Philippe Laurençot

Christoph Walker

Philippe Laurenc ¸ot

THE FRAGMENTATION EQUATION WITH SIZE DIFFUSION: WELL-POSEDNESS AND LONG-TERM BEHAVIOR

Keywords: Mathematics Subject Classification, 45K05 -47D06 -47B65 -47N50 -35B40, Key words and phrases, fragmentation -size diffusion -well-posedness -convergence -semigroup -perturbation

HAL is a multi-disciplinary

Introduction

The well-posedness of, along with the long-term behavior of solutions to, the fragmentation equation with size diffusion ∂ t φ(t, x) -D∂ 2

x φ(t, x) = -a(x)φ(t, x) + ∞ x a(y)b(x, y)φ(t, y) dy , (t, x) ∈ (0, ∞) 2 , (1.1a) φ(t, 0) = 0 , t > 0 , (1.1b) φ(0, x) = f (x) , x ∈ (0, ∞) .

(1.1c) is investigated by a semigroup approach. In (1.1), φ = φ(t, x) ≥ 0 denotes the size distribution function of particles of size x ∈ (0, ∞) at time t > 0, while a(x) ≥ 0 is the overall fragmentation rate of particles of size x, and b(x, y) is the daughter distribution function which describes the distribution of fragments resulting from the breakup of a particle of size y. Besides undergoing fragmentation events, particles are also assumed to modify their size by diffusion at a constant diffusion rate D > 0. Finally, nucleation is not taken into account in this model which leads to the homogeneous boundary condition (1.1b) at x = 0. An interplay between diffusion and fragmentation as depicted by (1.1) is met in the growth of ice crystals, see [START_REF] Ferkinghoff-Borg | Competition between diffusion and fragmentation: An important evolutionary process of nature[END_REF][START_REF] Mathiesen | Dynamics of crystal formation in the greenland NorthGRIP ice core[END_REF]. Indeed, on the one hand, ice crystals grow or shrink in a way which looks like diffusion and break apart due to internal stresses, the latter process being referred to as polygonization or rotation recrystallization in ice physics. The fragmentation equation with size diffusion (1.1) is also derived in [START_REF] Flyvbjerg | Stochastic dynamics of microtubules: A model for caps and catastrophes[END_REF] to describe the growth of microtubules. In the absence of diffusion (i.e. D = 0), Equation (1.1) is the spontaneous fragmentation equation which has a long and rich history and has been extensively studied in the mathematical and physical literature since the pioneering works of [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][START_REF] Ziff | The kinetics of cluster fragmentation and depolymerisation[END_REF], see [START_REF]Shattering and non-uniqueness in fragmentation models-an analytic approach[END_REF][START_REF] Banasiak | Perturbations of positive semigroups with applications[END_REF][START_REF] Banasiak | Analytic methods for coagulation-fragmentation models[END_REF][START_REF]Analytic methods for coagulation-fragmentation models[END_REF][START_REF] Bertoin | Random fragmentation and coagulation processes[END_REF], and the references therein.

An important role is played in the dynamics by the total mass of the particles' distribution

M 1 (φ(t)) := ∞ 0 xφ(t, x) dx , t ≥ 0 ,
which is expected to be conserved throughout time evolution when there is no loss of matter during fragmentation events; that is, when b satisfies y 0 xb(x, y) dx = y , y ∈ (0, ∞) .

Thus, X 1 := L 1 ((0, ∞), xdx) is a natural functional framework for the study of the fragmentation operator. We further observe that the homogeneous Dirichlet boundary condition (1.1b) corresponds actually to a no-flux boundary condition for the Laplace operator in X 1 , so that this space turns out to be also well-suited for diffusion. However, the analysis already performed on the fragmentation equation without diffusion reveals that a complete scale of weighted L 1 -spaces is needed besides X 1 .

In this regard, we introduce the spaces

X m := L 1 ((0, ∞), x m dx) and X 1,m := X 1 ∩ X m
for m ∈ R. We denote the positive cone of X 1,m by X + 1,m . For f ∈ X m and m ∈ R, we also define the moment M m (f ) of order m of f by

M m (f ) := ∞ 0 x m f (x) dx , so that f Xm = M m (|f |).
For definiteness, we equip X 1,m with the norm • X 1,m := • X 1 + • Xm and note that X 1 . = X 1,1 .

Our strategy to study the well-posedness and the long-term behavior of (1.1) is to write it as an abstract Cauchy problem in X 1,m for m ≥ 1 and show that the corresponding operator generates a semigroup with properties depending on m, a, and b. To this end, we assume throughout the paper that a ∈ L ∞,loc ([0, ∞)) , a ≥ 0 a.e. in (0, ∞) ,

and that the daughter distribution function b is a nonnegative measurable function on (0, ∞) 2 satisfying y 0 xb(x, y) dx = y , y ∈ (0, ∞) .

(1.3)

Moreover, the diffusion rate D is normalized to D = 1.

For m ≥ 1 we then define the (Schrödinger) operator A a,m on X 1,m by dom(A a,m ) := {f ∈ X 1,m :

f ′′ ∈ X 1,m , af ∈ X 1,m , f (0) = 0} , A a,m f := f ′′ -af , f ∈ dom(A a,m ) , (1.4
in X 1,m , and we shall investigate generation properties of the operator A m .

For concise statements we introduce the following notation: Given κ ≥ 1 and ω ∈ R, we write A ∈ G(X 1,m , κ, ω) if the (unbounded) linear operator A on X 1,m is the generator of a strongly continuous semigroup (e tA ) t≥0 on X 1,m and e tA L(X 1,m ) ≤ κe ωt , t ≥ 0 .

