t w(s)
Y (t) ≤ be -b 4 t/2 + b t 0 e -b 4 (t-s)/2 ds ≤ b ,
and completes the proof.

Owing to Lemma 3.6, we may argue as in the proof of Corollary 2.7, using in addition the exponential decay due to the positivity of δ, to obtain a Lipschitz estimate on w.

Corollary 3.7. Assume that M ∈ (0, 4πD). There is b 6 > 0 such that, for t ≥ 0,

w(t) W 1
∞ ≤ b 6 . Proof of Theorem 1.2 (a). From Lemma 3.6 and Corollary 3.7, we deduce that

u∇w ∈ L ∞ ((0, ∞); L 2 (Ω; R 2 )) .
Therefore, since u in ∈ L ∞ (Ω), we are in a position to apply [START_REF] Tao | Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity[END_REF]Lemma A.1] to conclude that

u(t) ∞ ≤ b , t ≥ 0 ,
see also [START_REF] Nagai | Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis[END_REF]Section 4]. Combining this last estimate with (1.4b) entails that v ∈ L ∞ ((0, ∞) × Ω) and completes the proof.

Proof of Theorem 1.2 (b). Since w is radially symmetric, it satisfies an improved version of Proposition 3.3: for all η > 0, there is K(η) > 0 depending only on η such that

Ω e w dx ≤ K(η) exp η + 1 16π ∇w 2 2 + 2 |Ω| w 1 , see [17, Theorem 2.1]. Consequently, -M ln e w 1 ≥ -M ln K(η) -M η + 1 16π ∇w 2 2 - 2M |Ω| w 1 ,
and it follows from (3.3) and (3.9) that

E 0 (u, w) ≥ 8πD -(1 + 16πη)M 16π ∇w 2 2 + δ 2 w 2 2 -b -M ln K(η) . Choosing η = (8πD -M)/(32πM) > 0 in the previous inequality gives E 0 (u, w) ≥ 8πD -M 32π ∇w 2 2 + δ 2 w 2 2 -b .
We then proceed as in Section 3.2 to complete the proof.

Stabilization.

Another useful consequence of the availability of a Liapunov functional is the stabilization of bounded solutions; that is, any cluster point as t → ∞ of a bounded solution to (1.4)-(1.5) is a stationary solution to (1.4). Not only does this information provide better insight into the dynamics of solutions to (1.4)-(1.5) when u in 1 < 4πD, since solutions are known to be bounded in that case by Theorem 1.2, but it is also an important step in the construction of unbounded solutions to be performed in Section 4. This fact has already been observed in [START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF] and used for similar purposes in [START_REF] Horstmann | The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results[END_REF][START_REF]On the existence of radially symmetric blow-up solutions for the Keller-Segel model[END_REF][START_REF] Horstmann | Blow-up in a chemotaxis model without symmetry assumptions[END_REF].

Proposition 3.8. Let θ ∈ (5/6, 1) and consider initial conditions (u in , v in , w in ) ∈ I M,θ . We denote the corresponding solution to (1.4)- (1.5) given by Theorem 1.1 by (u, v, w) and assume further that there is Λ > 0 such that u(t) ∞ ≤ Λ , t ≥ 0 .

(3.19) There are a sequence (t k ) k≥1 , t k → ∞, of positive real numbers and non-negative functions

(u * , v * , w * ) ∈ L ∞ (Ω; R 2 ) × W 2 2,B (Ω) such that lim k→∞ u(t k ) -u * 2 + v(t k ) -v * 2 + w(t k ) -w * W 1 2 = 0 , (3.20)
lim k→∞ E 0 (u(t k ), w(t k )) = E 0 (u * , w * ) , (3.21)
E(u * , v * , w * ) ≤ lim inf k→∞ E(u(t k ), v(t k ), w(t k )) , (3.22)
where

u * = v * = M e w * e w * 1 , (3.23
(t) ∞ + ∂ t v(t) ∞ ≤ 1 + 1 νε v in ∞ + Λ , t ≥ 0 . ( 3

Unbounded solutions

This section is devoted to the proof of Theorem 1.3 which is somewhat a direct consequence of the analysis performed in [START_REF] Horstmann | The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results[END_REF][START_REF]On the existence of radially symmetric blow-up solutions for the Keller-Segel model[END_REF][START_REF] Horstmann | Blow-up in a chemotaxis model without symmetry assumptions[END_REF]. The main step is actually to recast the problem in a way suitable to apply results from [START_REF] Horstmann | Blow-up in a chemotaxis model without symmetry assumptions[END_REF].

