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

STATIONARY SOLUTIONS TO COAGULATION-FRAGMENTATION EQUATIONS

Keywords: Mathematics Subject Classification. 45K05, 45M99. Key words and phrases. coagulation, fragmentation, stationary solution, mass conservation

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Stationary solutions to coagulation-fragmentation equations

Introduction

The coagulation-fragmentation equation is a mean-field model describing the time evolution of the size distribution function f of a system of particles increasing their size by pairwise merging or reducing it by splitting, no matter being lost during these processes. Denoting the coagulation kernel, the overall fragmentation rate, and the daughter distribution function by K, a, and b, respectively, the coagulation-fragmentation equation reads

∂ t f = Cf + F f , (t, x) ∈ (0, ∞) 2 , (1.1a
)

f (0) = f in , x ∈ (0, ∞) , (1.1b)
where the coagulation term Cf is given by Cf (x) := 1 2

x 0 K(y, x -y)f (x -y)f (y) dy -

∞ 0 K(x, y)f (x)f (y) dy , x > 0 , (1.1c)
and the fragmentation term F f by

F f (x) := -a(x)f (x) + ∞ x
a(y)b(x, y)f (y) dy , x > 0 .

(1.1d)

The first term in (1.1c) accounts for the formation of particles of size x > 0 as a consequence of the merging of two smaller particles with respective sizes y ∈ (0, x) and x -y. The second term in (1.1c) and the first term in (1.1d) describe the depletion of particles of size x > 0 due to coalescence with other particles and fragmentation, respectively. Finally, the breakup of a particle of size y > x produces fragments of various sizes ranging in (0, y), including fragments of size x according to the distribution b(x, y) as indicated by the second term in (1.1d). We further assume that there is no loss of matter during the breakage process, which amounts to require that b satisfies y 0 xb(x, y) dx = y , y > 0 , and b(x, y) = 0 , x > y > 0 .

(1.2)

Since there is also no loss of matter during coalescence, the total mass of the system is expected to be invariant throughout time evolution; that is,

∞ 0 xf (t, x) dx = ∞ 0 xf (0, x) dx , t ≥ 0 . (1.3)
Though this property may fail to be true when, either the coagulation is too strong compared to the fragmentation, a phenomenon known as gelation, or the overall fragmentation rate a is unbounded as x → 0, a phenomenon known as shattering, both are excluded in the forthcoming analysis and we refer to [START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF][START_REF] Escobedo | Gelation in coagulation and fragmentation models[END_REF][START_REF] Jeon | Existence of gelling solutions for coagulation-fragmentation equations[END_REF][START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF][START_REF] Leyvraz | Existence and properties of post-gel solutions for the kinetic equations of coagulation[END_REF][START_REF] Leyvraz | Singularities in the kinetics of coagulation processes[END_REF] and [START_REF] Arlotti | Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss[END_REF][START_REF] Filippov | On the distribution of the sizes of particles which undergo splitting[END_REF][START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF], respectively, for detailed information on these issues. Our interest in this paper is rather related to the possible balance between coagulation and fragmentation, which are competing mechanisms. Indeed, the latter increases the number of particles and reduces the mean size of particles, while the former acts in the opposite direction. It is then of interest to figure out the outcome of this competition and, in particular, whether it could lead to stationary solutions. This is the issue we aim at investigating herein.

The first example of coagulation-fragmentation equation featuring steady state solutions is the case of constant coefficients [START_REF] Aizenman | Convergence to equilibrium in a system of reacting polymers[END_REF]

∂ t f (t, x) = x 0 [f (t, x -y)f (t, y) -A 0 f (t, x)] dy -2 ∞ 0 [f (t, x)f (t, y) -A 0 f (t, x + y)] dy , (t, x) ∈ (0, ∞) 2 , (1.4)
which is obtained with the choice

K(x, y) = 2 , a(x) = A 0 x , b(x, y) = 2 y , 0 < x < y , (1.5)
in (1.1). For any z > 0, the function Q z defined by Q z (x) := A 0 e x ln z , x > 0, is a stationary solution to (1.4) and Q z has finite total mass if and only if z ∈ (0, 1). The example (1.5) is actually a particular case of coagulation and fragmentation coefficients satisfying the so-called detailed balance condition: there are a non-negative symmetric function F defined on (0, ∞) 2 and a non-negative function Q defined on (0, ∞) such that a(x) = 1 2

x 0 F (x * , x -x * ) dx * , a(y)b(x, y) = F (x, y -x) , 0 < x < y , (1.6a) K(x, y)Q(x)Q(y) = F (x, y)Q(x + y) , (x, y) ∈ (0, ∞) 2 .

(1.6b)

Note that we recover (1.5) from (1.6) by setting F ≡ 2A 0 and Q ≡ A 0 . Thanks to (1.6), the equation (1.1) reads

∂ t f (t, x) = 1 2
x 0 [K(x -y, y)f (t, x -y)f (t, y) -F (y, x -y)f (t, x)] dy

- ∞ 0 [K(x, y)f (t, x)f (t, y) -F (x, y)f (t, x + y)] dy , (t, x) ∈ (0, ∞) 2 , (1.7)
and Q z : x → Q(x)e x ln z is a stationary solution to (1.7) for all z ∈ (0, ∞). Whether Q z has finite total mass then depends on both the value of z and the integrability properties of Q. We refer to [START_REF] Carr | Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case[END_REF][START_REF] Carr | Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation[END_REF][START_REF] Laurenc | The continuous coagulation-fragmentation equations with diffusion[END_REF][START_REF]Convergence to equilibrium for the continuous coagulation-fragmentation equation[END_REF] for a more detailed account on the various situations that may happen. Coagulation and fragmentation coefficients satisfying the detailed balance condition (1.6) are however far from being generic and different approaches have to be designed to investigate the existence of stationary solutions to (1.1) when (1.6) fails to hold. When the coagulation and fragmentation coefficients are given by

K(x, y) = k 0 + k 1 (x + y) , a(x) = A 0 x , b(x, y) = 2 y , 0 < x < y , (1.8)
the existence of a stationary solution to (1.1) having total mass ̺ > 0 is proved in [START_REF] Dubovskiȋ | Trend to equilibrium for the coagulation-fragmentation equation[END_REF] for all ̺ > 0, the proof relying on a fixed point argument performed on the stationary version of (1.1a). It uses in an essential way the specific form of the coefficients and does not seem to extend to handle more general cases. Uniqueness and local stability of steady states are also established in [START_REF] Dubovskiȋ | Trend to equilibrium for the coagulation-fragmentation equation[END_REF]. In the same vein but with a completely different approach, a complete description of stationary solutions to (1.1) is obtained in [7, Theorem 5.1 & Remark 5.2] when K(x, y) = k 0 (xy) λ/2 , a(x) = A 0 x λ/2 , b(x, y) = 2 y , 0 < x < y , (1.9)

for some λ ∈ [0, 2], k 0 > 0, and A 0 > 0. Two steps are needed to obtain this result: first, when λ = 0, k 0 = 2, and A 0 = 1, given an integrable stationary solution f to (1.1), its Bernstein transform

U(s) := ∞ 0 1 -e -sx f (x) dx , s ≥ 0 , solves the integro-differential equation U(s) 2 + U(s) = 2 s s 0 U(r) dr , s > 0 , U(0) = 0 . (1.10)
This equation turns out to have an explicit solution U ⋆ which is the Bernstein transform of a nonnegative function

f ⋆ ∈ L 1 ((0, ∞), (1 + x)dx) satisfying ∞ 0 f ⋆ (x) dx = ∞ 0 xf ⋆ (x) dx = 1 , (1.11)
and any solution U to (1.10) is a dilation of U ⋆ ; that is, there is µ > 0 such that U(s) = U ⋆ (µs) for s ≥ 0. Moreover,

f ⋆ (x) ∼ x→0 x -2/3 Γ(1/3) and f ⋆ (x) ∼ x→∞ 9 8
x -3/2 Γ(1/2) e -4x/27 .

(1.12)

In particular, f ⋆ features an integrable singularity as x → 0. To handle the case λ > 0 in (1.9), it suffices to note that, if f is a stationary solution to (1.1) corresponding to coagulation and fragmentation rates given by (1.8) for some λ ∈ [0, 2], k 0 > 0, and A 0 > 0, then x → k 0 x λ/2 f (x)/2A 0 is a stationary solution to (1.1) corresponding to coagulation and fragmentation rates given by (1.8) with λ = 0, k 0 = 2, and A 0 = 1. Consequently, there is µ > 0 such that

f (x) = 2A 0 µ k 0 x -λ/2 f ⋆ (µx) , x ∈ (0, ∞) . (1.13)
It readily follows from (1.12) and (1.13) that f also features a singularity as x → 0 which is not integrable if λ > 2/3. However, the total mass of f is finite for all λ ∈ [0, 2]. Stability of stationary solutions is also investigated in [START_REF] Degond | Coagulation-fragmentation model for animal group-size statistics[END_REF] when λ = 0, k 0 = 2, and A 0 = 1. The just described results only deal with very specific coagulation and fragmentation coefficients, and the approaches used in both cases exploit their particular structure. They are thus rather unlikely to extend to a wider setting. As far as we know, the only result handling a fairly general class of coagulation and fragmentation coefficients is to be found in [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF], the coagulation and fragmentation coefficients being given by

K(x, y) = x -α y β + x β y -α , (x, y) ∈ (0, ∞) 2 , (1.14a)
and

a(x) = a 0 x γ , b(x, y) = 1 y B x y , 0 < x < y , (1.14b) where (α, β) ∈ [0, 1] 2 , β -α ∈ [0, 1) , γ ≥ 0 , a 0 > 0 , (1.14c) and
B is a non-negative function in L 1 ((0, 1), (z + z -2α )dz) .

(1.14d) Assuming further that (β, γ) = (1, 0) and (α, γ) = (0, 0), the existence of a non-negative stationary solution to (1.1) with total mass ̺ is shown in [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]Theorem 4.1] for all ̺ > 0. Furthermore, this stationary solution belongs to L 1 ((0, ∞), x m dx) for all m ≥ -2α and, under the additional assumption that B ∈ L ∞ (0, 1), it belongs to L p (0, ∞) for all p ∈ [1, ∞). The approach developed to prove this result is of a completely different nature and actually relies on a dynamical approach. Roughly speaking, the basic idea is to find a suitable functional setting in which the initial value problem (1.1) is well-posed, along with a closed and convex set Z which is compact for the associated topology and is positively invariant for the dynamical system associated to (1.1) (in the sense that f (t) ∈ Z for all t > 0 as soon as f (0) ∈ Z). If a fixed point theorem is available in this functional setting, then a classical argument guarantees the existence of at least one stationary solution, see [START_REF] Amann | Ordinary differential equations[END_REF]Theorem 16.5], [15, Proof of Theorem 5.2], and [12, Theorem 1.2], for instance. Though this method merely gives the existence of a steady state solution without any information on uniqueness or stability, it is far more flexible than the previous ones and we shall partially employ it in the forthcoming analysis. Let us mention that it is also the cornerstone of the construction of massconserving self-similar solutions to the coagulation equation [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Fournier | Existence of self-similar solutions to Smoluchowski's coagulation equation[END_REF][START_REF] Niethammer | Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels[END_REF].

According to the previous description, no result on the existence of steady state solutions seems to be available for the classical coagulation kernel

K(x, y) = K 0 x α y β + x β y α , (x, y) ∈ (0, ∞) 2 , (1.15a) with 0 ≤ α ≤ β ≤ 1 , λ := α + β ∈ [0, 1) , (1.15b
) and the purpose of this paper is to fill this gap for a rather large class of fragmentation coefficients. More precisely, we assume that there are

γ > 0 , a 0 > 0 , p 0 > 1 , (1.16a)
and a non-negative function

B ∈ L 1 ((0, 1), zdz) ∩ L p 0 (0, 1) , 1 0 zB(z) dz = 1 , (1.16b) such that a(x) = a 0 x γ , b(x, y) = 1 y B x y , 0 < x < y . (1.16c)
Note that the class of coagulation kernels (1.15) includes the sum kernels corresponding to α = 0 and β = λ ∈ [0, 1) and the product kernels corresponding to

α = β = λ/2 ∈ [0, 1/2
B(z) = B 1,ν (z) := (ν + 2)z ν , z ∈ (0, 1) , ν > -1 , (1.17)
and the parabolic breakup distribution

B(z) = B 2,ν (z) := (ν + 2)(ν + 1)z ν-1 (1 -z) , z ∈ (0, 1) , ν > 0 . (1.18)
Indeed, B 1,ν given by (1.17) satisfies (1.16b) for any p 0 > 1 when ν ≥ 0 and for any p 0 ∈ (1, 1/|ν|) when ν ∈ (-1, 0). Similarly, B 2,ν given by (1.18) satisfies (1.16b) for any p 0 > 1 when ν ≥ 1 and p 0 ∈ (1, 1/(1 -ν)) when ν ∈ (0, 1). Before stating the main result, let us introduce some notation. Throughout the paper, for m ∈ R, we set

X m := L 1 ((0, ∞), x m dx) , M m (h) := ∞ 0 x m h(x) dx , h ∈ X m , (1.19)
and denote the positive cone of X m by X + m . We also denote the space X m endowed with its weak topology by X m,w .

Theorem 1.1. Assume that the coagulation and fragmentation coefficients satisfy (1.15) and (1.16). Given ̺ > 0 there exists at least a stationary (weak) solution ϕ ∈ X + 1 to (1.1) with the following properties:

(s1) M 1 (ϕ) = ̺; (s2) there are p 1 ∈ (1, p 0 ) and m 1 ∈ (λ, 1) such that ϕ ∈ L p 1 ((0, ∞), x m 1 +γ dx) ∩ m>λ X m ; (s3) for all ϑ ∈ Θ 1 := {h ∈ W 1,∞ (0, ∞) : h(0) = 0}, 1 2 ∞ 0 ∞ 0 K(x, y) [ϑ(x + y) -ϑ(x) -ϑ(y)] ϕ(x)ϕ(y) dydx = ∞ 0 a(y)ϕ(y) ϑ(y) - y 0 ϑ(x)b(x, y) dx dy .
It is worth pointing out here that Theorem 1.1 (s2) does not exclude a non-integrable singularity of ϕ as x → 0, a situation which may indeed occur, as we shall see below. This feature is not encountered for the coagulation and fragmentation coefficients given by (1.14) and considered in [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF] when α < 0, as the unboundedness of the coagulation kernel for small sizes implies the vanishing of the stationary solution as x → 0. This possible singular behaviour for small sizes is actually the main difficulty to be overcome in the analysis carried out below and requires a more involved approach, which we describe now.