We set G(X 1,m ) := κ≥1 , ω∈R G(X 1,m , κ, ω) .

Moreover, we write A ∈ G + (X 1,m ) if the semigroup (e tA ) t≥0 is positive on the Banach lattice X 1,m . We denote the domain of the (unbounded) operator A in X 1,m by dom(A) and set

D(A) := dom(A), • A , where f A := f X 1,m + Af X 1,m for f ∈ dom(A) is the graph norm. Finally, we write A ∈ H(X 1,m ) if A ∈ G(X 1,m
) and the semigroup (e tA ) t≥0 is analytic.

With this notation we may formulate the generation result in X 1,m for m ≥ 1:

Theorem 1.1. Assume that a and b satisfy (1.2) and (1.

3). (a)

There is an extension

Ã1 ∈ G + (X 1 , 1, 0) of A 1 . (b) Assume further that there is δ 2 ∈ (0, 1) such that (1 -δ 2 )y 2 ≥ y 0 x 2 b(x, y) dx , y ∈ (0, ∞) . (1.8) If m > 1, then A m ∈ G + (X 1,m ) ∩ H(X 1,m ). In addition, for all f ∈ X 1,m , M 1 e tAm f = M 1 (f ) , t ≥ 0 . (1.9) (c) Assume that a satisfies lim x→∞ a(x) = ∞ (1.10)
and that b satisfies (1.8). Then, (e tAm ) t≥0 is immediately compact on X 1,m for m > 1.

Assumption (1.8) is commonly encountered in the investigation of the fragmentation equation and somehow excludes the concentration of b along the diagonal.

At this stage, the extension Ã1 of A 1 is not completely identified. In particular, we do not know whether or not Ã1 coincides with A 1 . Still, it is a question worth of investigation and we refer to [START_REF] Banasiak | Analytic methods for coagulation-fragmentation models[END_REF] for a thorough discussion of this issue for the fragmentation equation without diffusion. Anyway, a positive answer is straightforward when a ∈ L ∞ (0, ∞) and reported in the next result.

Proposition 1.2. Assume that a ∈ L ∞ (0, ∞) is non-negative and that b satisfies (1.3). (a) For m ≥ 1, A m ∈ G + (X 1,m ) ∩ H(X 1,m ) and (1.9) is satisfied. In addition, A 1 ∈ G + (X 1 , 1, 0). (b)
The semigroup (e tAm ) t≥0 is not compact on X 1,m .

We immediately obtain the well-posedness of the Cauchy problem (1.7) in X 1,m and, equivalently, of (1.1) in a classical sense. Since we shall see that D(A m ) . = D(A a,m ), we can formulate the result as follows:

Corollary 1.3. Assume that a and b satisfy (1.2) and (1.3). Assume further that, either m = 1 and a ∈ L ∞ (0, ∞), or m > 1 and b satisfies (1.8). Then, for any f ∈ X 1,m , there is a unique classical solution

φ ∈ C([0, ∞), X 1,m ) ∩ C 1 ((0, ∞), X 1,m ) ∩ C((0, ∞), D(A a,m
)) to (1.1) which is given by φ(t) = e tAm f for t ≥ 0 and satisfies

M 1 (φ(t)) = M 1 (f ) , t ≥ 0 . (1.11)
Moreover, if f ≥ 0 a.e. in (0, ∞), then φ(t) ≥ 0 for all t > 0.

Remark 1.4. Given f ∈ X + 1,m , the corresponding solution φ to (1.1) provided by Corollary 1.3 satisfies the mass conservation (1.11), a feature which is in particular due to the assumed boundedness (1.2) of a on (0, 1). Indeed, even when b satisfies (1.3), infringement of mass conservation is known to occur when the overall fragmentation a is unbounded for small sizes. In that case, the total mass is a decreasing function of time, a phenomenon usually referred to as shattering which is closely related to the honesty property of the associated semigroup, see [START_REF] Banasiak | Conservative and shattering solutions for some classes of fragmentation models[END_REF][START_REF] Banasiak | Analytic methods for coagulation-fragmentation models[END_REF][START_REF] Cheng | Kinetics of fragmentation[END_REF][START_REF] Filippov | On the distribution of the sizes of particles which undergo splitting[END_REF][START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF][START_REF] Jeon | Stochastic fragmentation and some sufficient conditions for shattering transition[END_REF][START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF].

Having settled the well-posedness of (1.1), we next turn to qualitative properties of its dynamics. As a guideline it was pointed out in [START_REF] Ferkinghoff-Borg | Competition between diffusion and fragmentation: An important evolutionary process of nature[END_REF] that the interplay between diffusion and fragmentation results in the stabilization of solutions to (1.1) to a stationary solution. This is in sharp contrast to the fragmentation equation without diffusion, since fragmentation is an irreversible process driving the particle distribution to zero. When diffusion is turned on, a closed-form stationary solution to (1.1) can be computed for the particular choice a(x) = x and b(x, y) = 2y -1 1 (0,y) (x), see [START_REF] Ferkinghoff-Borg | Competition between diffusion and fragmentation: An important evolutionary process of nature[END_REF]. The existence of stationary solutions is also established in [START_REF] Ph | Steady states for a fragmentation equation with size diffusion[END_REF] for an overall fragmentation rate a obeying a power law (a(x) = x γ , γ ≥ 0) and for a specific class of daughter distribution function b. Here we extend this existence result to a broader class of fragmentation coefficients a and b, see Proposition 1.6. In addition, when a diverges to infinity as x → ∞, we provide the exponential decay of the solution to (1.1) to the steady state with the total mass of the initial value.