Given M > 0, we define the set S M as follows: (a) If M ∈ (4πD, ∞) \ 4πDN, then

(u, v, w) ∈ S M if (u, v, w) ∈ W 2 2,B (Ω; R 3 ) , (4.1a)
u = v = M e w e w 1 ≥ 0 in Ω , w ≥ 0 in Ω , (4.1b)
µ M := inf (u,v,w)∈S M E(u, v, w) > -∞ . (b) If Ω = B r (0) for some r > 0 and M ∈ (8πD, ∞), then µ M := inf (u,v,w)∈S M,rad E(u, v, w) > -∞ ,
where S M,rad := {(u, v, w) ∈ S M : u, v, w are radially symmetric}.

Proof. -Case (a). Consider (u, v, w) ∈ S M . Introducing

E(w) := D 2 ∇w 2 2 + δ 2 w 2 2 -M ln e w 1 + M ln M -M + |Ω| , it follows from (3.9) that E(u, v, w) ≥ E 0 (u, w) ≥ E(w) , (4.2)
while the fact that (u, v, w) ∈ S M implies that w ≥ 0 in Ω and w

1 = M δ . (4.3)
Setting

W := w - M δ|Ω| = w - 1 |Ω| Ω w(x) dx ,
an alternative formula for E(w) in terms of W reads

E(w) = D 2 ∇W 2 2 + δ 2 W 2 2 -M ln e W 1 - M 2 2δ|Ω| + M ln M -M + |Ω| = M |Ω| F (W ) - M 2 2δ|Ω| + M ln M -M + |Ω| -M ln |Ω| , (4.4)
with

F (W ) := |Ω| 2M D ∇W 2 2 + δ W 2 2 -|Ω| ln e W 1 |Ω| . (4.5)
Furthermore, since w solves (4.1c), W is a solution to

-D∆W + δW = M |Ω| |Ω|e W e W 1 -1 in Ω , ∇W • n = 0 on ∂Ω .
Since M ∈ (4πD, ∞) \ 4πDN, we infer from [13, Lemma 3.5] that there is µ ≥ 0 which does not depend on W such that F (W ) ≥ -µ . -Case (b). The proof is the same as that for case (a), except that we use [13, Corollary 3.7 & Remark 3.8] instead of [START_REF] Horstmann | Blow-up in a chemotaxis model without symmetry assumptions[END_REF]Lemma 3.5] to obtain the lower bound (4.6) for all M > 8πD.

The second step is to show that E is not bounded from below on the set I M,θ of initial data to which Theorem 1.1 applies.

Proposition 4.2. Let θ ∈ (5/6, 1). (a) If M > 4πD, then inf (u,v,w)∈I M,θ E(u, v, w) = -∞ . (b) If Ω = B r (0) for some r > 0 and M > 8πD, then inf (u,v,w)∈I rad M,θ E(u, v, w) = -∞ ,
where I rad M,θ := {(u, v, w) ∈ I M,θ : u, v, w are radially symmetric}.

Proof. -Case (a). Fix x 0 ∈ ∂Ω. For η ∈ (0, ∞) and x ∈ Ω, define

Θ η (x) := 2 ln η (η 2 + π|x -x 0 | 2 ) - 2 |Ω| Ω ln η (η 2 + π|y -x 0 | 2 )
dy .

Then Θ η ∈ W 1 2 (Ω) with Θ η = 0 and satisfies lim

η ∈ (0, ∞). Since W 1 3 (Ω) • = [L 3 (Ω), W 2 3,B (Ω)] 1/2 by [3, Theorem 5.2] and W 1 3 (Ω) is densely embedded in W 1 2 (Ω), there is a sequence (W j,η ) j≥1 in W 2 3,B (Ω) such that W j,η W 1 2 ≤ 2 Θ η W 1 2 , W j,η = 0 , j ≥ 1 and lim j→∞ W j,η -Θ η W 1 2 = 0 . (4.8)
In addition, possibly after extracting a subsequence, we may assume that lim j→∞ W j,η (x) = Θ η (x) for a.e. x ∈ Ω . (4.9)

In addition, it follows from Hölder's inequality and Proposition 3.3 that, for any measurable subset ω of Ω, ω e W j,η dx ≤ |ω| |Ω| , which allows us to deduce from Dunford-Pettis' theorem that e W j,η j≥1 is relatively sequentially weakly compact in L 1 (Ω) . Since e W j,η j≥1 converges to e Θη a.e. in Ω by (4.9), we deduce from these two properties and Vitali's theorem that lim j→∞ e W j,η -e Θη 1 = 0 . We then infer from (4.12), (4.14), (4.15), and Hölder's inequality that

E(u η , v η , w η ) ≤ M |Ω| F (Θ η ) + ηM |Ω| + ε|Ω| 1/3 2 -D∆w η + δw η -v η 2 3 + χ ≤ M |Ω| F (Θ η ) + ηM |Ω| + εη 2 |Ω| 1/3 2 + χ .
Taking the limit η → 0 in the previous inequality and using (4.7) lead us to lim η→0 E(u η , v η , w η ) = -∞ , and completes the proof.