The proof of Theorem 1.1 is carried out in two steps. We fix ̺ > 0. Using the dynamical approach already alluded to, given ε ∈ (0, 1), we first construct a stationary solution ϕ ε ∈ X 1 to

∂ t f = C ε f + F ε f , (t, x) ∈ (0, ∞) 2 , f (0) = f in , x ∈ (0, ∞) , (1.20)
satisfying M 1 (ϕ ε ) = ̺, where the coagulation and fragmentation operators C ε and F ε are given by (1.1c) with K ε := K + 2εK 0 instead of K and (1.1d) with a ε := a + a 0 ε 2 instead of a, respectively. For this choice of coagulation and fragmentation coefficients, we actually build a closed convex and sequentially weakly compact subset Z ε of X 1 such that solutions to (1.20) starting from an initial condition in Z ε remain in Z ε for all positive times. Recalling that, according to the Dunford-Pettis theorem, sequential weak compactness in X 1 requires to prevent concentration and escape of mass for small and large sizes, finding Z ε amounts to derive time-independent estimates in X m 0 ∩ X m ∩ L p 2 (0, ∞) for some suitably chosen m 0 < 1 < m and p 2 > 1. While some of the moment estimates can be obtained directly for ε = 0 (Section 2.1), it does not seem to be possible to derive uniform integrability estimates without the positive lower bounds on K ε and a ε (Section 2.2). Besides the construction of Z ε (Section 3.2), we also show the well-posedness of (1.20) in Section 3.1, as well as the continuous dependence of solutions to (1.20) in X 1,w with respect to the initial condition (Section 3.3).

To justify rigorously the computations performed in Section 2, an additional approximation is needed and we shall actually work with truncated versions of K ε and a ε . Thanks to this analysis, it remains to apply [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]Theorem 1.2] to obtain the existence of a stationary solution ϕ ε ∈ Z ε to (1.20) (Section 3.4).

To complete the proof of Theorem 1.1, we are left with taking the limit ε → 0. To this end, we realize that, since we have payed special attention to the dependence on ε of the estimates derived in Section 2, there is a sequentially weakly compact subset Z in X 1 such that Z ε ⊂ Z for all ε ∈ (0, 1), see Section 3.5. Consequently, there are ϕ ∈ Z and a subsequence (ϕ ε k ) k≥1 of (ϕ ε ) ε∈(0,1) such that ϕ ε k ⇀ ϕ in X 1 . We finally combine this convergence with the properties of Z and (ϕ ε k ) k≥1 to prove that ϕ is a stationary weak solution to (1.1) as described in Theorem 1.1 (Section 3.5).

Theorem 1.1 only provides the finiteness of the moments of ϕ of order larger than λ and thus does not provide much information on its behaviour for small sizes. In fact, the small size behaviour described in Theorem 1.1 (s2) does not seem to be accurate. Indeed, formal asymptotics indicate that, if ϕ is a stationary weak solution to (1.1) satisfying the properties (s1)-(s3) stated in Theorem 1.1 and ϕ(x) ∼ Ax -τ as x → 0 (1.21a) for some A > 0 and τ > 0, then τ can be identified and depends on the values of α, β, γ, and possibly on B. Specifically,

-if γ > α, then τ = α + 1 + m ⋆ < α + 1 , (1.21b) where m ⋆ is defined in (1.22) below; -if γ = α < β, then τ = α + 1 ; (1.21c) -if γ = α = β and B = B 1,ν , see (1.17), then τ = α + 2 ν + 3 < α + 1 ; (1.21d) -if γ < α, then τ = λ + 1 -γ .
(1.21e) In particular, the prediction (1.21d) perfectly agrees with (1.13) when γ = α = β = λ/2 ∈ [0, 1/2) and ν = 0 (B = B 1,0 ). On the one hand, (1.21) implies that ϕ may have a non-integrable singularity as x → 0 and, in particular, it is not expected to belong to X α when γ < α. On the other hand, different behaviours are predicted in (1.21), which vary according to the sign of γ -α, and seem to be sensitive to the behaviour of B(z) as z → 0 when γ = α = β. We shall not attempt a complete proof of (1.21) herein but, as a first step in that direction, we provide additional integrability properties of ϕ which complies with (1.21).

m ⋆ := inf m ∈ R : B ∈ L 1 ((0, 1), z m dz) ≤ 1 -p 0 p 0 < 0 . (1.22) Moreover, if m ⋆ > -∞ and B / ∈ L 1 ((0, 1), z m⋆ dz), then ϕ / ∈ X α+m⋆ ; (m2) if γ = α < β, then ϕ ∈ X m for any m ≥ β; (m3) if γ = α = β, then ϕ ∈ X m for any m ≥ α; (m4) if γ < α, then ϕ ∈ X m for any m > λ -γ.
The proof of Proposition 1.2 is carried out in Section 4 and relies on the choice of suitable test functions in Theorem 1.1 (s3). Comparing (1.21) and Proposition 1.2 reveals that the properties (m2) and (m3) are not optimal. Improving Proposition 1.2 so that it matches (1.21) in these cases seems to require a finer analysis which we have yet been unable to set up. We however hope to return to this problem in the near future.

A truncated approximation

Let ̺ > 0 and assume that K, a, and b are coagulation and fragmentation coefficients satisfying (1.15) and (1.16). Also, let f in be an initial condition satisfying

f in ∈ X + 0 ∩ X 2+γ with M 1 (f in ) = ̺ . (2.1)
We now introduce the approximation to (1.1) we are going to work with in this section. Besides requiring a positive lower bound on the coagulation kernel and the overall fragmentation rate as already mentioned, we also truncate both of them as in [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]. Specifically, we fix a positive integer j ≥ 2 and a positive real number ε ∈ (0, 1) and set

K j,ε (x, y) := 2εK 0 + K (min{x, j}, min{y, j}) , (x, y) ∈ (0, ∞) 2 , (2.2)
a j,ε (x) := a 0 min{x, j} γ + ε 2 , x ∈ (0, ∞) . (2.3)
Since K j,ε and a j,ε are bounded, we may proceed as in [START_REF] Banasiak | Analytic Methods for Coagulation-Fragmentation Models[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF][START_REF] Ch | Coalescence and breakage processes[END_REF] to show, by a Banach fixed point argument in X 0 = L 1 (0, ∞), that there is a unique non-negative strong solution

f j,ε ∈ C 1 ([0, ∞); X 0 )
to the coagulation-fragmentation equation

∂ t f j,ε = C j,ε f j,ε + F j,ε f j,ε , (t, x) ∈ (0, ∞) 2 , (2.4a) f j,ε (0) = f in , x ∈ (0, ∞) , (2.4b)
where the coagulation and fragmentation operators C j,ε and F j,ε are given by (1.1c) with K j,ε instead of K and (1.1d) with a j,ε instead of a, respectively. A first consequence of (2.4a) is that, for t ≥ 0 and

ϑ ∈ L ∞ (0, ∞), d dt ∞ 0 ϑ(x)f j,ε (t, x) dx = 1 2 ∞ 0 ∞ 0 K j,ε (x, y)χ ϑ (x, y)f j,ε (t, x)f j,ε (t, y) dydx - ∞ 0 a j,ε (y)N ϑ (y)f j,ε (t, y) dy , (2.5)
where

χ ϑ (x, y) := ϑ(x + y) -ϑ(x) -ϑ(y) , (x, y) ∈ (0, ∞) 2 , (2.6a) N ϑ (y) := ϑ(y) - y 0 ϑ(x)b(x, y) dx , y ∈ (0, ∞) . (2.6b)
Owing to (1.16c), an alternative formula for N ϑ reads

N ϑ (y) = ϑ(y) - 1 0 ϑ(yz)B(z) dz , y ∈ (0, ∞) . (2.6c)
For the particular choice ϑ(x) = ϑ m (x) := x m , x > 0, for some m ∈ R, we set χ m := χ ϑm and N m := N ϑm for simplicity.

Owing to the boundedness of K j,ε and a j,ε and the integrability (1.16b) of B over (0, 1), we infer from (2.5) by an approximation argument that f j,ε is mass-conserving; that is,

f j,ε ∈ L ∞ ((0, ∞), X 1 ) and M 1 (f j,ε (t)) = ̺ , t ≥ 0 . (2.7)
Moreover, a similar approximation argument allows us to show that, if f in ∈ X m for some m > 1, then f j,ε ∈ L ∞ ((0, T ), X m ) for any T > 0. We shall refine this result in the next section.

We now derive several estimates for the family {f j,ε : j ≥ 2 , ε ∈ (0, 1)}, which do not depend on j ≥ 2. We also pay special attention to the dependence on ε ∈ (0, 1), if any. Throughout this section, C and C i , i ≥ 1, denote positive constants which depend only on K 0 , α, β, a 0 , γ, B, and ̺. Dependence upon additional parameters will be indicated explicitly. For further use, we set

b m := 1 0 z m B(z) dz for m > m ⋆ and B p p := 1 0 B(z) p dz for p ∈ [1, p 0 ] , (2.8a)
which are finite by (1.16b) and (1.22), and satisfy

b m < 1 ⇐⇒ m > 1 (2.8b)
due to (1.16b). Also, Young's inequality and (1.15) entail that

K(x, y) ≤ K 0 x λ + y λ , (x, y) ∈ (0, ∞) 2 . (2.9) 2.1. Moment Estimates. For m ∈ R we set M m,j,ε := sup t≥0 {M m (f j,ε (t))} ∈ [0, ∞] (2.10)
and begin with the behaviour of f j,ε for large sizes.

Lemma 2.1. Let m ≥ 2 and assume that f in ∈ X m . There is a positive constant µ m ≥ Γ(m + 1)̺ m depending only on K 0 , α, β, a 0 , γ, B, ̺, and m such that

M m,j,ε ≤ max{M m (f in ), µ m } .
Proof. We first recall that there is c m > 0 depending only on m such that

χ m (x, y) ≤ c m xy m-1 + x m-1 y , (x, y) ∈ (0, ∞) 2 , (2.11)
see [START_REF] Carr | Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I. The strong fragmentation case[END_REF]Lemma 2.3 (ii)] for instance. Let t > 0. We infer from (2.5) with ϑ = ϑ m , (2.8), (2.11), and the symmetry of K that

d dt M m (f j,ε (t)) ≤ c m 2 ∞ 0 ∞ 0 K j,ε (x, y) x m-1 y + xy m-1 f j,ε (t, x)f j,ε (t, y) dydx -(1 -b m ) ∞ 0 x m a j,ε (x)f j,ε (t, x) dx ≤ c m ∞ 0 ∞ 0 xy m-1 K j,ε (x, y)f j,ε (t, x)f j,ε (t, y) dydx -a 0 (1 -b m ) ∞ 0 x m min{x, j} γ f j,ε (t, x) dx .
On the one hand, by (2.7),

∞ 0 x m min{x, j} γ f j,ε (t, x) dx ≥ ∞ 1 x m min{x, j} γ f j,ε (t, x) dx ≥ ∞ 1 x m f j,ε (t, x) dx = M m (f j,ε (t)) - 1 0 x m f j,ε (t, x) dx ≥ M m (f j,ε (t)) -̺ .
On the other hand, it follows from (2.7) and Hölder's and Young's inequalities that

∞ 0 ∞ 0 xy m-1 f j,ε (t, x)f j,ε (t, y) dydx ≤ ̺M m-1 (f j,ε (t)) ≤ ̺M m (f j,ε (t)) (m-2)/(m-1) M 1 (f j,ε (t)) 1/(m-1) ≤ a 0 (1 -b m ) 8c m K 0 M m (f j,ε (t)) + C(m) .
Similarly,

∞ 0 ∞ 0 xy m-1 min{y, j} λ f j,ε (t, x)f j,ε (t, y) dydx ≤ ̺M m+λ-1 (f j,ε (t)) ≤ ̺M m (f j,ε (t)) (m+λ-2)/(m-1) M 1 (f j,ε (t)) (1-λ)/(m-1) ≤ a 0 (1 -b m ) 4c m K 0 M m (f j,ε (t)) + C(m) ,
and

∞ 0 ∞ 0 xy m-1 min{x, j} λ f j,ε (t, x)f j,ε (t, y) dydx ≤ M 1+λ (f j,ε (t))M m-1 (f j,ε (t)) ≤ ̺M m (f j,ε (t)) (m+λ-2)/(m-1) M 1 (f j,ε (t)) (m-λ)/(m-1) ≤ a 0 (1 -b m ) 4c m K 0 M m (f j,ε (t)) + C(m) .
Collecting the previous inequalities and using (2.9), we obtain

d dt M m (f j,ε (t)) ≤ 2 + ε 4 a 0 (1 -b m )M m (f j,ε (t)) + C(m) -a 0 (1 -b m ) (M m (f j,ε (t)) -̺) ≤ - a 0 4 (1 -b m )M m (f j,ε (t)) + C(m) .
Integrating the previous differential inequality gives

M m (f j,ε (t)) ≤ e -a 0 (1-bm)t/4 M m (f in ) + 4C(m) a 0 (1 -b m ) 1 -e -a 0 (1-bm)t/4
for t ≥ 0. Therefore,

M m (f j,ε (t)) ≤ max M m (f in ), 4C(m) a 0 (1 -b m ) , t ≥ 0 ,
from which Lemma 2.1 follows.