Theorem 1.5. Assume that a and b satisfy (1.2), (1.3), (1.8), and (1.10), and that a > 0 and b > 0.

There is a unique nonnegative

ψ 1 ∈ r≥1 dom(A r ) such that M 1 (ψ 1 ) = 1 and ker(A m ) = Rψ 1 for every m > 1.
In particular, for m > 1, the spectral bound s(A m ) = 0 is a dominant eigenvalue of A m and there are N m ≥ 1 and

ν m > 0 such that, for all f ∈ X 1,m , e tAm f -M 1 (f )ψ 1 X 1,m ≤ N m e -νmt f X 1,m , t ≥ 0 .
It is worth pointing out that the stationary solution ψ 1 decays faster than algebraically at infinity, a property which is perfectly consistent with the exponentially decaying tail experimentally observed in [START_REF] Mathiesen | Dynamics of crystal formation in the greenland NorthGRIP ice core[END_REF]. Also, combining Theorem 1.5 with Lemma 2.1 below implies that ψ 1 ∈ X r for all r > -1.

Theorem 1.5 provides a complete description of the long-term behavior of solutions to (1.1) when a diverges to infinity as x → ∞. However, the unboundedness of a at infinity is not a necessary condition for the existence of stationary solutions. In fact, when

a(x) = 1 , b(x, y) = 2 y 1 (0,y) (x) , 0 < x < y ,
we notice that Equation (1.1) has an explicit stationary solution ψ 1 (x) = xe -x , x > 0.1 This particular example is not peculiar and we actually obtain the existence of stationary solutions to (1.1) as soon as there is a positive lower bound for a as x → ∞.

Proposition 1.6. Assume that a and b satisfy (1.2), (1.3), and (1.8), and that a > 0 and b > 0.

Assume further that

α := 1 2 lim inf x→∞ a(x) ∈ (0, ∞) . (1.12)
There is a unique nonnegative

ψ 1 ∈ r≥1 dom(A r ) such that M 1 (ψ 1 ) = 1 and ker(A m ) = Rψ 1 for every m > 1.
Let us mention here that there is no loss of generality in assuming the finiteness of lim inf x→∞ a(x) in (1.12). Indeed, if lim inf x→∞ a(x) = ∞, then a satisfies (1.10), a situation which is dealt with in Theorem 1.5.

When a only satisfies (1.12), the associated semigroup e tAm t≥0 need not be compact, see Proposition 1.2. We thus take a different route to prove Proposition 1.6 an approximation procedure. This approach does not allow us to retrieve information on the long-term behavior and it is likely that, either a more precise study of the operator A 1 , or a different approach (such as the one developed in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]) is required to fully identify the long-term behavior when a only satisfies (1.12).

Let us end this introduction with a brief outline of the paper. Auxiliary results are gathered in the next section, which includes integrability properties of elements of dom(A 0,1 ) on the one hand, and a weighted version of Kato's inequality on the other hand. In Section 3, we recall some properties of the heat semigroup in the weighted L 1 -space X 1,m with m ≥ 1, relying on the explicit representation formula which is available in that case. Section 4 is devoted to the Schrödinger operator A a,m and the associated absorption semigroup and is mostly a consequence of the thorough study performed in [START_REF] Arendt | Absorption semigroups and Dirichlet boundary conditions[END_REF]. We then use a perturbation argument in Section 5 to study the full fragmentation-diffusion operator A m = A a,m + B m . On the one hand, for m = 1, the existence of an extension Ã1 ∈ G + (X 1 , 1, 0) of A 1 is a consequence of [START_REF] Voigt | On substochastic C 0 -semigroups and their generators[END_REF]. On the other hand, if m > 1, then we can use a Miyadera perturbation technique to prove that A m ∈ H(X 1,m ). We recall that this approach has already proved successful for the fragmentation equation without size diffusion, see [START_REF]Global solutions of continuous coagulation-fragmentation equations with unbounded coefficients[END_REF]. The remainder of the paper is then devoted to the long-term dynamics. As a preliminary step, we first establish in Section 6 the immediate compactness of the semigroup in X 1,m for m > 1 when a diverges to infinity as x → ∞. We then construct in Section 7 a bounded convex subset of X 1,m which is invariant with respect to the semigroup. This feature, along with the immediate compactness of the semigroup, implies the existence of at least one stationary solution for any given mass. After showing that the latter is unique, we perform a detailed study of the spectrum of A m and end up with the announced convergence at a (yet non-explicit) exponential rate. Building upon the analysis performed in Section 7, we turn to the proof of Proposition 1.6 in Section 8 which relies on an approximation procedure. Specifically, introducing a n (x) := a(x) + x/n for x > 0 and n ≥ 1, we deduce from Theorem 1.5 that there is a unique non-negative stationary solution ψ 1,n to (1.1) with a n instead of a. We then show that cluster points as n → ∞ of this sequence are stationary solutions to (1.1).

Auxiliary Results

According to the definition of dom(A 0,1 ), an important role is played in the forthcoming analysis by functions f ∈ X 1 such that f ′′ ∈ X 1 . We collect useful properties of this class of functions in the next lemma and show, in particular, that the boundary condition (1.1c) is well-defined for such functions.