-Case (b). The proof is the same as the previous one except that the starting point is [START_REF]On the existence of radially symmetric blow-up solutions for the Keller-Segel model[END_REF]Lemma 2], the function Θ η being defined as in Case (a) but with x 0 = 0.

Proof of Theorem 1.3. -Case (a). Consider M ∈ (4πD, ∞) \ 4πDN and θ ∈ (5/6, 1). According to Proposition 4.2, there is (u in , v in , w in ) ∈ I M,θ such that

E(u in , v in , w in ) < µ M , (4.16)
the parameter µ M being defined in Proposition 4.1. Assume for contradiction that there is Λ > 0 such that u(t) ∞ ≤ Λ for all t ≥ 0. We are then in a position to apply Proposition 3.

Related models

To finish with, we just point out in an informal way that the related models (1.1) and (1.3) also have a Liapunov functional which is closely related to that of (1.4). Specifically, the functional E 0 defined by (1.7a) is a Liapunov functional for (1.1) and, if (u, v, w) is a solution to (1.1), then d dt E 0 (u, w) = -Ω u|∇(ln u -w)| 2 dx -D ∇∂ t w 2 2 .

In the same vein, E 0 is also a Liapunov functional for (1. It is then likely that the analysis performed for (1.

. 10 )By [ 3 ,

 103 Theorem 8.5], the realization in W 2θ-2 3,B (Ω) of the Laplace operator with homogeneous Neumann boundary conditions generates an analytic semigroup in W 2θ-2 3,B (Ω) (with domain W 2θ 3,B (Ω)) and we infer from (1.4a), (2.4), (2.10), and[START_REF]Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems[END_REF] Theorem 10.1]

-D∆w + δw = M e w e w 1

 1 in Ω , ∇w • n = 0 on ∂Ω . (4.1c) Equivalently, S M is the set of non-negative stationary solutions (u, v, w) to (1.4) which belong to W 2 2,B (Ω; R 3 ) and for which u 1 = M. We begin with a lower bound of the Liapunov functional E (defined in (1.7)) on S M for suitable values of M. Proposition 4.1.

(4. 6 )

 6 Combining (4.2), (4.4), and (4.6) completes the proof.

η→0F

  (Θ η ) = -∞ and lim η→0 ∇Θ η 2 = ∞ (4.7)by [13, Section 3], since M > 4πD and the boundary of Ω is smooth. Let

(4. 10 )F 2 + 2 = 2 - 2 2

 102222 Recalling the definition (4.5) of F , we infer from (4.8) and (4.10) that lim j→∞ (W η,j ) = F (Θ η ) .(4.11)Owing to (4.8) and (4.11), there is j η ≥ 1 such that the function W η := W jη,η satisfiesW η ∈ W 2 3,B (Ω) and W η -Θ η W 1 |F (W η ) -F (Θ η )| ≤ η . (4.12)After this preparation, for η > 0, we setw η := W η + M δ|Ω| , u η := M e wη e wη 1 ,(4.13)and, since -D∆w η + δw η ∈ L 3 (Ω), we fixv η ∈ W 1 3 (Ω) such that v η + D∆w η -δw η 3 ≤ η . (4.14)Now, owing to (1.7a), (4.4), and (4.13),E 0 (u η , w η ) = M ln M -M ln e wη 1 -M + |Ω| + E(W η ) = M |Ω| F (W η ) + χ , with χ := -M 2 2δ|Ω| + M ln M -M + |Ω| -M ln |Ω| , so that E(u η , v η , w η ) = M |Ω| F (W η ) + ε D∆w η + δw η -v η + χ .(4.15)

  3 (a) and deduce that there are a sequence (t k ) k≥1 , t k → ∞, of positive real numbers and (u* , v * , w * ) ∈ S M such that E(u * , v * , w * ) ≤ lim inf k→∞ E(u(t k ), v(t k ), w(t k )) .Since E is a Liapunov function for (1.4) by Lemma 3.4 and (u * , v * , w * ) ∈ S M , we infer from Proposition 4.1 (a) and the previous identity thatµ M ≤ E(u * , v * , w * ) ≤ lim inf k→∞ E(u(t k ), v(t k ), w(t k )) ≤ E(u in , v in , w in ) ,which contradicts (4.16) and completes the proof.-Case (b). The proof is the same as the previous one and relies on Proposition 4.1 (b) and Proposition 4.2 (b).