From now on, we fix a positive real number

σ > max {1, ̺, µ 2 , µ 2+γ } (2.12a) such that max M 2 (f in ), M 2+γ (f in ) ≤ σ . (2.12b)
A first consequence of (2.7), (2.12), Lemma 2.1, and Hölder's inequality is that

M 1+γ,j,ε ≤ σ and M 2,j,ε ≤ σ . (2.13)
Next, owing to (2.7), (2.12), and (2.13), another application of Hölder's inequality provides a similar bound for moments of order m ∈ (1, 2), which we report now.

Corollary 2.2. For m ∈ (1, 2), M m,j,ε ≤ σ .
We next turn to the behaviour for small sizes and, to this end, derive estimates for moments of order smaller than one. Lemma 2.3. Let m ∈ (λ, 1). There is µ m ≥ Γ(m + 1)̺ m depending only K 0 , α, β, a 0 , γ, B, ̺, and m such that M m,j,ε ≤ max{M m (f in ), µ m + σ} .

Proof. Let m ∈ (λ, 1) and t > 0. We first argue as in [START_REF] Fournier | Existence of self-similar solutions to Smoluchowski's coagulation equation[END_REF]Lemma 3.1] to estimate the contribution of the coagulation term to the time evolution of M m (f j,ε ), see also [START_REF] Banasiak | Analytic Methods for Coagulation-Fragmentation Models[END_REF]Lemma 8.2.12]. More precisely, since j ≥ 2, χ m ≤ 0, and K j,ε (x, y) ≥ 2K 0 (xy) λ/2 for (x, y) ∈ (0, 1) 2 , we obtain

- 1 2K 0 ∞ 0 ∞ 0 K j,ε (x, y)χ m (x, y)f j,ε (t, x)f j,ε (t, y) dydx ≥ 1 2K 0 1 0 1 0 [x m + y m -(x + y) m ] K j,ε (x, y)f j,ε (t, x)f j,ε (t, y) dydx ≥ P j,ε (t) := 1 0 1 0 [x m + y m -(x + y) m ] (xy) λ/2 f j,ε (t, x)f j,ε (t, y) dydx . (2.14) Since m < 1, it follows from the convexity of x → x m-1 that, for (x, y) ∈ (0, ∞) 2 , x m + y m -(x + y) m = x x m-1 -(x + y) m-1 + y y m-1 -(x + y) m-1 ≥ 2(1 -m)xy(x + y) m-2 .
Therefore,

P j,ε (t) ≥ 2(1 -m) 1 0 1 0 (x + y) m-2 (xy) (2+λ)/2 f j,ε (t, x)f j,ε (t, y) dydx .

Introducing

x i := i -2/(m-λ) and P j,ε (t, i) :=

x i x i+1 x (2+λ)/2 f j,ε (t, x) dx , i ≥ 1 ,
we further obtain

P j,ε (t) ≥ 2(1 -m) ∞ i=1 x i x i+1 x i x i+1 (x + y) m-2 (xy) (2+λ)/2 f j,ε (t, x)f j,ε (t, y) dydx ≥ 2 m-1 (1 -m) ∞ i=1 x m-2 i P j,ε (t, i) 2 . (2.15)
It next follows from the Cauchy-Schwarz inequality that

1 0 x m f j,ε (t, x) dx = ∞ i=1 x i x i+1 x m f j,ε (t, x) dx ≤ ∞ i=1 x (2m-2-λ)/2 i+1 P j,ε (t, i) ≤ ∞ i=1 x 2m-2-λ i+1 x 2-m i 1/2 ∞ i=1 x m-2 i P j,ε (t, i) 2 1/2 . (2.16) Since x 2m-2-λ i+1 x 2-m i ≤ (2i) 2(2+λ-2m)/(m-λ) i -2(2-m)/(m-λ) = 4 (2+λ-2m)/(m-λ) i -2 ,
the series in the right-hand side of (2.16) converges and we deduce from (2.15) and (2.16) that

P j,ε (t) ≥ C 1 (m) 1 0 x m f j,ε (t, x) dx 2 .
(2.17)

Furthermore, as

M m (f j,ε (t)) = 1 0 x m f j,ε (t, x) dx + ∞ 1 x m f j,ε (t, x) dx ≤ 1 0 x m f j,ε (t, x) dx + ∞ 1 xf j,ε (t, x) dx ≤ 1 0
x m f j,ε (t, x) dx + ̺ by (2.7), we infer from Young's inequality that

1 0 x m f j,ε (t, x) dx 2 ≥ M m (f j,ε (t)) 2 2 -̺ 2 . (2.18)
Combining (2.14), (2.17), and (2.18) provides the existence of two positive constants C 2 (m) and

C 3 (m) such that 1 2 ∞ 0 ∞ 0 K j,ε (x, y)χ m (x, y)f j,ε (t, x)f j,ε (t, y) dydx ≤ C 2 (m) -C 3 (m)M m (f j,ε (t)) 2 . (2.19)
Consequently, recalling that b m > 1 by (2.8) as m < 1, it follows from (2.5) with ϑ = ϑ m , (2.13), (2.19), and Young's inequality that

d dt M m (f j,ε (t)) = 1 2 ∞ 0 ∞ 0 K j,ε (x, y)χ m (x, y)f j,ε (t, x)f j,ε (t, y) dydx + a 0 (b m -1) ∞ 0 x m ε 2 + min{x, j} γ f j,ε (t, x) dx ≤ C 2 (m) -C 3 (m)M m (f j,ε (t)) 2 + a 0 b m M m+γ (f j,ε (t)) + a 0 b m ε 2 M m (f j,ε (t)) ≤ C 2 (m) -C 3 (m)M m (f j,ε (t)) 2 + a 0 b m γ γ + 1 -m M γ+1 (f j,ε (t)) + a 0 b m 1 -m γ + 1 -m + 1 M m (f j,ε (t)) ≤ C 4 (m) [1 + M m (f j,ε (t)) + σ] -C 3 (m)M m (f j,ε (t)) 2 .
As

M m (f j,ε (t)) ≤ C 3 (m) 2C 4 (m) M m (f j,ε (t)) 2 + C 4 (m) 2C 3 (m) ,
we finally obtain

d dt M m (f j,ε (t)) ≤ -C 4 (m)M m (f j,ε (t)) + C 4 (m) 1 + C 4 (m) C 3 (m) + σ , t ≥ 0 .
Integrating the previous differential inequality gives

M m (f j,ε (t)) ≤ e -C 4 (m)t M m (f in ) + 1 + C 4 (m) C 3 (m) + σ 1 -e -C 4 (m)t , t ≥ 0 .
Therefore,

M m (f j,ε (t)) ≤ max M m (f in ), 1 + C 4 (m) C 3 (m) + σ , t ≥ 0 , from which Lemma 2.3 follows.
The next step is devoted to the derivation of additional estimates for small sizes but now with a strong dependence on ε.

Lemma 2.4. There is µ 0 ≥ 1 depending only on K 0 , a 0 , B, and ̺ such that

M 0,j,ε ≤ max M 0 (f in ), σ + µ 0 ε .
Proof. It follows from (2.5) with ϑ ≡ 1, (2.8), (2.13), and Young's inequality that, for t ≥ 0,

d dt M 0 (f j,ε (t)) = - 1 2 ∞ 0 ∞ 0 K j,ε (x, y)f j,ε (t, x)f j,ε (t, y) dydx -a 0 (1 -b 0 ) ∞ 0 ε 2 + min{x, j} γ f j,ε (t, x) dx ≤ -εK 0 M 0 (f j,ε (t)) 2 + a 0 b 0 ε 2 M 0 (f j,ε (t)) + a 0 b 0 M γ (f j,ε (t)) ≤ -εK 0 M 0 (f j,ε (t)) 2 + γa 0 b 0 1 + γ M γ+1 (f j,ε (t)) + a 0 b 0 1 1 + γ + 1 M 0 (f j,ε (t)) ≤ -εK 0 M 0 (f j,ε (t)) 2 + a 0 b 0 σ + 2a 0 b 0 M 0 (f j,ε (t)) .
By the Cauchy-Schwarz inequality,

4a 0 b 0 M 0 (f j,ε (t)) ≤ εK 0 M 0 (f j,ε (t)) 2 + 4a 2 0 b 2 0 εK 0 . Hence d dt M 0 (f j,ε (t)) + 2a 0 b 0 M 0 (f j,ε (t)) ≤ a 0 b 0 σ + 4a 2 0 b 2 0 εK 0 , t ≥ 0 .
Integrating this differential inequality, we find

M 0 (f j,ε (t)) ≤ M 0 (f in )e -2a 0 b 0 t + σ 2 + 2a 0 b 0 εK 0 1 -e -2a 0 b 0 t ≤ max M 0 (f in ), σ + 2a 0 b 0 εK 0 for t ≥ 0, as claimed.
The previous result actually extends to some moments of negative order.

Lemma 2.5. Let m ∈ (m ⋆ , 0) and set

ε m,σ := 1 σ min 1, K 0 ̺ 2 4a 0 b m , (2.20)
where m ⋆ and σ are defined in (1.22) and (2.12), respectively. There is µ m > 0 depending only on K 0 , a 0 , ̺, B, and

m such that, if f in ∈ X m and ε ∈ (0, ε m,σ ), then M m,j,ε ≤ max M m (f in ), µ m σ 2 ε -(γ+2-2m)/γ .
We may also assume that µ m ≥ Γ(m + 1)̺ m when m > -1.

Proof. For δ ∈ (0, 1), we set ϑ m,δ (x) := (x + δ) m , x > 0, and notice that

χ ϑ m,δ (x, y) ≤ -(x + δ) m ≤ 0 , (x, y) ∈ (0, ∞) 2 .
Let ε ∈ (0, ε m,σ ) and t > 0. We infer from (2.5) with ϑ = ϑ m,δ that

d dt ∞ 0 ϑ m,δ (x)f j,ε (t, x) dx ≤ -εK 0 ∞ 0 ∞ 0 (x + δ) m f j,ε (t, x)f j,ε (t, y) dydx + ∞ 0 a j,ε (y)f j,ε (t, y) y 0 ϑ m,δ (x)b(x, y) dxdy .
On the one hand, by (2.7), (2.13), and the Cauchy-Schwarz inequality,

̺ 2 = M 1 (f j,ε (t)) 2 ≤ M 0 (f j,ε (t))M 2 (f j,ε (t)) ≤ σM 0 (f j,ε (t)) , so that U j,ε (t) := ∞ 0 ∞ 0 (x + δ) m f j,ε (t, x)f j,ε (t, y) dydx = M 0 (f j,ε (t)) ∞ 0 (x + δ) m f j,ε (t, x) dx ≥ ̺ 2 σ ∞ 0 ϑ m,δ (x)f j,ε (t, x) dx .
On the other hand, we infer from (1.16c), (2.13), and the negativity of m that

V j,ε (t) := ∞ 0 a j,ε (y)f j,ε (t, y) y 0 ϑ m,δ (x)b(x, y) dxdy = ∞ 0 a j,ε (y)f j,ε (t, y) 1 0 (yz + δ) m B(z) dzdy ≤ ∞ 0 a j,ε (y)f j,ε (t, y) 1 0 (yz + δz) m B(z) dzdy ≤ a 0 b m ∞ 0 (x + δ) m ε 2 + min{x, j} γ f j,ε (t, x) dx . Since ∞ 0 (x + δ) m min{x, j} γ f j,ε (t, x) dx ≤ ε 2 ε 2/γ 0 (x + δ) m f j,ε (t, x) dx + ε 2(m-1)/γ ∞ ε 2/γ x γ+1 f j,ε (t, x) dx ≤ ε 2 ∞ 0 (x + δ) m f j,ε (t, x) dx + ε 2(m-1)/γ M γ+1,j,ε ≤ ε 2 ∞ 0 (x + δ) m f j,ε (t, x) dx + σε 2(m-1)/γ
by (2.13), we further obtain

V j,ε (t) ≤ a 0 b m 2ε 2 ∞ 0 ϑ m,δ (x)f j,ε (t, x) dx + σε 2(m-1)/γ .
Collecting the previous estimates and using the definition (2.20) of ε m,σ lead us to the differential inequality

d dt ∞ 0 ϑ m,δ (x)f j,ε (t, x) dx ≤ - εK 0 ̺ 2 σ ∞ 0 ϑ m,δ (x)f j,ε (t, x) dx + 2a 0 b m ε 2 ∞ 0 ϑ m,δ (x)f j,ε (t, x) dx + a 0 b m σε 2(m-1)/γ ≤ 2a 0 b m ε(ε -2ε m,σ ) ∞ 0 ϑ m,δ (x)f j,ε (t, x) dx + a 0 b m σε 2(m-1)/γ ≤ -2a 0 b m εε m,σ ∞ 0 ϑ m,δ (x)f j,ε (t, x) dx + a 0 b m σε 2(m-1)/γ .
After integration with respect to time, we end up with

∞ 0 ϑ m,δ (x)f j,ε (t, x) dx ≤ e -2a 0 bmεεm,σt ∞ 0 ϑ m,δ (x)f in (x) dx + σε -(γ+2-2m)/γ 2ε m,σ 1 -e -2a 0 bmεεm,σt ≤ max M m (f in ), µ m σ 2 ε -(γ+2-2m)/γ , t ≥ 0 .
Since the right-hand side of the previous inequality does not depend on δ ∈ (0, 1) and is finite, we may pass to the limit as δ → 0 and thereby complete the proof of Lemma 2.5.

Remark 2.6. It is worth mentioning here that the positivity of γ is only used in the proof of Lemma 2.5.

Integrability

Estimates. We now turn to weighted L p -estimates and actually derive two different estimates, one depending on ε but not on t, and the other one depending on t but not on ε.

For m ≥ 0, p ≥ 1, and h ∈ L p ((0, ∞), x m dx), we set

L m,p (h) := ∞ 0 x m |h(x)| p dx . (2.21)
Lemma 2.7. Consider m ∈ (λ, 1) and p ∈ (1, p 0 ] satisfying

1 < p < m + 1 λ + 1 and p ≤ m + γ γ , (2.22)
and assume that f in ∈ L p ((0, ∞), x m dx). Then

L m,p (f j,ε (t)) ≤ max L m,p (f in ), S j,ε (m, p) ε 2 (2.23
)

and 1 t t 0 ∞ 0 x m min{x, j} γ (f j,ε (s, x)) p dxds ≤ 1 a 0 t L m,p (f in ) + S j,ε (m, p) , (2.24)
where

S j,ε (m, p) := 2 p B p p M p (m+1+γ-p)/p,j,ε + M p (m+1+γp-p)/p,j,ε + ε 2 M p (m+1-p)/p,j,ε
and B p is defined in (2.8a).