Lemma 2.1. Consider f ∈ X 1 such that f ′′ ∈ X 1 . Then f ∈ BC([0, ∞))∩C 1 ((0, ∞)), f ′ ∈ L 1 (0, ∞), and, for x > 0, |f (x)| ≤ f ′′ X 1 , x|f ′ (x)| ≤ f ′′ X 1 , f ′ L 1 (0,∞) ≤ f ′′ X 1 . (2.1a) Moreover, lim x→∞ xf (x) = lim x→∞ xf ′ (x) = 0 . (2.1b)
In fact, f and f ′ are given by

f (x) = - ∞ x f ′ (y) dy , f ′ (x) = - ∞ x f ′′ (y) dy , x ∈ (0, ∞) . (2.1c)
Also, f ∈ X m for any m ∈ (-1, 1) and, for all ε > 0,

f Xm ≤ ε m+1 m + 1 f ′′ X 1 + ε m-1 f X 1 . (2.2)
Equivalently,

f Xm ≤ 2(1 -m) (m-1)/2 m + 1 f ′′ (1-m)/2 X 1 f (m+1)/2 X 1 , m ∈ (-1, 1) . (2.3) Proof. Introducing F (x) := ∞ x (y -x)f ′′ (y) dy , x ∈ (0, ∞) , it follows from the integrability of f ′′ that -f ′′ X 1 ≤ - ∞ x (y -x)|f ′′ (y)|dy ≤ F (x) ≤ ∞ x (y -x)|f ′′ (y)|dy ≤ f ′′ X 1 , so that F (x) is well-defined for x ≥ 0. Moreover, F belongs to BC([0, ∞))∩C 1 ((0, ∞))∩W 2 1,loc ((0, ∞)) and satisfies F ′ (x) = - ∞ x f ′′ (y) dy , F ′′ (x) = f ′′ (x) , x ∈ (0, ∞) , (2.4)
and lim

x→∞ xF ′ (x) = lim x→∞ F (x) = 0 . (2.5)
In particular, we infer from (2.4) that there is (α,

β) ∈ R 2 such that (f -F )(x) = α + βx for x > 0. Moreover, since f ∈ X 1 , it follows from (2.5) that lim x→∞ x+1 x |α + βy| dy ≤ lim x→∞ x+1 x (|f (y)| + |F (y)|) dy = 0 ,
which readily gives α = β = 0 and F = f , thereby establishing (2.1), except for the limiting behavior of f at infinity. To this end, we observe that, since

f ∈ L ∞ (0, ∞) and f ′ ∈ L 1 (0, ∞), a formula for f (x) 2 reads f (x) 2 = -2 ∞ x f (y)f ′ (y) dy , x ≥ 0 .
We then deduce from (2.1a) that

f (x) 2 ≤ 2 ∞ x y|f ′ (y)||f (y)| dy y ≤ 2 x 2 f ′′ X 1 ∞ x y|f (y)| dy , which implies that xf (x) → 0 as x → ∞ due to f ∈ X 1 . Next, let ε > δ > 0. It follows from (2.1c) that ∞ δ x m |f (x)| dx ≤ ε δ x m |f (x)| dx + ε m-1 ∞ ε x |f (x)| dx ≤ ε δ x m ∞ x |f ′ (y)| dydx + ε m-1 f X 1 ≤ ε δ x m ε x |f ′ (y)| dydx + ε m+1 m + 1 ∞ ε |f ′ (y)| dy + ε m-1 f X 1 . By Fubini's theorem, ε δ x m ε x |f ′ (y)| dydx = 1 m + 1 ε δ y m+1 -δ m+1 |f ′ (y)| dy ≤ ε m+1 m + 1 ε 0 |f ′ (y)| dy .
Combining the above inequalities with (2.1a) gives

∞ δ x m |f (x)| dx ≤ ε m+1 m + 1 ∞ 0 |f ′ (y)| dy + ε m-1 f X 1 ≤ ε m+1 m + 1 f ′′ X 1 + ε m-1 f X 1 .
Letting δ → 0 completes the proof of (2.2). We next optimize (2.2) with respect to ε ∈ (0, ∞) to derive (2.3).

We next state for the sake of completeness the density in X 1,m of smooth functions with compact support in (0, ∞), which is actually a straightforward consequence of the density of

C ∞ c ((0, ∞)) in L 1 (0, ∞). Lemma 2.2. The space C ∞ c ((0, ∞)) is dense in X 1,m for m ≥ 1. We finally recall a variant of the celebrated inequality of Kato [20, Lemma A]. Lemma 2.3. Let ℓ be a nonnegative function in W 1 ∞,loc ([0, ∞)) and consider f ∈ W 2 1,loc (0, ∞) such that f ′ ∈ L 1 ((0, ∞), ℓ ′ (x)dx) and f ′′ ∈ L 1 ((0, ∞), ℓ(x)dx). Then - ∞ 0 ℓ(x) sign(f (x))f ′′ (x) dx ≥ ∞ 0 ℓ ′ (x) |f | ′ (x) dx .
(2.6)

Proof. For ε ∈ (0, 1), we define

β ε ∈ W 2 ∞,loc (R) by β ε (0) = 0 and β ′ ε (r) := sign(r) , |r| > ε , and β ′ ε (r) := r ε , r ∈ [-ε, ε] .
Since β ′′ ε ≥ 0 a.e. in R, integration by parts gives

- ∞ 0 ℓ(x) β ′ ε (f (x))f ′′ (x) dx = ∞ 0 ℓ(x) β ′′ ε (f (x))|f ′ (x)| 2 + ℓ ′ (x)β ′ ε (f (x))f ′ (x) dx ≥ ∞ 0 ℓ ′ (x)[β ε (f )] ′ (x) dx .
Since |β ε (r) -r| ≤ ε for r ∈ R and β ′ ε converges pointwise to the sign function in R as ε → 0, we may pass to the limit as ε → 0 in the previous inequality to complete the proof.