  3) and, if (u, v, w) is a solution to (1.3), then d dt E 0 (u, w) = -Ω u|∇(ln u -w)| 2 dx -νε D ∇∂ t w 2 2 + δ ∂ t w 2 2 .

4 )

 4 adapts to (1.1) and (1.3), possibly with slightly different statements for (1.1). Indeed, Proposition 4.1 (b) is only valid for M ∈ (8πD, ∞) \ 8πDN in that case, see [11, Lemma 3].

  ). Even though the parabolic equation solved by U involves a nonlocal term, it turns out that such a powerful tool as the comparison principle becomes available, see [23, Lemma 4.2], and the proofs of [23, Theorems 1.2 & 1.3] rely on the construction of suitable supersolutions and subsolutions.

  2 ds -b 4 t ) 2 ds * -b 4 (t -s) ≤

	Owing to Lemma 3.5, we infer from Young's inequality that, for 0 ≤ s ≤ t, b 5 t s ∂ t w(s t s b 4 2 + b 2 5 2b 4 ∂ t w(s * ) 2 2 ds * -b 4 (t -s)
			≤	b 2 b 2 5 2b 4	-	b 4 2	(t -s) .	(3.18)
		t		t
	+ b 5	0	exp b 5	s	∂ t w(s * ) 2 ds * -b 4 (t -s) ds .	(3.17)

*

  The remainder of the proof is then the same as that of[START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF] Theorem 5.2], see also[11, Lemma 1]. The only additional point is to derive(3.22). To this end, we notice that (3.20) and (3.27) entail that (-D∆w(t k ) + δw(t k ) -v(t k )) k≥1 converges weakly towards -D∆w * + δw * -v * in L 2 (Ω), which implies that -D∆w * + δw * -v * 2 ≤ lim inf D∆w(t k ) + δw(t k ) -v(t k ) 2by a weak lower semicontinuity argument. Combining this property with (3.21) gives(3.22) .

	Also, (1.4c), (3.25), and (3.26) entail that
	D ∆w(t) 2 ≤ ν ∂ t w(t) 2 + δ w(t) 2 + v(t) 2 ≤ b(1 + Λ) , from which we deduce that	t ≥ 0 ,
	w(t) W 2 2 ≤ b(1 + Λ) ,	t ≥ 0 .	(3.27)
			.26)

k→∞

-

Acknowledgments

I gratefully thank Christoph Walker for helpful discussions during the preparation of this paper. Part of this work was done while enjoying the hospitality and support of the Institute of Mathematics of the Polish Academy of Sciences in Warsaw.

We next derive L r -estimates for (u, v), the starting point being the just proved L ln L-estimate on u and the following fundamental inequality established in [START_REF] Biler | The Debye system: existence and large time behavior of solutions[END_REF]Equation (22)]: given η > 0, there is a positive constant κ η > 0 depending only on η and Ω such that

Lemma 3.6. Assume that M ∈ (0, 4πD). There is b 3 > 0 such that, for t ≥ 0,

Proof. On the one hand, it follows from (1.4a), (1.4d), Hölder's inequality, and the non-negativity of u and w that 1 2

Using the Gagliardo-Nirenberg inequality (2.19) and Young's inequality, we further obtain

Hence, thanks to Lemma 3.5,

On the other hand, we infer from (1.4b) and Hölder's and Young's inequalities that

Hence, .15)


Anti Rohumaa
email: anti.rohumaa@gmail.com

Joffrey Viguier

Stéphane Girardon

Michael Krebs

Louis Denaud

Lathe check development and properties: effect of log soaking temperature, compression rate, cutting radius and cutting speed during peeling process of European beech (Fagus sylvatica L.) veneer