Proof. We first note that (1.16a) and (2.22) ensure that

m + 1 + γp -p p ≥ m + 1 + γ -p p ≥ m + 1 -p p > λ ,
so that S j,ε (m, p) is well-defined and finite by Lemma 2.3.

Let t > 0. We first deal with the contribution of the coagulation term. As already observed in [START_REF] Banasiak | Analytic Methods for Coagulation-Fragmentation Models[END_REF][START_REF] Dubovskii | Mathematical theory of coagulation[END_REF][START_REF] Laurenc | The continuous coagulation-fragmentation equations with diffusion[END_REF][START_REF] Mischler | Existence globale pour l'équation de Smoluchowski continue non homogène et comportement asymptotique des solutions[END_REF], the sublinearity of x → x m and the monotonicity of x → K j,ε (x, y) for all y > 0 allow us to show that this contribution is negative. Indeed, it follows from the inequality

(x + y) m ≤ x m + y m , (x, y) ∈ (0, ∞) 2 ,
the symmetry of K j,ε , and Fubini's theorem that

P j,ε (t) := p ∞ 0 x m (f j,ε (t, x)) p-1 (C j,ε f j,ε )(t, x) dx = p 2 ∞ 0 ∞ 0 (x + y) m K j,ε (x, y)(f j,ε (t, x + y)) p-1 f j,ε (t, x)f j,ε (t, y) dydx -p ∞ 0 ∞ 0 x m K j,ε (x, y)(f j,ε (t, x)) p f j,ε (t, y) dydx ≤ p 2 ∞ 0 ∞ 0 (x m + y m ) K j,ε (x, y)(f j,ε (t, x + y)) p-1 f j,ε (t, x)f j,ε (t, y) dydx -p ∞ 0 ∞ 0 x m K j,ε (x, y)(f j,ε (t, x)) p f j,ε (t, y) dydx = p ∞ 0 ∞ 0 x m K j,ε (x, y)(f j,ε (t, x + y)) p-1 f j,ε (t, x)f j,ε (t, y) dydx -p ∞ 0 ∞ 0 x m K j,ε (x, y)(f j,ε (t, x)) p f j,ε (t, y) dydx .
We next deduce from the convexity inequality

pU p-1 V ≤ (p -1)U p + V p , (U, V ) ∈ [0, ∞) 2 , that P j,ε (t) ≤ (p -1) ∞ 0 ∞ 0 x m K j,ε (x, y)(f j,ε (t, x + y)) p f j,ε (t, y) dydx -(p -1) ∞ 0 ∞ 0 x m K j,ε (x, y)(f j,ε (t, x)) p f j,ε (t, y) dydx ≤ (p -1) ∞ 0 ∞ y (x -y) m K j,ε (x -y, y)(f j,ε (t, x)) p f j,ε (t, y) dxdy -(p -1) ∞ 0 ∞ y x m K j,ε (x, y)(f j,ε (t, x)) p f j,ε (t, y) dxdy .
Now, the monotonicity of x → x m and x → K j,ε (x, y) implies that

(x -y) m K j,ε (x -y, y) ≤ x m K j,ε (x, y) , 0 < y < x .
Consequently, P j,ε (t) ≤ 0 .

(2.25)

Concerning the contribution of the fragmentation term, it reads

Q j,ε (t) := p ∞ 0 x m (f j,ε (t, x)) p-1 (F j,ε f j,ε )(t, x) dx = -pa 0 Λ j (f j,ε (t)) -pa 0 ε 2 L m,p (f j,ε (t)) + R j,ε (t) , (2.26)
where

Λ j (f j,ε (t)) := ∞ 0 x m min{x, j} γ (f j,ε (t, x)) p dx and R j,ε (t) := p ∞ 0 a j,ε (y)f j,ε (t, y) y 0 x m b(x, y)(f j,ε (t, x)) p-1 dxdy = pa 0 ∞ 0 min{y, j} γ y -1 f j,ε (t, y) y 0 x m B xy -1 (f j,ε (t, x)) p-1 dxdy + pa 0 ε 2 ∞ 0 y -1 f j,ε (t, y) y 0 x m B xy -1 (f j,ε (t, x)) p-1 dxdy .
We infer from Hölder's inequality that y 0

x m B xy -1 (f j,ε (t, x)) p-1 dx = y 0

x m/p min{x, j} -γ(p-1)/p B xy -1 x m(p-1)/p min{x, j} γ(p-1)/p (f j,ε (t, x)) p-1 dx

≤ y 0 x m min{x, j} -γ(p-1) B xy -1 p dx 1/p y 0 x m min{x, j} γ (f j,ε (t, x)) p dx (p-1)/p . Since y 0 x m min{x, j} -γ(p-1) B xy -1 p dx 1/p ≤ y 0 x m-γ(p-1) + x m B xy -1 p dx 1/p ≤ y (m+1-γ(p-1))/p 1 0 z m-γ(p-1) B(z) p dz 1/p + y (m+1)/p 1 0 z m B(z) p dz 1/p , we further obtain y 0 x m B xy -1 (f j,ε (t, x)) p-1 dx ≤ y (m+1-γ(p-1))/p 1 0 z m-γ(p-1) B(z) p dz 1/p Λ j (f j,ε (t)) (p-1)/p + y (m+1)/p 1 0 z m B(z) p dz 1/p Λ j (f j,ε (t)) (p-1)/p .
Similarly, by Hölder's inequality,

y 0 x m B xy -1 (f j,ε (t, x)) p-1 dx = y 0 x m/p B xy -1 x m(p-1)/p (f j,ε (t, x)) p-1 dx ≤ y 0 x m B xy -1 p dx 1/p y 0 x m (f j,ε (t, x)) p dx (p-1)/p ≤ y (m+1)/p 1 0 z m B(z) p dz 1/p [L m,p (f j,ε (t))] (p-1)/p .
Since 0 ≤ m -γ(p -1) ≤ m and p ∈ [1, p 0 ] by (2.22), we infer from (2.8a) that

1 0 z m B(z) p dz ≤ 1 0 z m-γ(p-1) B(z) p dz ≤ B p p < ∞ .
Gathering the above estimates and using Young's inequality, we end up with

R j,ε (t) ≤ pa 0 B p M (m+1+γ-p)/p (f j,ε (t))Λ j (f j,ε (t)) (p-1)/p + pa 0 B p M (m+1+γp-p)/p (f j,ε (t))Λ j (f j,ε (t)) (p-1)/p + pa 0 ε 2 B p M (m+1-p)/p (f j,ε (t)) [L m,p (f j,ε (t))] (p-1)/p ≤ p -1 2 a 0 Λ j (f j,ε (t)) + 2 p-1 a 0 B p p M p (m+1+γ-p)/p,j,ε + p -1 2 a 0 Λ j (f j,ε (t)) + 2 p-1 a 0 B p p M p (m+1+γp-p)/p,j,ε + (p -1)a 0 ε 2 L m,p (f j,ε (t)) + a 0 ε 2 B p p M p (m+1-p)/p,j,ε . (2.27)
We then deduce from (2.26) and (2.27) that

Q j,ε (t) ≤ -a 0 Λ j (f j,ε (t)) + ε 2 L m,p (f j,ε (t)) + a 0 S j,ε (m, p) . ( 2
(t)) + a 0 Λ j (f j,ε (t)) + ε 2 L m,p (f j,ε (t)) ≤ a 0 S j,ε (m, p) (2.29)
for t > 0. We first infer from (2.29) that, for t > 0,

d dt L m,p (f j,ε (t)) + a 0 ε 2 L m,p (f j,ε (t)) ≤ a 0 S j,ε (m, p) .
Hence, after integration with respect to time,

L m,p (f j,ε (t)) ≤ e -a 0 ε 2 t L m,p (f in ) + S j,ε (m, p) ε 2 1 -e -a 0 ε 2 t ≤ max L m,p (f in ), S j,ε (m, p) ε 2 ,
from which (2.23) follows. We also infer from (2.29) that, for t > 0,

d dt L m,p (f j,ε (t)) + a 0 Λ j (f j,ε (t)) ≤ a 0 S j,ε (m, p) .
Integrating with respect to time and using the non-negativity of L m,p (f j,ε (t)), we obtain

a 0 t 0 Λ j (f j,ε (s)) ds ≤ L m,p (f in ) + a 0 tS j,ε (m, p)
for t > 0. Dividing the above inequality by a 0 t gives (2.24).

Combining the outcome of Lemma 2.5 and Lemma 2.7 leads to an ε-dependent L p -estimate for (f j,ε ) j≥2 for a suitable value of p.

Corollary 2.8. Let m 0 ∈ (m ⋆ , 0), m 1 ∈ (λ, 1), and p 1 ∈ (1, p 0 ) be such that

1 < p 1 < m 1 + 1 λ + 1 and p 1 ≤ m 1 + γ γ , (2.30)
and assume that f in ∈ X m 0 ∩ L p 1 ((0, ∞), x m 1 dx). For ε ∈ (0, ε m 0 ,σ ) and t ≥ 0,

L 0,p 2 (f j,ε (t)) ≤ max M m 0 (f in ), µ m 0 σ 2 ε -(γ+2-2m 0 )/γ + max L m 1 ,p 1 (f in ), S j,ε (m 1 , p 1 ) ε 2 ,
where

p 2 := m 1 m 1 -m 0 + p 1 |m 0 | m 1 -m 0 ∈ (1, p 1 ) . Proof. Since m 1 m 1 -m 0 m 0 + |m 0 | m 1 -m 0 m 1 = 0 , we infer from Young's inequality that, if h ∈ X m 0 ∩ L p 1 ((0, ∞), x m 1 dx), then h ∈ L p 2 (0, ∞) and L 0,p 2 (h) = h p 2 p 2 = ∞ 0 (x m 0 |h(x)|) m 1 /(m 1 -m 0 ) (x m 1 |h(x)|) |m 0 |/(m 1 -m 0 ) dx ≤ m 1 m 1 -m 0 ∞ 0 x m 0 |h(x)| dx + |m 0 | m 1 -m 0 ∞ 0 x m 1 |h(x)| p 1 dx ≤ M m 0 (|h|) + L m 1 ,p 1 (h) .
(2.31)

Now, consider t ≥ 0. As ε ∈ (0, ε m 0 ,σ ) and p 1 satisfies (2.30), Corollary 2.8 readily follows from Lemma 2.5 (with m = m 0 ), Lemma 2.7 (with (m, p) = (m 1 , p 1 )), and (2.31) (with h = f j,ε (t)).

Time Equicontinuity.

The last estimate to be derived in this section provides the time equicontinuity of the sequence (f j,ε ) j≥2 in L 1 (0, ∞), which is needed later to apply a variant of the Arzelà-Ascoli theorem.

Lemma 2.9. There is a positive constant C 5 > 0 such that

∂ t f j,ε (t) 1 ≤ C 5 σ + M 2 0,j,ε , t ≥ 0 .
Proof. Let t > 0. It follows from (2.4a), (2.9), and Fubini's theorem that

∂ t f j,ε (t) 1 ≤ 3 2 ∞ 0 ∞ 0 K j,ε (x, y)f j,ε (t, x)f j,ε (t, y) dydx + (1 + b 0 ) ∞ 0 a j,ε (x)f j,ε (t, x) dx ≤ 3K 0 2 ∞ 0 ∞ 0 x λ + y λ + 2ε f j,ε (t, x)f j,ε (t, y) dydx + a 0 (1 + b 0 ) ∞ 0 x γ + ε 2 f j,ε (t, x) dx ≤ 3K 0 M λ (f j,ε (t))M 0 (f j,ε (t)) + M 0 (f j,ε (t)) 2 + a 0 (1 + b 0 ) [M γ (f j,ε (t)) + M 0 (f j,ε (t))] .
We then infer from (2.7), (2.12a), (2.13), and the inequalities

x λ ≤ 1 + x , x γ ≤ 1 + x 1+γ , x ≥ 0 , that ∂ t f j,ε (t) 1 ≤ 3K 0 M 0 (f j,ε (t))M 1 (f j,ε (t)) + 2M 0 (f j,ε (t)) 2 + a 0 (1 + b 0 ) [M γ+1 (f j,ε (t)) + 2M 0 (f j,ε (t))] ≤ 3K 0 ̺ 2 + 3M 2 0,j,ε + a 0 (1 + b 0 ) σ + 1 + M 2 0,j,ε ≤ [3K 0 (̺ + 3) + 2a 0 (1 + b 0 )] σ + M 2 0,j,ε
, and the proof is complete.

3. Stationary solutions by a dynamical approach: ε ∈ (0, 1)

In this section, we fix ε ∈ (0, 1) and study the coagulation-fragmentation equation (1.20) with coagulation kernel K ε and overall fragmentation rate a ε given by

K ε (x, y) = K(x, y) + 2εK 0 , a ε (x) = a(x) + a 0 ε 2 , (x, y) ∈ (0, ∞) 2 ; (3.1)
that is,

∂ t f = C ε f + F ε f , (t, x) ∈ (0, ∞) 2 , (3.2a) f (0) = f in , x ∈ (0, ∞) , (3.2b)
where the coagulation and fragmentation operators C ε and F ε are defined in (1.20).

Several results are established in this section. We begin with the well-posedness of (3.2) for a suitable class of initial conditions, the existence of solutions being obtained by passing to the limit as j → ∞ in (2.4) (Section 3.1). We also establish the continuity of the solutions to (3.2) with respect to the initial condition for the weak topology of X 1 (Section 3.3) and construct an invariant set for the dynamics of (3.2) (Section 3.2). Combining the outcome of this analysis with a consequence of Tychonov's fixed point theorem provides the existence of a stationary solution to (3.2a) (Section 3.4). The estimates derived in the previous section are of course at the heart of the proofs of the results of this section.