The Heat Semigroup

It is well-known that the solution to the heat equation with homogeneous Dirichlet boundary conditions at x = 0,

∂ t w -∂ 2 x w = 0 , (t, x) ∈ (0, ∞) × (0, ∞) , w(t, 0) = 0 , t ∈ (0, ∞) , w(0, x) = f (x) , x ∈ (0, ∞) ,
is given by the representation formula

w(t, x) = W (t)f (x) := ∞ 0 [k(t, x -y) -k(t, x + y)]f (y) dy , (t, x) ∈ (0, ∞) 2 , (3.1)
where

k(t, x) := 1 √ 4πt e -|x| 2 /4t (t, x) ∈ (0, ∞) × R . (3.2) Moreover, (W (t)) t≥0 (with W (0) := id L 1 (0,∞) ) is a positive analytic semigroup of contractions on L 1 (0, ∞) with generator G given by dom(G) := {f ∈ L 1 (0, ∞) : f ′′ ∈ L 1 (0, ∞) and f (0) = 0} , Gf := f ′′ , f ∈ dom(G) . That is, G ∈ G + (L 1 (0, ∞), 1, 0) ∩ H(L 1 (0, ∞)
) and e tG = W (t) for t ≥ 0. We now consider this semigroup in the weighted space X 1,m . More precisely, for m ≥ 1, we recall the definition (1.4) (with a = 0) of the unbounded operator A 0,m on X 1,m , given by

dom(A 0,m ) = {f ∈ X 1,m : f ′′ ∈ X 1,m and f (0) = 0} , A 0,m f = f ′′ , f ∈ dom(A 0,m ) , (3.3)
and show that it is the generator of the heat semigroup in X 1,m .

Proposition 3.1. Let m ≥ 1. There is ω m ≥ 0 such A 0,m ∈ G + (X 1,m , 1, ω m ) ∩ H(X 1,m ) with (0, ∞) ⊂ ρ(A 0,m ).
The semigroup e tA 0,m t≥0 is given by

e tA 0,m f (x) = ∞ 0 [k(t, x -y) -k(t, x + y)]f (y) dy , (t, x) ∈ (0, ∞) × (0, ∞) , (3.4
)

for f ∈ X 1,m , where k is defined in (3.2). Moreover, e tA 0,m = e tA 0,1 | X 1,m for t ≥ 0 and (λ -A 0,m ) -1 = (λ -A 0,1 ) -1 | X 1,m , λ > 0 .
Two steps are needed to show Proposition 3.1. We first establish Proposition 3.1 for m = 1 and m ≥ 3, see Lemma 3.2 below. An interpolation argument then completes the proof for m ∈ (1, 3).

Lemma 3.2. Let m ∈ {1} ∪ (3, ∞). Then A 0,m ∈ G + (X 1,m , 1, ω m ) ∩ H(X 1,m ) with ω 1 := 0 and ω m := 4 1/(m-1) m(m -3) (m-3)/(m-1) , m ≥ 3 .
Moreover, e tA 0,m = e tA 0,1 | X 1,m for t ≥ 0, (0, ∞) ⊂ ρ(A 0,m ), and

(λ -A 0,m ) -1 = (λ -A 0,1 ) -1 | X 1,m , λ > 0 .
Proof. We first note that A 0,m is a closed operator on X 1,m and that its domain is dense in X 1,m due to Lemma 2.1 and Lemma 2.2. We divide the remainder of the proof into several steps.

Step 1. We first show the dissipativity of A 0,m -ω m on X 1,m . To this end, let λ > 0 and f ∈ dom(A 0,m ). By Kato's inequality (2.6) (with ℓ(x) = x m ), Lemma 2.1, and the boundary condition

f (0) = 0, λf -(A 0,m -ω m )f Xm ≥ ∞ 0 x m sign(f (x))[(λ + ω m )f (x) -f ′′ (x)] dx ≥ (λ + ω m ) f Xm + m ∞ 0 x m-1 |f | ′ (x) dx = (λ + ω m ) f Xm -m(m -1) f X m-2 .
In particular, when m = 1,

λf -A 0,1 f X 1 ≥ λ f X 1 , (3.5) so that A 0,1 is dissipative on X 1 . We next handle the case m ≥ 3. Then m -2 ∈ [1, m) and we infer from Young's inequality that, for ε > 0, m(m -1) f X m-2 ≤ m(m -3)ε f Xm + 2mε (3-m)/2 f X 1 . Hence, (λ + ω m ) f Xm ≤ m(m -3)ε f Xm + 2mε (3-m)/2 f X 1 + λf -(A 0,m -ω m )f Xm .