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Adhesive bond formation and performance in veneer-based products is highly dependent on the veneer quality used, which is formed through the multistage mechanical processing of the wood material. One of these process stages is soaking, where the logs are commonly heated prior to peeling in order to soften the wood [START_REF] Sellers | Plywood and adhesive technology[END_REF][START_REF] Marchal | Cutting forces in wood machiningbasics and applications in industrial processes. A review COST Action E35 2004-2008: wood machining-micromechanics and fracture[END_REF][START_REF] Dupleix | The effect of log heating temperature on the peeling process and veneer quality: beech, birch, douglas-fir and spruce case studies[END_REF]) so as to obtain smoother veneer and minimise the severity of knife checks [START_REF] Meriluoto | The Influence of Raw Material Factors on the Quantity and Quality of Rotary Cut Brick Veneer[END_REF][START_REF] Dupleix | The effect of log heating temperature on the peeling process and veneer quality: beech, birch, douglas-fir and spruce case studies[END_REF]. Generally, at higher temperature during peeling the formation of deep lathe checks will be reduced [START_REF] Meriluoto | The Influence of Raw Material Factors on the Quantity and Quality of Rotary Cut Brick Veneer[END_REF]Dupleix et al.2013) and the integrity of the veneer will be improved (Rohumaa et al. 2016a). The depth of checks and integrity of veneer have been shown to be critical factors affecting the bonding process [START_REF] Rohumaa | The influence of lathe check depth and orientation on the bond quality of phenol-formaldehyde-bonded birch plywood[END_REF][START_REF] Rohumaa | The influence of felling season and log-soaking temperature on the wetting and phenol formaldehyde adhesive bonding characteristics of birch veneer[END_REF][START_REF] Rohumaa | Effect of log soaking and the temperature of peeling on the properties of rotary-cut birch (Betula pendula Roth) veneer bonded with phenol-formaldehyde adhesive[END_REF], but also have effect on veneer based products shape stability and mechanical properties [START_REF] Blomqvist | Influence of veneer orientation on shape stability of plane laminated veneer products[END_REF][START_REF] Pot | Numerical study of the influence of veneer lathe checks on the elastic mechanical properties of laminated veneer lumber (LVL) made of beech[END_REF].

The effect of soaking on veneer quality cannot be observed as a separate process in the veneer production, because the subsequent peeling process forms the veneer surface. In the peeling process, it has been noted that the compression rate affects the depth and frequency of lathe checks [START_REF] Lutz | Techniques for Peeling, Slicing, and Drying Veneer[END_REF]; with low compression the rough veneer and deep lathe checks could be formed and with high pressure the spin outs and the tearing of grain could occur [START_REF] Spelter | Recent developments in veneer peeling confront quality variables[END_REF]. According to the literature [START_REF] Leney | A photographic study of veneer formation[END_REF], the lathe check phenomenon is almost constant when it is well established, and for homogeneous wood species it could be periodical [START_REF] Denaud | Wood peeling process monitoring: A comparison of signal processing methods to estimate veneer average lathe check frequency[END_REF]Palubicki et al. 2010). Moreover, the lathe checks are supposed to not vary too much as a function of peeling radius except for very low radius where the presence of juvenile wood (exhibiting different mechanical properties) could change the behaviour of the veneer [START_REF] Darmawan | Lathe check characteristics of fast growing sengon veneers and their effect on LVL glue-bond and bending strength[END_REF].

Generally, lathe check properties such as depth and frequency correlate with each other; deeper checks tend to be less frequent than shallower checks [START_REF] Mcmillin | The relation of mechanical properties of wood and nosebar pressure in the production of veneer[END_REF][START_REF] Denaud | Online control of wood peeling process: acoustical and vibratory measurements of lathe checks frequency[END_REF][START_REF] Pałubicki | A Method of Lathe Checks Measurement; SMOF device and its software[END_REF]. However, some studies show opposite results [START_REF] Darmawan | Lathe check characteristics of fast growing sengon veneers and their effect on LVL glue-bond and bending strength[END_REF] suggesting that the check frequency will decrease with higher soaking temperature. These contradictions in published results highlight that there are still open questions about the role of processing in lathe check formation. Variation and contradiction in results could be caused by many variables, such as different measuring techniques, processing parameters, wood species, etc.

Lathe check formation and measurement during peeling process have received a lot of attention in the past as well as more recently [START_REF] Denaud | Online control of wood peeling process: acoustical and vibratory measurements of lathe checks frequency[END_REF][START_REF] Denaud | Wood peeling process monitoring: A comparison of signal processing methods to estimate veneer average lathe check frequency[END_REF][START_REF] Tomppo | Ultrasound evaluation of lathe check depth in birch veneer[END_REF][START_REF] Pałubicki | A Method of Lathe Checks Measurement; SMOF device and its software[END_REF][START_REF] Dupleix | The effect of log heating temperature on the peeling process and veneer quality: beech, birch, douglas-fir and spruce case studies[END_REF][START_REF] Antikainen | Simultaneous measurement of lathe check depth and the grain angle of birch (Betula pendula Roth) veneers using laser transillumination imaging[END_REF][START_REF] Darmawan | Lathe check characteristics of fast growing sengon veneers and their effect on LVL glue-bond and bending strength[END_REF]. Lathe check depth (LCD) has been measured traditionally by dyeing and evaluating under the microscope [START_REF] Lutz | Heating veneer bolts to improve quality of Douglas-fir plywood[END_REF][START_REF] Lutz | Wood veneer: Log selection, cutting, and drying[END_REF] in order to identify their effect on product quality [START_REF] Rohumaa | The influence of lathe check depth and orientation on the bond quality of phenol-formaldehyde-bonded birch plywood[END_REF][START_REF] Pot | Numerical study of the influence of veneer lathe checks on the elastic mechanical properties of laminated veneer lumber (LVL) made of beech[END_REF]. However, microscopic measurements are time-consuming, laborious, and typically only a limited amount of specimens is evaluated during study, which might explain the dispersion in obtained results.