We fix

m 0 ∈ (m ⋆ , 0) ∩ (-1, 0) , m 1 ∈ (λ, 1) , p 1 ∈ (1, p 0 ) , (3.3a) such that 1 < p 1 < m 1 + 1 λ + 1 and p 1 ≤ m 1 + γ γ . (3.3b)
We recall that (3.3) implies that

m 2 ∈ (λ, 1) and m 2 < m 1 + 1 + γ -p 1 p 1 < m 1 + 1 + γp 1 -p 1 p 1 ≤ 1 + γ , (3.4a)
where

m 2 := m 1 + 1 -p 1 p 1 < 1 . (3.4b)
We also fix ̺ > 0 and σ > 0 satisfying

σ > max {1, ̺, µ 2 , µ 2+γ } , (3.5) recalling that µ m is defined in Lemma 2.1 for m ≥ 2. We next define a subset Y ε of X + 1 as follows: h ∈ Y ε if and only if h ∈ X + 1 ∩ X m 0 ∩ X 2+γ , M 1 (h) = ̺ , (3.6a) max{M 2 (h), M 2+γ (h)} ≤ σ , M m 2 (h) ≤ µ m 2 + σ , (3.6b) M 0 (h) ≤ σ + µ 0 ε -1 , M m 0 (h) ≤ µ m 0 σ 2 ε -(γ+2-2m 0 )/γ , (3.6c) L m 1 ,p 1 (h) ≤ σ 1 ε -2 , (3.6d)
where

σ 1 := 2 p 1 B p 1 p 1 [2σ p 1 + 3(µ m 2 + σ) p 1 ] (3.7)
and B p 1 is defined in (2.8a).

3.1.

Well-posedness of (3.2). We begin with the well-posedness of (3.2) in Y ε , along with several estimates for its solutions.

Proposition 3.1. Consider ε ∈ (0, ε m 0 ,σ ) and f in ∈ Y ε , recalling that ε m 0 ,σ = 1 σ min 1, K 0 ̺ 2 4a 0 b m 0 is defined in (2.20
) with b m 0 given by (2.8a). There is a unique weak solution

Ψ ε (•, f in ) = f ε ∈ C([0, ∞), X + 0 ) ∩ C([0, ∞), X 1,w ) to (3.2) which satisfies d dt ∞ 0 ϑ(x)f ε (t, x) dx = 1 2 ∞ 0 ∞ 0 K ε (x, y)χ ϑ (x, y)f ε (t, x)f ε (t, y) dydx - ∞ 0 a ε (y)N ϑ (y)f ε (t, y) dy , (3.8)
for all t ≥ 0 and ϑ ∈ L ∞ (0, ∞), the functions χ ϑ and N ϑ being defined in (2.6), and possesses the following properties:

M 1 (f ε (t)) = ̺ , t ≥ 0 , (3.9a) sup t≥0 M m (f ε (t)) ≤ σ , m ∈ (1, 2 + γ] , (3.9b) sup t≥0 M m (f ε (t)) ≤ max M m (f in ), σ + µ m , m ∈ (λ, 1) , (3.9c) sup t≥0 M 0 (f ε (t)) ≤ σ + µ 0 ε -1 , (3.10a) sup t≥0 M m 0 (f ε (t)) ≤ µ m 0 σ 2 ε -(γ+2-2m 0 )/γ , (3.10b) sup t≥0 L m 1 ,p 1 (f ε (t)) ≤ σ 1 ε -2 , (3.11a) sup t≥0 L 0,p 2 (f ε (t)) ≤ κ ε := µ m 0 σ 2 ε -(γ+2-2m 0 )/γ + σ 1 ε -2 , (3.11b) and 1 t t 0 L m 1 +γ,p 1 (f ε (s)) ds ≤ 1 a 0 t L m 1 ,p 1 (f in ) + σ 1 , t > 0 . (3.12) Moreover, if f in ∈ X m for some m > 2 + γ, then f ε ∈ L ∞ ((0, ∞), X m ) and sup t≥0 M m (f ε (t)) ≤ max{M m (f in ), µ m } , (3.13)
the constant µ m being defined in Lemma 2.1.

Proof.

Step 1: Existence. Let j ≥ 2 and recall that f j,ε is the strong solution to the coagulationfragmentation equation (2.4), see Section 2. Since f in ∈ Y ε , it follows from (2.7) that

M 1 (f j,ε (t)) = ̺ , t ≥ 0 , j ≥ 2 , ( 3
M m 2 (f j,ε (t)) ≤ µ m 2 + σ , j ≥ 2 , (3.16)
while, since ε ∈ (0, ε m 0 ,σ ), we deduce from (2.12), (3.5), (3.6c), Lemma 2.4, and Lemma 2.5 that

sup t≥0 M 0 (f j,ε (t)) ≤ σ + µ 0 ε -1 , j ≥ 2 , (3.17)
sup t≥0 M m 0 (f j,ε (t)) ≤ µ m 0 σ 2 ε -(γ+2-2m 0 )/γ , j ≥ 2 . (3.18)
Finally, by (3.4), (3.15), and Hölder's and Young's inequalities,

M (m+1+γ-p 1 )/p 1 (f j,ε (t)) p 1 ≤ γ p 1 (1 + γ -m 2 ) M 1+γ (f j,ε (t)) p 1 + p 1 (1 + γ -m 2 ) -γ p 1 (1 + γ -m 2 ) M m 2 (f j,ε (t)) p 1 ≤ σ p 1 + M m 2 (f j,ε (t)) p 1 , and
M (m+1+γp 1 -p 1 )/p 1 (f j,ε (t)) p 1 ≤ γ 1 + γ -m 2 M 1+γ (f j,ε (t)) p 1 + 1 -m 2 1 + γ -m 2 M m 2 (f j,ε (t)) p 1 ≤ σ p 1 + M m 2 (f j,ε (t)) p 1
for t ≥ 0 and j ≥ 2, so that, using also (3.7) and (3.16),

S j,ε (m 1 , p 1 ) = 2 p 1 B p 1 p 1 sup t≥0 M (m+1+γ-p 1 )/p 1 (f j,ε (t)) p 1 + 2 p 1 B p 1 p 1 sup t≥0 M (m+1+γp 1 -p 1 )/p 1 (f j,ε (t)) p 1 + 2 p 1 B p 1 p 1 ε 2 sup t≥0 M m 2 (f j,ε (t)) p 1 ≤ 2 p 1 B p 1 p 1 [2σ p 1 + 3 (µ m 2 + σ) p 1 ] = σ 1 . ( 3
t≥0 L m 1 ,p 1 (f j,ε (t)) ≤ σ 1 ε -2 , j ≥ 2 . (3.20)
A straightforward consequence of (3.3b), (3.5), (3.9), (3.18), (3.20), and Corollary 2.8 is the bound

sup t≥0 L 0,p 2 (f j,ε (t)) ≤ µ m 0 σ 2 ε -(γ+2-2m 0 )/γ + σ 1 ε -2 = κ ε , j ≥ 2 . (3.21)
Now, introducing the set

W ε :=    h ∈ X m 0 ∩ X 2+γ ∩ L p 2 (0, ∞) : M 2+γ (h) ≤ σ , max{M m 0 (h), L 0,p 2 (h)} ≤ κ ε    , (3.22)
it readily follows from (3.15), (3.18), and (3.21) that

f j,ε (t) ∈ W ε , t ≥ 0 , j ≥ 2 , (3.23)
while the Dunford-Pettis theorem ensures that W ε is a relatively sequentially weakly compact subset of X m (

for any m ∈ (m 0 , 2 + γ), and in particular of X 0 . Moreover, it follows from (3.17) and Lemma 2.9 that, for 0 ≤ t 1 ≤ t 2 and j ≥ 2,

f j,ε (t 2 ) -f j,ε (t 1 ) 1 ≤ t 2 t 1 ∂ t f j,ε (t) 1 dt ≤ C 5 σ + σ + µ 0 ε -1 2 (t 2 -t 1 ) . (3.25)
Consequently, (f j,ε ) j≥2 is equicontinuous at each t ≥ 0 for the norm-topology of L 1 (0, ∞), and thus it is also equicontinuous for the weak topology of L 1 (0, ∞). This property, along with (3.23) and the relative compactness (3.24) of W ε , allows us to apply a variant of the Arzelà-Ascoli theorem [START_REF] Vrabie | C 0 -semigroups and applications[END_REF]Theorem A.3.1] to conclude that there are a subsequence of (f j,ε ) j≥2 (possibly depending on ε but not relabeled) and

f ε ∈ C([0, ∞), X 0,w ) such that f j,ε -→ f ε in C([0, T ], X 0,w ) for all T > 0 . (3.26)
A first consequence of (3.26) is that f ε (t) ∈ X + 0 for all t ≥ 0. It next follows from (3.14), (3.15), (3.17), (3.18), and (3.26) by a weak lower semicontinuity argument that f ε satisfies (3.9b), (3.10a), (3.10b), and

M 1 (f ε (t)) ≤ ̺ , t ≥
f j,ε -→ f ε in C([0, T ], X 1,w ∩ X γ,w ) for all T > 0 . (3.27)
Recalling (3.14), we readily infer from (3.27) that f ε satisfies the mass conservation (3.9a). We employ again weak lower semicontinuity arguments to deduce (3.11) and [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF], see also [START_REF] Banasiak | Analytic Methods for Coagulation-Fragmentation Models[END_REF][START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF]From the discrete to the continuous coagulation-fragmentation equations[END_REF], to deduce from (2.5) that f ε is a weak solution to (3.2), in the sense that it satisfies (3.8). Furthermore, we may argue as in the proof of Lemma 2.9 with the help of (3.9a), (3.9b), and (3.10a) to show that ∂ t f ε (t) belongs to X 0 for any t ≥ 0 and satisfies

1 t t 0 R 0 x m+γ (f ε (s, x)) p dxds ≤ 1 a 0 t L m 1 ,p 1 (f in ) + σ 1 , t > 0 , R ≥ 1 , (3.28
∂ t f ε (t) 1 ≤ C 5 σ + σ + µ 0 ε -1 2 , t ≥ 0 , (3.29)
the constant C 5 being defined in Lemma 2.9.

Step 2: Uniqueness. It is a consequence of [4, Theorem 8.2.55] (with ℓ(x) = 1 + x max{1,γ} , x > 0, and ζ = 1), see also [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF].

Step 3: Higher moments. Finally, if f in ∈ X m for some m > 2+γ, then the proof of (3.13) relies on a weak lower semicontinuity argument as that of (3.9b) and follows from (3.26) and Lemma 2.1.

Invariant Set.

As a consequence of the various estimates derived in Proposition 3.1, we construct a subset

Z ε of Y ε which is left invariant by Ψ ε . Specifically, h ∈ Z ε if and only if h ∈ Y ε ∩ m>2+γ X m , (3.30a)
M m (h) ≤ µ m , m > 2 + γ , (3.30b)
M m (h) ≤ σ , m ∈ (1, 2 + γ] , (3.30c)
M m (h) ≤ σ + µ m , m ∈ (λ, 1) . (3.30d) Proposition 3.2. Consider ε ∈ (0, ε m 0 ,σ ) and f in ∈ Z ε . Then Ψ ε (t, f in ) ∈ Z ε for all t ≥ 0.
Proof. Set f ε := Ψ ε (•, f in ) and consider t > 0. We first deduce from (3.9a), (3.9b) (with m = 2 and m = 2 + γ), (3.9c) (with m = m 2 ), (3.10), and (3.11a) that f ε (t) ∈ Y ε . In addition, f ε (t) ∈ X m for all m > 2 + γ and satisfies (3.30b) by (3.13), while (3.30c) and (3.30d) follow from (3.9b) and (3.9c), respectively.

3.3. Dynamical System in X 1,w . We go on with the continuity properties of the map f in → Ψ ε (., f in ) defined in Proposition 3.1 and actually show that Ψ ε is a dynamical system on Y ε for the weak topology of X 1 .