Combining (3.5) with this inequality gives

(λ + ω m ) f X 1,m ≤ λf -(A 0,m -ω m )f X 1,m + m(m -3)ε f Xm + 2mε (3-m)/2 f X 1 . We now choose ε = ε m := (2/(m -3)) 2/(m-1) . Since ω m = m(m -3)ε m = mε (3-m)/2 m , we readily conclude that λ f X 1,m ≤ λf -(A 0,m -ω m )f X 1,m , so that A 0,m -ω m I is a dissipative operator on X 1,m .
Step 2. We next show that rg(λ

-A 0,m ) = X 1,m for λ > 0. Consider g ∈ X 1,m . According to Lemma 2.2, there is a sequence (g n ) n≥1 in C ∞ c ((0, ∞)) such that lim n→∞ g n -g X 1,m = 0 . (3.6) Since g n ∈ L 1 (0, ∞) and (0, ∞) ⊂ ρ(G), there is a unique f n ∈ dom(G) such that λf n -Gf n = g n ; that is, λf n -f ′′ n = g n in (0, ∞) , f n (0) = 0 . (3.7)
Now, let R > 1. We multiply (3.7) by (x ∧ R) m sign(f n (x)) and integrate over (0, ∞). Using Kato's inequality (2.6) (with ℓ(x) = (x ∧ R) m ), we obtain

g n Xm ≥ ∞ 0 (x ∧ R) m sign(f n (x))[λf n (x) -f ′′ n (x)] dx ≥ λ ∞ 0 (x ∧ R) m |f n (x)| dx + m R 0 x m-1 |f n | ′ (x) dx = λ ∞ 0 (x ∧ R) m |f n (x)| dx -m(m -1) R 0 x m-2 f n (x) dx .
In particular, for m = 1 we get

g n X 1 ≥ λ ∞ 0 (x ∧ R) |f n (x)| dx ,
so that, letting R → ∞ and using Fatou's lemma,

λ f n X 1 ≤ g n X 1 . (3.8)
The same argument entails that

λ f n -f l X 1 ≤ g n -g l X 1 , n, l ≥ 1 . (3.9) If m ≥ 3, then m -2 ∈ [1, m)
, and we use Young's inequality to deduce that, for ε > 0,

λ ∞ 0 (x ∧ R) m |f n (x)| dx ≤ g n Xm + m(m -3)ε R 0 x m |f n (x)| dx + 2mε (3-m)/2 R 0 x |f n (x)| dx ≤ g n Xm + m(m -3)ε ∞ 0 (x ∧ R) m |f n (x)| dx + 2mε (3-m)/2 f n X 1 .
Choosing ε = λ/(2m(m -3)), we combine (3.8) and the above inequality to conclude that

λ 2 ∞ 0 (x ∧ R) m |f n (x)| dx ≤ g n Xm + 2m λ 2m(m -3) λ (m-3)/2 g n X 1 .
We now let R → ∞ in this inequality and deduce from Fatou's lemma that

f n ∈ X m with λ f n Xm ≤ 2 g n Xm + 4m λ 2m(m -3) λ (m-3)/2 g n X 1 . (3.10)
The same argument entails that

λ f n -f l Xm ≤ 2 g n -g l Xm + 4m λ 2m(m -3) λ (m-3)/2 g n -g l X 1 (3.11)
for n ≥ 1 and l ≥ 1. Therefore, for m ∈ {1} ∪ [3, ∞), the estimates (3.9) and (3.11) along with (3.6) guarantee that (f n ) n≥1 is a Cauchy sequence in X 1,m and that there is f ∈ X 1,m such that

lim n→∞ f n -f X 1,m = 0 . (3.12)
Since f ′′ n = λf n -g n for all n ≥ 1 by (3.7), it readily follows from (3.6) and (3.12) that (f ′′ n ) n≥1 converges to λf -g in X 1,m and to f ′′ in the sense of distributions. Therefore,

f ′′ ∈ X 1,m , f ′′ = λf -g, and f ′′ n -f ′′ X 1,m → 0 as n → ∞. Finally, by (2.1c), |f (0)| = |f (0) -f n (0)| ≤ ∞ 0 |(f ′ -f ′ n )(x)| dx ≤ ∞ 0 ∞ x |(f ′′ -f ′′ n )(y)| dydx = ∞ 0 y |(f ′′ -f ′′ n )(y)| dy = f ′′ -f ′′ n X 1 ,
from which we deduce that f (0) = 0. Consequently, f ∈ dom(A 0,m ) and λf -A 0,m f = g. Since λ -A 0,m is one-to-one by (3.5), we have thus shown for any λ > 0 that

rg(λ -A 0,m ) = X 1,m and (λ -A 0,m ) -1 g = (λ -A 0,1 ) -1 g for all g ∈ X 1,m . (3.13) Step 3. For m ∈ {1} ∪ [3, ∞), we infer from Step 1 -Step 2 and the Lumer-Phillips theorem [26, Theorem 1.4.3] that A 0,m -ω m belongs to G(X 1,m , 1, 0). Hence, A 0,m ∈ G(X 1,m , 1, ω m ). Also, it readily follows from (3.13) that (0, ∞) ⊂ ρ(A 0,m ) and (λ -A 0,m ) -1 = (λ -A 0,1 ) -1 | X 1,m .
In particular, the latter, along with the exponential formula [26, Theorem 1.8.3], entails that e tA 0,m = e tA 0,1 | X 1,m for all t ≥ 0.