To automatize the LCD measurement process for online measurement, the checks have to be measured from the green veneer since measuring checks from the dry veneer is complicated due to the waviness of the veneer [START_REF] Tomppo | Ultrasound evaluation of lathe check depth in birch veneer[END_REF]. Moreover, check properties should be measured immediately after peeling to give a comprehensive overview on check properties and to enable quick response back to the process. During this study, the further development of SMOF, a semi-automated system designed to measure lathe checks in veneer ribbon, presented by [START_REF] Pałubicki | A Method of Lathe Checks Measurement; SMOF device and its software[END_REF] has been done, which will enable to evaluate remarkable amount of checks automatically immediately after the peeling process. Moreover, it will enable to measure precisely all checks and their parameters through the whole mat of veneer peeled.

This study shows how beech veneer lathe check frequency and depth are affected by soaking temperature, compression rate, cutting radius and cutting speed during the peeling process in wellcontrolled laboratory conditions with a large number of samples evaluated with SMOF.

Materials and methods

Veneer material preparation

In this study, nineteen (19) European beech (Fagus sylvatica L.) logs with a maximum diameter of 520 mm were prepared from two trees freshly felled from the public forest of Cluny (France).

The material was selected carefully and only material without visual defects was used. Table 1 shows the full experimental design. The logs were soaked in a water tank until their whole volume reached the target soaking temperature, which was set to 50, 60, 70 or 80°C. This range of temperature corresponds to the common distribution also used in industry. Following soaking, the logs were immediately peeled. An instrumented light packaging scale lathe (SEM S500 -Knife length 900 mm) equipped with an angular pressure bar was used to peel 3.5 mm thick veneer with a cutting speed of 1, 2 or 3 m s -1 . The thickness was set to 3.5 mm, since it is a common thickness used to produce Laminated Veneer Lumber. The compression rates used in this study were 0, 5, 10, or 15% of veneer nominal thickness and this range of compression rates from 0 to 15% is expected to form remarkable differences in lathe check properties. Immediately after peeling, veneer ribbons were cut along the whole obtained veneer mat with a width of approximately 80 mm. One edge of the veneer ribbon was planed to reveal the checks and to enable their reliable detection and measurement.

The LaBoMaP has developed a specific apparatus, called SMOF (Système de Mesure d'Ouverture des Fissures), to measure lathe check parameters. The principle of SMOF operation is described in detail by Palubicki et al. (2010). The SMOF enables to measure automatically and precisely the position and the depth of a large number of checks by slightly bending the veneer on a wheel with previously approved diameter. Below Figure 1 presents the SMOF specifically adapted for this study. Different to the SMOF used by Palubicki et al. (2010), modified SMOF has a larger wheel (d = 180 mm), which was chosen to measure shallow checks without increasing their depth.

Moreover, the objective of the camera has also been changed to improve the field depth. Finally, the initial algorithm of measurement was completely modified to optimize the automatic detection of the check parameters as presented in Figure 2. New software was designed to improve the robustness of the data treatment. The resolution in length and width of the veneer is equal to 0.08mm.

In addition, SMOF will enable the precise measurement of veneer thickness performed with the linear variable differential transformer (LVDT) sensor shown in Figure 1.

Data Analysis

Lathe check depth (LCD) is calculated by comparing the detection of the end of the check to the thickness measured by the LVDT sensor. Distances between checks are calculated only on the horizontal component of the end of the checks. Everything was therefore performed using Matlab (MathWorks, R2012a, Natick, USA). Statistical analysis was made using the Matlab built-in functions "anova1" and "multcompare".