Proposition 3.3. Consider ε ∈ (0, ε m 0 ,σ ), f in ∈ Y ε , and a sequence (f in n ) n≥1 of initial conditions in Y ε such that f in n ⇀ f in in X 1 . (3.31)
Then, for any

T > 0, Ψ ε (•, f in n ) -→ Ψ ε (•, f in ) in C([0, T ], X 1,w ) .
Proof. For n ≥ 1 we put f ε,n := Ψ ε (•, f in n ). On the one hand, it follows from (3.9b), (3.10b), and

(3.11b) that f ε,n (t) ∈ W ε , t ≥ 0 , n ≥ 1 , (3.32)
recalling that the set W ε is defined in (3.22). On the other hand, let 0 ≤ t 1 < t 2 and n ≥ 1. We infer from (3.29) that

f ε,n (t 2 ) -f ε,n (t 1 ) 1 ≤ C 5 σ + σ + µ 0 ε -1 2 (t 2 -t 1 ) .
Combining this estimate with (3.9b) gives, for R > 0,

∞ 0 x |f ε,n (t 2 , x) -f ε,n (t 1 , x)| dx ≤ R f ε,n (t 2 ) -f ε,n (t 1 ) 1 + 1 R ∞ R x 2 (f ε,n (t 2 , x) + f ε,n (t 1 , x)) dx ≤ C 5 σ + σ + µ 0 ε -1 2 (t 2 -t 1 ) + 2σ R . Now, taking R = 1/ √ t 2 -t 1 in the previous inequality, we end up with ∞ 0 x |f ε,n (t 2 , x) -f ε,n (t 1 , x)| dx ≤ C 5 σ + σ + µ 0 ε -1 2 + 2σ √ t 2 -t 1 .
Consequently, the sequence (f ε,n ) n≥1 is equicontinuous at each t ≥ 0 for the norm-topology of X 1 and thus also for the weak topology of X 1 . Recalling (3.24) and (3.32), we are again in a position to use the variant of the Arzelà-Ascoli theorem stated in [START_REF] Vrabie | C 0 -semigroups and applications[END_REF]Theorem A.3.1] to deduce that there are

F ε ∈ C([0, ∞), X 1,w ) and a subsequence (f ε,n k ) k≥1 of (f ε,n ) n≥1 (possibly depending on ε) such that f ε,n k -→ F ε in C([0, T ], X 1,w ) (3.33)
for any T > 0. Since f ε,n k satisfies (3.9), (3.10), (3.11), (3.12), and (3.33) for k ≥ 1, we can argue as in Step 1 of the proof of Proposition 3.1 to establish that F ε is a weak solution to (3.2) with initial condition f in and also satisfies (3.9), (3.10), (3.11), and (3.12), along with

f ε,n k -→ F ε in C([0, T ], X 0,w ∩ X γ,w )
for any T > 0. The uniqueness assertion in Proposition 3.1 then guarantees that

F ε = Ψ ε (•, f in ).
A consequence of the above analysis is that Ψ ε (•, f in ) is the only cluster point of the sequence (f ε,n ) n≥1 in the space C([0, T ], X 1,w ), whatever the value of T > 0. Together with the compactness of (f ε,n ) n≥1 , this observation ensures that it is the whole sequence (f ε,n ) n≥1 which converges to Ψ ε (•, f in ) in C([0, T ], X 1,w ) for any T > 0, thereby completing the proof of Proposition 3.3. (3.2). Thanks to the outcome of Sections 3.1-3.3, we are now in a position to prove the existence of at least one stationary weak solution ϕ ε to the coagulationfragmentation equation (3.2) for ε ∈ (0, ε m 0 ,σ ), along with some estimates on ϕ ε which will be needed in Section 3.5 to carry out the limit ε → 0. Theorem 3.4. For ε ∈ (0, ε m 0 ,σ ), the coagulation-fragmentation equation (3.2a) has a stationary weak solution

Stationary Solution to

ϕ ε ∈ Z ε ∩ L p 1 ((0, ∞), x m 1 +γ dx) satisfying 1 2 ∞ 0 ∞ 0 K ε (x, y)χ ϑ (x, y)ϕ ε (x)ϕ ε (y) dydx = ∞ 0 a ε (x)N ϑ (x)ϕ ε (x) dx (3.34)
for all ϑ ∈ L ∞ (0, ∞) and

L m 1 +γ,p 1 (ϕ ε ) ≤ σ 1 , (3.35)
the constant σ 1 being defined in (3.7).

Proof. Let ε ∈ (0, ε m 0 ,σ ). By Propositions 3.1 and 3.3, Ψ ε is a dynamical system on Y ε for the weak topology of X 1 and, according to Proposition 3.2, the subset Z ε of Y ε is invariant under the action of Ψ ε ; that is, Ψ ε (t, Z ε ) ⊂ Z ε for all t ≥ 0. Since x → ̺ -1 e -x/̺ belongs to Z ε , the set Z ε is a non-empty convex and closed subset of X 1 . In addition, owing to the Dunford-Pettis theorem, Z ε is a sequentially weakly compact subset of X 1 . Thanks to these properties, we infer from [12, Theorem 1.2] that there is ϕ ε ∈ Z ε such that Ψ ε (t, ϕ ε ) = ϕ ε for all t ≥ 0. In other words, ϕ ε is a stationary solution to (3.2) as described in Proposition 3.1, and the weak formulation (3.34) readily follows from (3.8). We also deduce from (3.12) that, for t > 0,

L m 1 +γ,p 1 (ϕ ε ) = 1 t t 0 L m 1 +γ,p 1 (ϕ ε ) ds ≤ 1 a 0 t L m 1 ,p 1 (ϕ ε ) + σ 1 .
Letting t → ∞ in the above inequality gives (3.35) and completes the proof of Theorem 3.4.

3.5. Proof of Theorem 1.1. We are left with investigating the limit ε → 0 (if any) of the family (ϕ ε ) ε∈(0,εm 0 ,σ ) of stationary weak solutions to (1.20) constructed in Theorem 3.4. To this end, we first observe that, since ϕ ε ∈ Z ε for all ε ∈ (0, ε m 0 ,σ ), it satisfies

M 1 (ϕ ε ) = ̺ , ε ∈ (0, ε m 0 ,σ ) , (3.36)
M m (ϕ ε ) ≤ µ m , m > 2 + γ , ε ∈ (0, ε m 0 ,σ ) , (3.37)
M m (ϕ ε ) ≤ σ , m ∈ (1, 2 + γ] , ε ∈ (0, ε m 0 ,σ ) , (3.38)
M m (ϕ ε ) ≤ σ + µ m , m ∈ (λ, 1) , ε ∈ (0, ε m 0 ,σ ) , (3.39)
and

L m 1 +γ,p 1 (ϕ ε ) ≤ σ 1 , ε ∈ (0, ε m 0 ,σ ) , (3.40)
see the definition (3.30) of Z ε . We claim that these estimates guarantee that (ϕ ε ) ε∈(0,εm 0 ,σ ) is relatively sequentially weakly compact in X m for any m > λ .

(3.41) Indeed, let E be a measurable subset of (0, ∞) with finite measure and R > 1. We infer from Hölder's inequality that, for ε ∈ (0, ε m 0 ,σ ),

E x m ϕ ε (x) dx ≤ 1/R 0 x m ϕ ε (x) dx + R 1/R x m 1 E (x)ϕ ε (x) dx + ∞ R x m ϕ ε (x) dx ≤ R (λ-m)/2 1/R 0 x (m+λ)/2 ϕ ε (x) dx + R m |E| (p 1 -1)/p 1 R 1/R ϕ ε (x) p 1 dx 1/p 1 + R -2-γ ∞ R x m+2+γ ϕ ε (x) dx ≤ R (λ-m)/2 M (m+λ)/2 (ϕ ε ) + R (mp 1 +m 1 +γ)/p 1 |E| (p 1 -1)/p 1 L m 1 +γ,p 1 (ϕ ε ) 1/p 1 + R -2-γ M m+2+γ (ϕ ε ) .
We now infer from (3.37), (3.38), (3.39), and (3.40) that

E x m ϕ ε (x) dx ≤ A m,σ R (λ-m)/2 + R (mp 1 +m 1 +γ)/p 1 |E| (p 1 -1)/p 1 + R -2-γ , (3.42)
with A m,σ := sup ε∈(0,εm 0 ,σ )

M (m+λ)/2 (ϕ ε ) + σ 1/p 1 1 + µ m+2+γ < ∞ . Introducing η Xm (δ) := sup E x m ϕ ε (x) dx : |E| < δ , ε ∈ (0, ε m 0 ,σ ) , δ ∈ (0, 1) ,
we deduce from (3.42) that

η Xm (δ) ≤ A m,σ R (λ-m)/2 + R (mp 1 +m 1 +γ)/p 1 δ (p 1 -1)/p 1 + R -2-γ . Hence, since p 1 > 1, lim sup δ→0 η Xm (δ) ≤ A m,σ R (λ-m)/2 + R -2-γ .
We finally let R → ∞ to conclude that We now infer from (3.41) and the reflexivity of L p 1 ((0, ∞), x m 1 +γ dx) that there are a subsequence (ϕ ε k ) k≥1 of the family (ϕ ε ) ε∈(0,εm 0 ,σ ) and

ϕ ∈ X + 1 ∩ L p 1 ((0, ∞), x m 1 +γ dx) ∩ m>λ X m (3.45) such that ϕ ε k ⇀ ϕ in X m , m > λ , (3.46)
ϕ ε k ⇀ ϕ in L p 1 ((0, ∞), x m 1 +γ dx) .
A straightforward consequence of (3.36) and (3.46) (with m = 1) is that

M 1 (ϕ) = ̺ . (3.47)
Let us now check that ϕ is a stationary weak solution to (1.1), as described in Theorem 1.1 (s3). To this end, we consider ϑ ∈ Θ 1 and first note that

|χ ϑ (x, y)| ≤ 2 ϑ ′ ∞ min{x, y} , (x, y) ∈ (0, ∞) 2 , (3.48)
and

|N ϑ (x)| ≤ 2 ϑ ′ ∞ x , x > 0 , (3.49
) by (1.16c) and (1.16b).

Let us begin with the coagulation term. By (3.36), (3.39), and Hölder's inequality,

2ε k K 0 ∞ 0 ∞ 0 χ ϑ (x, y)ϕ ε k (x)ϕ ε k (y) dydx ≤ 4ε k K 0 ϑ ′ ∞ M (λ+1)/2 (ϕ ε k )M (1-λ)/2 (ϕ ε k ) ≤ 4ε k K 0 ϑ ′ ∞ M (λ+1)/2 (ϕ ε k )M 1 (ϕ ε k ) (1-λ)/2 M 0 (ϕ ε k ) (1+λ)/2 ≤ 4ε k K 0 ϑ ′ ∞ (σ + µ (λ+1)/2 )̺ (1-λ)/2 M 0 (ϕ ε k ) (1+λ)/2 . Since ϕ ε k ∈ Z ε k ⊂ Y ε k , we further deduce from (3.6c) that 2ε k K 0 ∞ 0 ∞ 0 χ ϑ (x, y)ϕ ε k (x)ϕ ε k (y) dydx ≤ 4ε k K 0 ϑ ′ ∞ (σ + µ (λ+1)/2 )̺ (1-λ)/2 (σ + µ 0 ε -1 k ) (1+λ)/2 ≤ 4K 0 ϑ ′ ∞ (σ + µ (λ+1)/2 )̺ (1-λ)/2 (σ + µ 0 ) (1+λ)/2 ε (1-λ)/2 k . Consequently, lim k→∞ 2ε k K 0 ∞ 0 ∞ 0 χ ϑ (x, y)ϕ ε k (x)ϕ ε k (y) dydx = 0 . (3.50) Next, by (3.48), |χ ϑ (x, y)| x (2β+1-λ)/2 y (2α+1-λ)/2 ≤ 2 ϑ ′ ∞ , (x, y) ∈ (0, ∞) ,
and, since

χ ϑ (x, y)x α y β ϕ ε k (x)ϕ ε k (y) = χ ϑ (x, y) x (2β+1-λ)/2 y (2α+1-λ)/2 x (1+λ)/2 ϕ ε k (x)y (1+λ)/2 ϕ ε k (y) , it follows from (3.46) (with m = (1 + λ)/2) that lim k→∞ ∞ 0 ∞ 0 χ ϑ (x, y)x α y β ϕ ε k (x)ϕ ε k (y) dydx = lim k→∞ ∞ 0 ∞ 0 χ ϑ (x, y) x (2β+1-λ)/2 y (2α+1-λ)/2 x (1+λ)/2 ϕ ε k (x)y (1+λ)/2 ϕ ε k (y) dydx = ∞ 0 ∞ 0 χ ϑ (x, y) x (2β+1-λ)/2 y (2α+1-λ)/2 x (1+λ)/2 ϕ(x)y (1+λ)/2 ϕ(y) dydx = ∞ 0 ∞ 0 χ ϑ (x, y)x α y β ϕ(x)ϕ(y) dydx . (3.51) Similarly, lim k→∞ ∞ 0 ∞ 0 χ ϑ (x, y)x β y α ϕ ε k (x)ϕ ε k (y) dydx = ∞ 0 ∞ 0 χ ϑ (x, y)x β y α ϕ(x)ϕ(y) dydx . (3.52)
For the fragmentation term, it readily follows from (3.36) and (3.49) that

a 0 ε 2 k ∞ 0 N ϑ (x)ϕ ε k (x) dx ≤ 2a 0 ε 2 k ϑ ′ ∞ M 1 (ϕ ε k ) = 2a 0 ε 2 k ϑ ′ ∞ ̺ . Hence, lim k→∞ a 0 ε 2 k ∞ 0 N ϑ (x)ϕ ε k (x) dx = 0 . (3.53)
We finally infer from (3.46) (with m = 1 + γ) and (3.49) that

lim k→∞ ∞ 0 x γ N ϑ (x)ϕ ε k (x) dx = lim k→∞ ∞ 0 N ϑ (x) x x 1+γ ϕ ε k (x) dx = ∞ 0 N ϑ (x) x x 1+γ ϕ(x) dx = ∞ 0 x γ N ϑ (x)ϕ(x) dx . ( 3

Small Size Behaviour

This section is devoted to the proof of Proposition 1.2. The starting point is the finiteness of some moments of order lower than λ when γ ≥ α. -

If γ > α, then ϕ ∈ X α ; -If γ = α, then ϕ ∈ X β .
Proof. For δ ∈ (0, 1), we set ζ 0,δ (x) = x max{x, δ} -1 , x > 0. Then ζ 0,δ ∈ Θ 1 and satisfies

-χ ζ 0,δ (x, y) ≥ 1 (δ,∞) 2 (x, y) , (x, y) ∈ (0, ∞) 2 , -N ζ 0,δ (x) ≤ b 0 1 (δ,∞) (x) , x > 0 .
It then follows from Theorem 1.1 (s3) that

K 0 ∞ δ x α ϕ(x) dx ∞ δ y β ϕ(y) dy = 1 2 ∞ δ ∞ δ K(x, y)ϕ(x)ϕ(y) dydx ≤ - 1 2 ∞ 0 ∞ 0 K(x, y)χ ζ 0,δ (x, y)ϕ(x)ϕ(y) dydx = - ∞ 0 a(x)N ζ 0,δ (x)ϕ(x) dx ≤ a 0 b 0 ∞ δ x γ ϕ(x) dx . (4.1)
-If γ > α, then we infer from Theorem 1.1 (s2) and Hölder's inequality that

∞ δ x γ ϕ(x) dx ≤ ∞ δ x α ϕ(x) dx 1/(1+γ-α) ∞ δ x 1+γ ϕ(x) dx (γ-α)/(1+γ-α) ≤ M 1+γ (ϕ) (γ-α)/(1+γ-α) ∞ δ x α ϕ(x) dx 1/(1+γ-α) and ∞ δ yϕ(y) dy 2-β ≤ ∞ δ y β ϕ(y) dy ∞ δ y 2 ϕ(y) dy 1-β ≤ M 2 (ϕ) 1-β ∞ δ y β ϕ(y) dy .
Combining (4.1) and the above inequalities gives

M 2 (ϕ) β-1 ∞ δ yϕ(y) dy 2-β ∞ δ x α ϕ(x) dx (γ-α)/(1+γ-α) ≤ a 0 b 0 K 0 M 1+γ (ϕ) (γ-α)/(1+γ-α) . Consequently, ∞ δ yϕ(y) dy (2-β)(1+γ-α)/(γ-α) ∞ δ x α ϕ(x) dx ≤ M 1+γ (ϕ) a 0 b 0 M 2 (ϕ) 1-β K 0 (1+γ-α)/(γ-α)
.