Step 4. We next derive the representation formulation for (e tA 0,1 ) t≥0 . To this end, we first note that, for t > 0, the operator W (t) defined in (3.1) is a bounded operator on X 1 . Indeed, since |x -y| ≤ |x

+ y| = x + y for (x, y) ∈ (0, ∞) 2 , k(t, x -y) -k(t, x + y) = 1 √ 4πt e -|x-y| 2 /4t -e -|x+y| 2 /4t ≥ 0 , (x, y) ∈ (0, ∞) 2 . (3.14)
By Fubini-Tonelli's theorem and (3.14),

W (t)f X 1 ≤ ∞ 0 x ∞ 0 [k(t, x -y) -k(t, x + y)]|f (y)| dydx = 1 √ π ∞ 0 |f (y)| ∞ -y/2 √ t (y + 2z √ t)e -z 2 dz - -y/2 √ t -∞ (-y -2z √ t)e -z 2 dz dy = 1 √ π ∞ 0 |f (y)| ∞ -∞ (y + 2z √ t)e -z 2 dz dy = f X 1 .
Thus, W (t) is a contraction on X 1 . Since (λ -G) -1 = (λ -A 0,1 ) -1 on L 1 (0, ∞) ∩ X 1 for all λ > 0, it follows from the exponential formula, see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Theorem 1.8.3], that e tG = e tA 0,1 on L 1 (0, ∞) ∩ X 1 for all t ≥ 0. Therefore, W (t) = e tA 0,1 on L 1 (0, ∞) ∩ X 1 for all t ≥ 0. Since L 1 (0, ∞) ∩ X 1 is dense in X 1 by Lemma 2.2 and e tA 0,1 and W (t) are both bounded operators on X 1 , we conclude that e tA 0,1 = W (t) on X 1 for all t ≥ 0. Together with the outcome of Step 3, this identity proves (3.4).

In particular, (e tA 0,m ) t≥0 is a positive semigroup according to (3.14).

Step 5. We are left with showing the analyticity of (e tA 0,m ) t≥0 on X 1,m . To this end, let f ∈ X 1,m . Clearly,

t -→ ∞ 0 [k(t, x -y) -k(t, x + y)]f (y) dy is differentiable on (0, ∞) and, since 2t∂ t k(t, x) = -1 + |x| 2 2t k(t, x) , (t, x) ∈ (0, ∞) × R , we derive for (t, x) ∈ (0, ∞) 2 that 2t d dt e tA 0,m f (x) = -e tA 0,m f (x) + 2 ∞ 0 |x -y| 2 4t k(t, x -y) - |x + y| 2 4t k(t, x + y) f (y) dy = -3e tA 0,m f (x) + 2 ∞ 0 1 + |x -y| 2 4t k(t, x -y) -1 + |x + y| 2 4t k(t, x + y) f (y) dy .
Let t > 0. Since z → (1 + z)e -z is non-increasing on (0, ∞) and |x -y| ≤ x + y for (x, y) ∈ (0, ∞) 2 , we see that

1 + |x -y| 2 4t k(t, x -y) -1 + |x + y| 2 4t k(t, x + y) ≥ 0 , (x, y) ∈ (0, ∞) 2 , (3.15)
and infer from Fubini-Tonelli's theorem that

∞ 0 x m ∞ 0 1 + |x -y| 2 4t k(t, x -y) -1 + |x + y| 2 4t k(t, x + y) f (y) dy dx ≤ ∞ 0 x m ∞ 0 1 + |x -y| 2 4t k(t, x -y) -1 + |x + y| 2 4t k(t, x + y) |f (y)| dydx = 1 √ π ∞ 0 |f (y)| ∞ -y/2 √ t y + 2z √ t m (1 + z 2 )e -z 2 dz - -y/2 √ t -∞ -y -2z √ t m (1 + z 2 )e -z 2 dz dy = 1 √ π ∞ 0 |f (y)| ∞ -∞ y + 2z √ t m-1 y + 2z √ t (1 + z 2 )e -z 2 dzdy . Consequently, 2t d dt e tA 0,m f X 1,m ≤ 3 e tA 0,m f X 1,m + 2 √ π ∞ 0 |f (y)| ∞ -∞ y + 2z √ t (1 + z 2 )e -z 2 dzdy + 2 √ π ∞ 0 |f (y)| ∞ -∞ y + 2z √ t m-1 y + 2z √ t (1 + z 2 )e -z 2 dz dy so that lim sup t→0 t d dt e tA 0,m f X 1,m ≤ 3 f X 1,m
by Lebesgue's convergence theorem. It is well-known that this property implies the analyticity of (e tA 0,m ) t≥0 on X 1,m . Thus, the proof is complete.

The proof of Proposition 3.1 is now a consequence of the previous lemma and an interpolation argument as shown next.

Proof of Proposition 3.1. We only have to consider the case m ∈ (1, 3). Since A 0,1 ∈ H(X 1,1 ) and A 0,3 ∈ H(X 1,3 ) according to Lemma 3.2, Hille's characterization implies that there are λ 0 > 0 and κ ≥ 1 such that

(λ -A 0,1 ) -1 L(X 1,1 ) + (λ -A 0,3 ) -1 L(X 1,3 ) ≤ κ |λ -λ 0 | , Re λ > λ 0 .
Since (λ -A 0,3 ) -1 = (λ -A 0,1 ) -1 | X 1,3 , it readily follows by interpolation that

(λ -A 0,1 ) -1 | X 1,m L(X 1,m ) ≤ κ |λ -λ 0 | , Re λ > λ 0 . (3.16)
Obviously, A 0,m is closed and densely defined in X 1,m . Let g ∈ X 1,m be arbitrary and Re λ > λ 0 .