Results and discussion

The effect of log cutting radius on lathe check properties

In first part of the study, preliminary test was performed with the wood material soaked at 70°C and peeled with a compression rate of 0% to 3.5 mm thick veneer. The veneer mat was cut into sheets with dimensions 600 x 750 mm 2 . From each veneer sheet, a veneer ribbon was sawn with a width of approximately 80 mm and characterized with the SMOF. The results show that the median check interval (Figure 3a) and the check depths (Figure 3b) are almost constant through the whole length of the veneer mat. Plot box markers from top to bottom represent the maximum, the third quartile (q3), the median, the first quartile (q1) and the minimum value. Values considered as outliers are marked with the cross. Points are considered as outliers if they are larger than q3 + w(q3 -q1) or smaller than q1w(q3 -q1), where the factor w is equal to 1.5 and it corresponds to +/-2.7 times the standard deviation and 99.3% coverage of data. The coefficient of variation for all measured lathe check intervals is low (3.8%) and for the lathe check depths the coefficient of variation is 7.6%. This check test contains more than 4000 check evaluations and can be considered statistically representative of the process.

The results obtained from the whole length of the veneer ribbon also show remarkable scattering in individual measurement results. Depending on the techniques used, all checks might not be visible and properly revealed in order to detect and measure; hence this could also be one of the reasons for remarkable scattering in interval and the depth results obtained previously by other researchers [START_REF] Dupleix | The effect of log heating temperature on the peeling process and veneer quality: beech, birch, douglas-fir and spruce case studies[END_REF][START_REF] Darmawan | Lathe check characteristics of fast growing sengon veneers and their effect on LVL glue-bond and bending strength[END_REF].

The effect of log cutting speed on lathe check properties

In second part of the study, the effect of cutting speed on lathe check properties was evaluated.

The veneer was prepared under the same conditions as the material described in the previous section for the log radius study. The only variable was cutting speed, which was 1, 2 or 3 m s -1 .

Further, in this part of the study, the whole length of the veneer mat was examined for lathe check properties.

The results of the study show that cutting speed does not have a critical effect on lathe check parameters (Figure 4). As shown in Figure 4, lathe check depth and frequency show a slight decrease in mean values with increasing peeling speed; however, statistically there are no differences based on ANOVA analysis. These results support previous observations made by [START_REF] Lutz | Wood veneer: Log selection, cutting, and drying[END_REF] where cutting speed in the range of 50 to 150 m min -1 has shown not to affect veneer quality.

Moreover, the results of this study show that there are no differences in check properties below the cutting speed 180 m min -1 .

The effect of soaking temperature and compression rate on lathe check properties

In third part of the study, the effect of soaking temperature and compression rate on lathe check properties was evaluated. An image of the side of the veneer was acquired (Figure 5), and the check depth and frequency were automatically evaluated by SMOF (Figure 2).

Figure 5 shows that the lathe check depth is highly influenced by compression rate where the higher pressure shows to form shallower checks, which could be sometimes hard to detect visually.

Comparison of obtained images (Figure 5a-5d) also shows that at higher soaking temperature (Figure 5c,5d) and higher compression rate (10 -15%) checks are visually almost undetectable and non-periodical. On the other hand, at lower soaking temperature (50 or 60°C) and at higher compression rate deeper and periodical checks are formed. These images also support previously described finding, where some of the checks could not be revealed for visual or automatic detection and therefore, increase scattering of measured results. [START_REF] Buchelt | Quantification of cracks and cross-section weakening in sliced veneers[END_REF] noticed that the crack propagation could be influenced by structural elements, like large vessels or rays. In addition, the observations of this study show that check propagation in beech is affected by anatomical features like rays. On visual observation it was noticed that at lower soaking and peeling temperature checks might often change direction and start to propagate along the rays (Figure 6a); oppositely at the higher temperature the checks often pass through the rays as demonstrated in Figure 6b.

The correlation between lathe check depth and frequency was validated using SMOF. The results show that for deeply checked veneers (depth about 2/3 of the thickness) a relatively low number of checks per veneer meter was observed, contrary to shallow checks (depth about 1/4 of the thickness) where numerous checks were created. Figure 7 shows that strong correlation between check depth and frequency was observed, where the coefficient of determination R 2 between the median check interval and the median check depth was equal to 0.81.

Figure 8a shows lathe check interval measured between two consecutive checks and Figure 8b shows lathe check depth measurement results for all ribbons of veneers. The distribution of depths is large since the effect of the anatomy and the detection accuracy seems to be important. However, the number of checks measured in this study (about 20 000) largely exceeds the amount of checks measured in previous studies by other researchers and due to that constitutes a very strong reference data base.