Owing to Theorem 1.1 (s1) and the positivity of ̺, we can take the limit δ → 0 in the previous inequality to deduce that ϕ ∈ X α .

-If γ = α, then (4.1) gives, since ϕ ≡ 0 by Theorem 1.1 (s1), ∞ δ y β ϕ(y) dy ≤ a 0 b 0 K 0 for δ small enough, which obviously implies that ϕ ∈ X β after taking the limit δ → 0.

< M β (ϕ) < ∞ and M γ (ϕ) < ∞ . (4.2)
This implies that there is δ 0 ∈ (0, 1) such that

r δ := K 0 2a 0 b m ∞ δ y β ϕ(y) dy 1/(γ-α) > δ , δ ∈ [0, δ 0 ) . (4.3)
Next, for δ ∈ (0, δ 0 ), we define the function ζ m,δ by ζ m,δ (x) := x max{x, δ} m-1 , x > 0, and note that ζ m,δ belongs to Θ 1 . Moreover, since m < 0,

-for (x, y) ∈ (δ, ∞) 2 , -χ ζ m,δ (x, y) = x m + y m -(x + y) m ≥ x m ; -for (x, y) ∈ (δ, ∞) × (0, δ), -χ ζ m,δ (x, y) = x m + yδ m-1 -(x + y) m ≥ 0 ; -for (x, y) ∈ (0, δ) × (δ, ∞), -χ ζ m,δ (x, y) = xδ m-1 + y m -(x + y) m ≥ 0 ; -for (x, y) ∈ (0, δ) 2 such that x + y > δ, -χ ζ m,δ (x, y) = xδ m-1 + yδ m-1 -(x + y) m ≥ (x + y) δ m-1 -(x + y) m-1 ≥ 0 ; -for (x, y) ∈ (0, δ) 2 such that x + y < δ, -χ ζ m,δ (x, y) = xδ m-1 + yδ m-1 -(x + y)δ m-1 = 0 .
Also, by (1.16c) and (1.16b), -for x ∈ (0, δ), Thanks to (4.2), we may let δ → 0 in the above inequality and use Fatou's lemma to find

-N ζ m,δ (x) = δ m-1 x 0 yb(y, x) dy -δ m-1 x = 0 ; -for x > δ, -N ζ m,δ (x)
K 0 M β (ϕ) 2 ∞ 0
x α+m ϕ(x) dx ≤ a 0 b m r m 0 M γ (ϕ) .

Hence, ϕ ∈ X α+m for any m ∈ (m ⋆ , 0) which, together with Theorem 1.1 (s2) and an interpolation argument implies that ϕ ∈ X α+m for any m > m ⋆ .

To prove the second assertion in (m1) when m ⋆ > -∞ and b m⋆ = ∞, we argue by contradiction and assume that ϕ ∈ X α+m⋆ . Then, owing to (1.15b) and the assumption γ > α, M := max {M α+m⋆ (ϕ), M β+m⋆ (ϕ), M α (ϕ), M β (ϕ), M γ+m⋆ (ϕ)} < ∞ .

≤ 2K 0 M 2 .
Hence, using again (4.5),

a 0 R ∞ √ δ x γ+m⋆ ϕ(x) dx ≤ a 0 M γ+m⋆ (ϕ) + 2K 0 M 2 ≤ a 0 M + 2K 0 M 2 .
Taking the limit δ → 0 gives a 0 RM γ+m⋆ (ϕ) ≤ a 0 M + 2K 0 M 2 .

The above inequality being valid for all R > 1, we let R → ∞ to conclude that M γ+m⋆ (ϕ) = 0; that is, ϕ ≡ 0, which contradicts Theorem 1.1 (s1). (m4): α > γ. As in the proof of Lemma 2.3, we use a decomposition technique in the spirit of [START_REF] Fournier | Existence of self-similar solutions to Smoluchowski's coagulation equation[END_REF]Lemma 3.1] and [4, Lemma 8.2.12], along with a truncation procedure, to estimate the contribution of the coagulation term. More precisely, for m ∈ (λ -γ, λ), we deduce from (1.15b) and the assumption α > γ > 0 that 0 ≤ λ -2α < λ -2γ < m -γ < m < λ < 1 .

We define

ω := 2/(m + γ -λ) > 0 , y i := i -ω , i ≥ 1 , and set ζ i (x) := x max{x, y i } m-γ , x > 0, i ≥ 2. Clearly, ζ i ∈ Θ 1 for all i ≥ 2 and we infer from the convexity and monotonicity of x → x m-γ-1 that, -for (x, y) ∈ (y i , ∞) We then infer from (4.8) and (4.9) that there is c 1 (m) > 0 depending only on K 0 , α, β, a 0 , γ, B, ̺, and m such that The above inequality being valid for any I ≥ 2 with a right-hand side which does not depend on I ≥ 2, we may take the limit I → ∞ to conclude that ϕ ∈ X m and complete the proof of Proposition 1.2.

K 0 (1 + γ -m)

Proposition 1 . 2 .

 12 Consider ̺ > 0 and let ϕ be a stationary weak solution to (1.1) satisfying the properties (s1)-(s3) stated in Theorem 1.1. (m1) If γ > α, then ϕ ∈ X m for any m > α + m ⋆ , where

  0 . A similar argument allows us to deduce (3.9c) from Lemma 2.3 and (3.26). We then combine the just established property (3.9b) with (3.15) and (3.26) to improve the convergence (3.26) to

lim δ→0 η

 δ→0 Xm (δ) = 0 . (3.43) Similarly, for ε ∈ (0, ε m 0 ,σ ) and R > 1, it follows from (3.37) that ∞ R x m ϕ ε (x) dx ≤ R -2-γ µ m+2+γ ,and thus lim R→∞ sup ε∈(0,εm 0 ,σ ) ∞ R x m ϕ ε (x) dx = 0 . (3.44) The claim (3.41) is then a consequence of (3.43), (3.44), and the Dunford-Pettis theorem.

  .54) Collecting (3.50), (3.51), (3.52), (3.53), and (3.54) allows us to take the limit ε k → 0 in (3.34) and conclude that ϕ is a stationary weak solution to (1.1) in the sense of Theorem 1.1 (s3). Recalling (3.45) and (3.47), we have shown that ϕ satisfies the properties (s1)-(s3) stated in Theorem 1.1.

Lemma 4 . 1 .

 41 Let ̺ > 0 and consider a stationary weak solution ϕ to (1.1) satisfying the properties (s1)-(s3) stated in Theorem 1.1.

= δ 0 0 yx 0 ∞ 0 χχxx

 0000 yδ m-1 b(y, x) dy + x δ y m b(y, x) dy -x m ≤ x m b(y, x) dy = b m x m .We infer from Theorem 1.1 (s3) and the previous inequalities that α+m y β ϕ(x)ϕ(y) dydx ≤ K 0 ∞ ζ m,δ (x, y)x α y β ϕ(x)ϕ(y) ζ m,δ (x, y)K(x, y)ϕ(x)ϕ(y) dydx = a 0 ∞ 0 x γ N ζ m,δ (x)ϕ(x) dx ≤ a 0 b m ∞ δ x γ+m ϕ(x) dx . α+m ϕ(x) dx ≤ a 0 b m ∞ δ x γ+m ϕ(x) dx . (4.4)Now, since γ > α, it follows from (4.2) and (4.3) thata 0 b m ∞ δ x γ+m ϕ(x) dx ≤ a 0 b m r γ-α δ r δ δ x α+m ϕ(x) dx + a 0 b m r m δ ∞ r δ x γ ϕ(x) dx ≤ a 0 b m r γ-α δ ∞ δ x α+m ϕ(x) dx + a 0 b m r m δ M γ (ϕ) .Combining this inequality with (4.3) and (4α+m ϕ(x) dx ≤ a 0 b m r m δ M γ (ϕ) .

(4. 5 ) 0 ∞ 0 x

 500 Consider next R > 1. Since b m⋆ = ∞, there is δ R ∈ (0, 1) such that 1 √ δ z m⋆ B(z) dz ≥ R , δ ∈ (0, δ R ) .(4.6)Fix δ ∈ (0, δ R ). It follows from the negativity of m ⋆ and the definition ofζ m⋆,δ that 0 ≤ -χ ζ m⋆ ,δ (x, y) ≤ x m⋆ + y m⋆ , (x, y) ∈ (0, ∞) 2 ,and-N ζ m⋆ ,δ (x) ≥ 0 , x > 0 , while (4.6) entails that, for x > √ δ, -N ζ m⋆ ,δ (x) ≥ 1 δ/x z m⋆ B(z) dz -1 x m⋆ ≥ 1 √ δ z m⋆ B(z) dz -1 x m⋆ ≥ (R -1)x m⋆ .Since ζ m⋆,δ ∈ Θ 1 , we infer from (4.5), Theorem 1.1 (s3), and the previous inequalities thata 0 (R -1) , y)χ ζ m⋆,δ (x, y)ϕ(x)ϕ(y) dydx = -K 0 ∞ α y β χ ζ m⋆,δ (x, y)ϕ(x)ϕ(y) dydx ≤ K 0 [M α+m⋆ (ϕ)M β (ϕ) + M β+m⋆ (ϕ)M α (ϕ)]

2 , 1 i- 1 i+ yy m-γ- 1 i- 1 ≥ 1 i- 1 i+

 2111111 -χ ζ i (x, y) = x m-γ + y m-γ -(x + y) m-γ = x x m-γ-1 -(x + y) m-γ-1 + y y m-γ-1 -(x + y) m-γ-1 ≥ 2(1 + γ -m)xy(x + y) m-γ-2 ; -for (x, y) ∈ (y i , ∞) × (0, y i ), -χ ζ i (x, y) = x m-γ + yy m-γ-1 i -(x + y) m-γ = x x m-γ-1 -(x + y) m-γ-1 + y y m-γ-(x + y) m-γ-1 ≥ 0 ; -for (x, y) ∈ (0, y i ) × (y i , ∞), -χ ζ i (x, y) = xy m-γ-1 i + y m-γ -(x + y) m-γ ≥ 0 ; -for (x, y) ∈ (0, y i ) 2 such that x + y > y i , -χ ζ i (x, y) = xy m-γ-(x + y) m-γ-(x + y) y m-γ-(x + y) m-γ-1 ≥ 0 ; -for (x, y) ∈ (0, y i ) 2 such that x + y < y i , -χ ζ i (x, y) = xy m-γ-yy m-γ-1 i -(x + y)y m-γ-1 i = 0 .Also, by (1.16c) and (1.16b),-for x ∈ (0, y i ),-N ζ i (x) = y m-γγ b(y, x) dy -x m-γ ≤ x 0 y m-γ b(y, x) dy = b m-γ x m-γ .Next, since m < (λ + 2)/2, it follows from the Cauchy-Schwarz inequality that

1 y I x m ϕ(x) dx 2 .( 4 . 10 )c 1

 124101 (λ+2)/2 (x + y) m-γ-2 ϕ(x)ϕ(y) dydx ≥ c 1 (m)a 0 b m-γIn addition, since m < 1, we infer from Theorem 1.1 (s1) that 7), (4.10), and (4.11) and using the Cauchy-Schwarz inequality, we end up with (m) 2 ̺ 2 1/2 .

  ) from (2.23), (2.24), (3.3b), (3.6d), (3.19), (3.20), (3.21), and (3.26). As the right-hand side of (3.28) does not depend on R, we may let R → ∞ in (3.28) and use Fatou's lemma to obtain (3.12). Now, owing to (1.15), (1.16), (3.26), and (3.27), we may proceed as in

  Proof of Proposition 1.2. First, the integrability properties (m2) and (m3) stated in Proposition 1.2 readily follow from Lemma 4.1 and Theorem 1.1 (s2) by interpolation.

(m1): γ > α. Consider m ∈ (m ⋆ , 0) and recall that b m ∈ (1, ∞) by (1.22) and (2.8b). We first observe that, since γ > α, β ∈ [α, 1), ϕ ∈ X α ∩ X 1+γ , and ϕ ≡ 0 by (1.15b), Theorem 1.1, and Lemma 4.1, one has 0

Let I ≥ 2. Since (xy) λ/2 ≤ 1 2

x α y β + x β y α = K(x, y) 2K 0 , (x, y) ∈ (0, ∞) 2 , we deduce from Theorem 1.1 (s3) and the above properties of ζ I , χ ζ I , and

Next, (y I , 1) = 1≤i≤I-1 (y i+1 , y i ), so that

where

x (λ+2)/2 ϕ(x) dx , i ≥ 1 .


Waqas Ellahi

Toinon Vigier

Patrick Le Callet

HMM-BASED FRAMEWORK TO MEASURE THE VISUAL FIDELITY OF TONE MAPPING OPERATORS

Keywords: scanpath, eye movements, visual attention, hidden markov model, tone mapping operators

Recent advancements in the image capturing techniques and post processing software generate High Dynamic Range (HDR) images. Such images retain maximum information of the scene by capturing more realistic visual contents, which are often missed in traditional image capturing techniques. In this regard, tone mapping operators (TMOs) play a significant role in displaying HDR image contents on a traditional Low Dynamic Range (LDR) display. These operators tend to introduce artifacts in the original HDR image to change its brightness and contrast in such a way that it can destroy the important textures and information of the image. The assessment of these TMOs is a challenging topic to select best the best technique considering different perceptual and quality dimensions. In this paper, we propose to compare TMOs through their impact on visual behavior in comparison with HDR condition. This study is the first of its kind to utilize hidden markov model (HMM) as a similarity measure to evaluate perceived quality of TMO. The findings suggest that the proposed HMM-based method which emphasizes on temporal information produce better evaluation metric than the traditional approaches which are based only on visual spatial information.