The results show that the effect of compression rate on checks is absolutely clear: the higher the pressure, the shallower and more frequent the checks are. The pressure bar has definitely a decreasing effect on lathe check depths. It is also interesting to note that the distribution of measured intervals seems to increase with the use of pressure bar. Hence, when the lathe checks are shallow, as in the test group soaked at 80°C and peeled with 15% compression, some areas could be left unchecked or checks are not easily revealed. On the other hand, when cutting conditions are favourable to lathe checking (soaking at 50°C and peeling with 0% compression) the checks are easily revealed and are periodical with smaller distribution (Figure 8a). The effect of soaking temperature seems to be not remarkable on lathe check interval (Figure 8a). However, comparing this study with other studies conducted under similar conditions [START_REF] Dupleix | The effect of log heating temperature on the peeling process and veneer quality: beech, birch, douglas-fir and spruce case studies[END_REF]Rohumaa et al. 2016a), some interesting observations could be made. The results show that the rise of soaking temperature from 50°C to 80°C will have a decreasing effect on lathe check depths, which is approximately 15-20%. Similar values for beech were also obtained by [START_REF] Dupleix | The effect of log heating temperature on the peeling process and veneer quality: beech, birch, douglas-fir and spruce case studies[END_REF]. Interestingly, much higher decrease of lathe check depths was observed on birch [START_REF] Dupleix | The effect of log heating temperature on the peeling process and veneer quality: beech, birch, douglas-fir and spruce case studies[END_REF]Rohumaa et al. 2016a), showing that soaking temperature has a much more significant effect on birch than on beech by decreasing the depth of checks by approximately 30-35% in the same range of temperature. The reason might lie in the different anatomical structure, where beech has much higher volume of rays (approx. 20%) compared to birch (approx. 10%). Rays have been shown to resist crack growth in tangential direction [START_REF] Boatright | The effect of microstructure and stress state on the fracture behaviour of wood[END_REF] but act as weak planes in radial direction [START_REF] Ashby | The fracture and toughness of woods[END_REF]. These findings also support the results obtained in this study, where the lathe checks will more favourably propagate along the rays (radial direction) at lower temperature (Figure 6a) and will pass through the rays (Figure 6b) at higher temperature. This weaker plane caused by rays in radial direction will also explain the deeper checks compared to the results on birch wood obtained by [START_REF] Dupleix | The effect of log heating temperature on the peeling process and veneer quality: beech, birch, douglas-fir and spruce case studies[END_REF] and Rohumaa et al. (2016a).

Conclusion

The correlation between lathe check depth and frequency was validated using SMOF device with a very large amount of data (over 20 000). Check properties are varying inside the same veneer, but with a remarkable amount of obtained data the check parameters can be reliably measured and properties evaluated. In well-established peeling process the strong correlation between checks depth and frequency was shown.

At higher temperature shallower and more frequent checks are created, compared to lower temperature. However, the effect of soaking temperature (between 50-80°C) on veneer checking is much smaller than the effect of compression during peeling process, where the higher compression produced veneer with shallower and more frequent checks. When focussing only on lathe check phenomenon, optimal peeling setting was obtained at 70 and 80°C of soaking and at a compression rate of 15%. Proposed settings will produce very shallow checks. However, lathe checks are not the only criteria to produce good quality veneer.

The results also show that check parameters are varying inside the same veneer sheet and it can be challenging to measure all checks. Moreover, some of the checks just cannot be visible and due to that could increase deviation in results obtained by different authors and methods that are used previously.

The results of the study also show that in beech veneer the direction of lathe check propagation could be affected by the rays. Rays resist crack growth in tangential direction but act as weak planes in radial direction. Moreover, at higher temperature checks most likely will pass through the rays, but at lower temperature the checks will change their direction and start to follow the rays, which will affect check depth and shape and in turn also bonding quality.

Fig. 1

 1 Fig. 1 SMOF, optical lathe check measuring system. LVDT is the linear variable differential transformer.

Fig. 3

 3 Fig. 3 a) Lathe check interval and b) lathe check depth according to the cutting radius

Fig. 5

 5 Fig. 5 Effect of soaking temperature and compression rate on lathe check properties (camera images from the SMOF). Where the compression rate varies from 0% to 15% of the veneer thickness for a) soaking at 50°C, b) soaking at 60°C, c) soaking at 70°C, d) soaking at 80°C.

Fig. 7

 7 Fig. 7 Relationship between median check interval and median check depth.

Fig. 8

 8 Fig. 8 Effect of soaking temperature and compression rate on lathe check a) interval and b) depth (boxplot and distribution of checks)

Acknowledgements

The authors gratefully acknowledge the financial support from the Bourgogne Franche-Comté region and Pollmeier. The authors would also like to thank the Xylomat Technical Platform from the Xylomat Scientific Network funded by ANR-10-EQPX-16 XYLOFOREST.