INTRODUCTION AND BACKGROUND

High Dynamic Range (HDR) imaging has gained significant attention in recent years both in industry and academia. The is mainly due to the fact that HDR images are visually more realistic and appealing as they represent a larger dynamic range of the visual content present in the real world. The HDR technologies are arguably considered a major development in display technologies since the transition from black and white to color displays [START_REF] Oǧuz | Do hdr displays support ldr content? a psychophysical evaluation[END_REF][START_REF] Banterle | Advanced high dynamic range imaging: Theory and practice[END_REF]. However, most of the common imaging devices are only capable to work only with standard dynamic range (SDR) images, leading to a loss of luminance and contrast.

When the dynamic range is lower than the one of the content, the source image is processed with a tone mapping operator (TMO) to compress the dynamic range before rendering. The goal of this technique is to reproduce HDR content on SDR display while preserving details, contrasts and quality. As a consequence, several techniques have been developed in the past utilizing simple global operations such as linear scaling and clipping [START_REF] Drago | Adaptive logarithmic mapping for displaying high contrast scenes[END_REF][START_REF] Tumblin | Two methods for display of high contrast images[END_REF] to more local advance operators that exploit properties of the human visual system [START_REF] Chiu | Spatially nonuniform scaling functions for high contrast images[END_REF][START_REF] Durand | Fast bilateral filtering for the display of high-dynamic-range images[END_REF][START_REF] Michael Ashikhmin | A tone mapping algorithm for high contrast images[END_REF][START_REF] Kuang | icam06: A refined image appearance model for hdr image rendering[END_REF][START_REF] Reinhard | Photographic tone reproduction for digital images[END_REF]. TMOs can be also perceptually optimized using visual saliency and visual attention to locally adjust the contrast, preserve details and chromoticity [START_REF] Lin | Attention-based high dynamic range imaging[END_REF]. The quality evaluation of tone mapping operators is a challenging topic due to the consideration of multi-perceptual dimensions such as naturalness, colorfulness, brightness, fidelity, and artistic intent. Several studies have assessed these dimensions through subjective experiments [START_REF] Narwaria | Effect of tone mapping operators on visual attention deployment[END_REF][START_REF] Čadík | Evaluation of hdr tone mapping methods using essential perceptual attributes[END_REF][START_REF] Petit | Assessment of video tone-mapping: Are cameras' s-shaped tonecurves good enough?[END_REF] and different objective metrics for HDR content have been proposed [START_REF] Krasula | Objective evaluation of naturalness, contrast, and colorfulness of tone-mapped images[END_REF][START_REF] Yeganeh | Objective quality assessment of tone-mapped images[END_REF][START_REF] Granados | Contrast-use metrics for tone mapping images[END_REF].

The use of eye tracking experiment is proven to be a good methodology in measuring visual fidelity. Indeed, Narwaria et al. [START_REF] Narwaria | Effect of tone mapping operators on visual attention deployment[END_REF] showed that tone mapping operators modify visual attention by comparing the similarity between saliency maps (computed from eye data obtained during the visualisation of HDR and tone-mapped images). The current study performs the comparison of saliency maps and investigates their impact on pre-and post tone mapping. The findings suggest that some salient areas in HDR image are not seen after tone-mapping (and vice versa). However, the dynamic aspect of visual behavior, e.g. scanpaths, is not considered with saliency-based framework. Indeed, temporal aspect in visual attention can reveal the way a content is visually perceived, semantically processed and qualitatively assessed.

Scanpath modeling is a challenging task which has been more recently raised with new analytical tools to compare visual behaviour of different conditions or populations. To cater this issue, probability based approaches are developed by assuming that eye movement parameters are random variables generated by underlying stochastic processes. The set of gaze positions is modeled as bivariate gaussian distribution and hidden markov models (HMMs) are utilized to introduce 978-1-7281-1485-9/20/$31.00 c 2020 IEEE the temporal component of gaze behavior [START_REF] Chuk | Understanding eye movements in face recognition using hid-den markov models[END_REF][START_REF] David | Predicting artificial visual field losses: a gaze-based inference study[END_REF][START_REF] Coutrot | Scanpath modeling and classification with hidden markov models[END_REF]. This in turn will determine the percentage of transitions from one ROI to another ROI. These techniques have been used for the study of scanpath in face recognition [START_REF] Chuk | Understanding eye movements in face recognition using hid-den markov models[END_REF], the classification of scanpath according to different visual task [START_REF] Coutrot | Scanpath modeling and classification with hidden markov models[END_REF] or different population with visual field losses [START_REF] David | Predicting artificial visual field losses: a gaze-based inference study[END_REF].

The main contribution of the paper is a new HMM based similarly measure in order to evaluate the quality of HDR and TMO generated scanpaths. This paper is structured as follows. First, we explain HMMs in the context of gaze behavior modeling. Then, we illustrate the application of this technique for TMOs bencharmarking. Finally, we conclude our method in the last section.

HMM TOOLS FOR THE MEASURE OF VISUAL SCANPATH SIMILARITY

HMM used as eye movement modeling

The proposed work utilizes HMMs as a statistical approach to produce the sequence of observation through the set of stochastic process. These observed sequences are then predicted by estimating the transition among different states. it is a data-driven technique and integrates influences stemming from bottom-up mechanisms,top-down mechanisms, and viewing biases into a single model [START_REF] Coutrot | Scanpath modeling and classification with hidden markov models[END_REF]. In this work, the eye position is observed using estimated ROI of the image which is set as the hidden state of HMM. Following the technique used in Chuk et al. [START_REF] Chuk | Understanding eye movements in face recognition using hid-den markov models[END_REF], the transition matrix represents the saccadic movements of the participant. The emission densities (the distribution of fixation points in each ROI) are modeled as two-dimensional Gaussian distributions. The initial state of the model, i.e., the probability distribution of the first fixation, is modeled with the prior values. The parameter K which represents the number of ROI in context of scanpath must be selected according to the gaze data instead of setting the parameter with prior information. The variational approach to Bayesian inference allows simultaneous estimation of complexity of model and its parameters [START_REF] Mcgrory | Variational bayesian analysis for hidden markov models[END_REF], which automates the process of finding number of states (K) in HMM model [START_REF] Chuk | Understanding eye movements in face recognition using hid-den markov models[END_REF]. An HMM is trained on eye movement data which enables to compare and visualize the gaze behavior of different groups of observers, in different experimental conditions. The individual HMMs of the similar gaze pattern behaviour is clustered by VHEM [START_REF] Coviello | The variational hierarchical em algorithm for clustering hidden markov models[END_REF][START_REF] Coviello | Clustering hidden markov models with variational hem[END_REF] into joint HMM. For each cluster, the VHEM algorithm produced a representative HMM, which summarizes the common ROIs and transitions in the cluster.

Measure of similarity

In the proposed work, similarity is evaluated using sequence of steps. The steps involved in this process are presented in Fig. 1. Initially, each of the observers eye movement is modeled using HMM for one condition (C1). The HMM of each observer is then clustered into single Joint HMM by using VHEM. The similarity between the scanpath (i.e. sequence of fixations) of second condition (C2) and the joint clustered is determined by computing the likelihood using Viterbi decorder. The mean likelihood of all observers is considered as a similarity score between the two conditions. The publicly available dataset "Eye-Tracking on High dYnamic range iMAges" (ETHyma) [START_REF] Narwaria | Effect of tone mapping operators on visual attention deployment[END_REF] is analyzed in this study. The dataset contains 11 HDR images, captured in different environmental conditions, i.e. indoor and outdoor scenes with varying illumination conditions. A total of 88 TMOs images were computed using 8 different TMO methods. These methods are referred in Table 1, column 4-11, where each column presents the output of these TMO methods using proposed and state-of-the-art techniques. The HDR stimuli were displayed on an HDR display (Sim2 HDR47E S 4K) and the tone mapped contents are presented on on a stan-

Similarity metrics

In this section, we present the obtained scores of similarity for our framework compared with state-of-the-art saliency-based metrics: Kullback-Liebler divergence (KLD), 2D correlations (Corr) and Normalized scanpath saliency (NSS) [START_REF] Emami | Selection of a best metric and evaluation of bottom-up visual saliency models[END_REF]. These metrics are calculated after the computation of fixation maps. These maps are computed by convolving a Gaussian kernel with standard deviation of one degree of visual angle across the observer's fixation locations. The proposed framework is applied with C1 as the HDR condition compared with the different tone mapping techniques. The HMM model is applied on eye fixations extracted from raw gaze data for each observer during HDR experiment [START_REF] Asim H Dar | Remodnav: Robust eye movement detection for natural viewing[END_REF]. In addition, each HMM is then clustered into single joint HMM by using VHEM. The likelihood is calculated using Viterbi decoder by passing joint HMM parameters and fixation points recorded during TMOs experiment. The similarity score between HDR and TMOs is estimated by taking mean of likelihood of all observers for each TMOs. In this study, the maximum value of states in HMM, K, is set to 7. The scores of the proposed framework with the ones of all the other similarity / dissimilarity metrics are presented Table 2. In addition, a baseline likelihood score on individual HDR scanpaths is also calculated to validate this method. The baseline likelihood is estimated by providing the fixation points of each observer for HDR condition and joint HMM parameter to the Viterbi decoder. The final score is taking a mean of all likelihood for all observers. This baseline can be interpreted as the similarity score between one observer and a model of all scanpaths for the same condition. It is thus expected that the likelihood score for the TM condition would be lower than the baseline.

Validation of the framework

The similarity metrics are compared between each other through the computation of Pearson's correlation coefficients (PCC) presented Table 3. The proposed framework shows a strong significant negative correlations with KLD metric. The relationship between KLD and HMM likelihood scores is also presented Fig. 2. These results ensure the validity of the proposed framework which is correlated with existing metrics used to measure visual fidelity.

Concerning the baseline, the obtained scores seem to validate the previous assumption except for a few values (highlighted in red in Table 2). These specific cases could be explained by a low inter-observer congruency for HDR baseline condition and further analysis should be done to better understand these outcomes.

Benchmarking of TMOs through dynamic visual fidelity

The similarity scores detailed in Table 2 show a strong correlation between saliency-based metrics and the proposed HMM-based one, except for some specific tone-mapped contents. The moto and rend02 oC95 are selected as test images to investigate these results through the comparison of the performance between HMM and KLD. The performed analysis of KLD suggests that Drago and Tumblin methods offer the best performance among all for moto and rend02 oC95 images respectively. Whereas with the HMM-based framework, iCAM06 method offers best performance for moto and Chiu method for rend02 oC95 image (see Table 2-highlighted green cells). Saliency maps, HDR joint HMM states and transition matrix as well as scanpaths for TM0 condition are depicted on Fig 3 and4. We can see that most performant TMOs with our framework seem to lead to more similar saliency maps than the best ones with KLD metric. In addition, the scanpaths for iCAM06 (resp. Chiu) TMO fit better the joint HMM on HDR condition than Drago (resp. Tumblin) TMO. From these observations, we can see that HMM-based framework outperfoms KLD metric and better reveals visual attention similarity for both spatial (i.e salient) and temporal (i.e. scanpath) information.

One limitation of this proposed framework is that HMM based on Chuck methodology perform well with image con- tent with clear regions of interest and, thus, low visual attention dispersion. Therefore, future work should be needed to deeper assess the impact of content, inter-observer congruency and HMM meta-parameters as number of states (i.e regions of interest) on the similarity metric performance. Different HMM techniques can be also further tested and compared for similarity evaluation through this framework.

CONCLUSION

In this paper, an HMM based framework is proposed to measure the similarity of visual behaviour between two conditions of experiment. The framework is then applied and validated for the comparison of visual attention in HDR and tonemapped images. The proposed framework provides promising results due to the capability of HMM to analyse eye data considering both the spatial and sequential aspects. On the contrary, other saliency models do not reflect the dynamic (saccadic) nature of visual scanpaths.

Fig. 1 :

 1 Fig. 1: Scanpath similarity evaluation framework.

Fig. 2 :

 2 Fig. 2: Relationship between scores of KLD and HMM metrics

Fig. 3 :

 3 Fig. 3: Tone-mapped images, saliency maps, scanpaths and HDR joint HMM model for moto content.

  The work in this paper was funded from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 765911, European Training Network on Real Vision project.

Fig. 4 :

 4 Fig. 4: Tone-mapped images, saliency maps, scanpaths and HDR joint HMM model for rend02 oC95 content.

Table 1 :

 1 Characteristics of subjective experiment

	Features	HDR test	TMO test
	# of images	11	88
	Image resolution	760x900 to 1620x1080
	Image duration		10 seconds
	Distance	3 times of display height
	# of Observers	37	48
	Display model	Sim2 HDR47E TvLogic LVM401
	Display resolution	1920x1080 pixels
	Eyetracker	SMI Hi-Speed
	3. APPLICATION OF BENCHMARKING TMOS
	3.1. Dataset

Table 2 :

 2 Similarity scores for the studied dataset. (Red cells represent similarity scores higher than the baseline, discussed Section 3.3. Results of green cells are discussed in Section 3.4 to highlight the interest of the framework.) Each image was viewed by the participant during 10 seconds. The eye movements were recorded using SMI Hi-Speed tracker with a frequency of 500Hz. Thirty seven observers participated in the HDR experiment and 48 did the TMO experiment. The characteristics of this subjective experiment are presented in Table.1.

	TMOs algorithms

Table 3 :

 3 Pearson correlation coefficients (PCC) between similarity metrics. (* indicates a p-value < 0.05 and ** indicates a p-value < 0.001).

	PCC	KLD	HMM	Corr	NSS
	KLD	1.0000	-0.3957 * * -0.2755 *	0.0581
	HMM -0.3975 * *	1.0000	0.1300	-0.1472
	Corr	-0.2755 *	0.1300	1.0000	0.6811 * *
	NSS	0.0581	-0.1472	0.6811 * *	1.0000