There is a unique f ∈ dom(A 0,1 ) such that (λ

-A 0,1 )f = g. By (3.16), f = (λ -A 0,1 ) -1 | X 1,m g ∈ X 1,m and f ′′ = A 0,m f = λf -g ∈ X 1,m . From this identity, we deduce f ∈ dom(A 0,m ) with A 0,m f = A 0,1 f and (λ -A 0,m )f = g. Since f is unique, we conclude that λ ∈ ρ(A 0,m ) and (λ -A 0,m ) -1 g = f = (λ -A 0,1 ) -1 | X 1,m g ; that is, (λ -A 0,m ) -1 = (λ -A 0,1 ) -1 | X 1,m . Invoking (3.16) we derive that (λ -A 0,m ) -1 L(X 1,m ) ≤ κ |λ -λ 0 | , Re λ > λ 0 ,
and thus conclude that A 0,m ∈ H(X 1,m ) with e tA 0,m = e tA 0,1 | X 1,m for t ≥ 0. The fact that A 0,m ∈ G(X 1,m , 1, ω m ) follows again by interpolation using Lemma 3.2.

The Absorption Semigroup

Let a ∈ L ∞,loc ([0, ∞)) be a non-negative function and m ≥ 1. We recall the definition (1.4) of the Schrödinger operator A a,m on X 1,m given by

dom(A a,m ) = {f ∈ X 1,m : f ∈ dom(A 0,m ) , af ∈ X 1,m } , A a,m f = f ′′ -af , f ∈ dom(A a,m ) .
The main result of this section is that

A a,m ∈ G + (X 1,m , 1, ω m ) ∩ H(X 1,m
), the parameter ω m being defined in Proposition 3.1. The proof relies on [START_REF] Arendt | Absorption semigroups and Dirichlet boundary conditions[END_REF].

Proposition 4.1. Assume that a satisfies (1.2) and let m ≥ 1. Then A a,m ∈ G + (X 1,m , 1, ω m ) ∩ H(X 1,m ). Moreover, e tAa,m = e tA a,1 | X 1,m for t ≥ 0.
Proof. Let us recall that X 1,m is a Banach lattice with order-continuous norm and introduce Y := L 1 ((0, ∞), (x + x m )a(x)dx). We first observe that (dom(A 0,m ) ∩ Y ) ⊥ = {0}, where the disjoint complement F ⊥ of a subset F of X 1,m is given by

F ⊥ := {g ∈ X 1,m : min{|f |, |g|} = 0 for all f ∈ F } . Indeed, since C ∞ c ((0, ∞)
) is a subset of dom(A 0,m ) ∩ Y , we readily deduce that g ≡ 0 for g ∈ (dom(A 0,m ) ∩ Y ) ⊥ . Consequently, we are in a position to apply [2, Proposition 4.3] and conclude that there is an extension Âa,m

∈ G + (X 1,m ) of A a,m with domain dom( Âa,m ) :=    f ∈ X 1,m : there exist (f n ) n≥1 in dom(A 0,m ) and g ∈ X 1,m such that lim n→∞ f n -f X 1,m + A 0,m f n -(a ∧ n)f n + g X 1,m = 0    .
It first follow from Lemma 4.2 below that dom( Âa,m ) = dom(A a,m ) and therefore Âa,m = A a,m . Moreover, 0 ≤ e tAa,m = e t Âa,m ≤ e tA 0,m for t ≥ 0 by [2, p.432]. Since A 0,m ∈ G + (X 1,m , 1, ω m ) due to Proposition 3.1, this ordering property, along with [6, Remark 2.68], implies e t Âa,m L(X 1,m ) ≤ e tA 0,m L(X 1,m ) ≤ e ωmt , t ≥ 0 .

Hence Âa,m ∈ G + (X 1,m , 1, ω m ). Finally, recalling that

(-ω m + A 0,m ) ∈ G + (X 1,m , 1, 0) ∩ H(X 1,m ) by Proposition 3.1, we infer from [2, Theorem 6.1] that A a,m ∈ H(X 1,m ).
It remains to check that dom( Âa,m ) = dom(A a,m ). This property actually follows from the monotonicity of the Laplace operator A 0,m and the multiplication f → af . Proof. Pick f ∈ dom( Âa,m ). Then there are a sequence (f n ) n≥1 in dom(A 0,m ) and g ∈ X 1,m such that

lim n→∞ f n -f X 1,m + g n -g X 1,m = 0 , (4.1)
with g n := -A 0,m f n + (a ∧ n)f n for n ≥ 1. In particular,

κ := sup n≥1 f n X 1,m + g n X 1,m < ∞ .
Step 1. Let us first prove that af ∈ X 1,m . Indeed, we infer from Lemma 2.3 that, for n ≥ 1,

g n Xm ≥ ∞ 0 x m sign(f n (x)) g n (x) dx ≥ m ∞ 0 x m-1 |f n | ′ (x) dx + ∞ 0 x m (a(x) ∧ n)|f n (x)| dx = -m(m -1) f n X m-2 + ∞ 0 x m (a(x) ∧ n)|f n (x)| dx .
In particular, we derive

∞ 0 x (a(x) ∧ n)|f n (x)| dx ≤ g n X 1 ≤ κ . (4.2)
Next, if m ≥ 3, then it follows from Young's inequality that

∞ 0 x m (a(x) ∧ n)|f n (x)| dx ≤ m(m -1) m -3 m -1 f n Xm + 2 m -1 f n X 1 + g n Xm ≤ [m(m -1) + 1]κ . Likewise, if m ∈ (1, 3), then Lemma 2.1 implies that ∞ 0 x m (a(x) ∧ n)|f n (x)| dx ≤ m(m -1) 1 m -1 f ′′ n X 1 + f n X 1 + g n Xm ≤ m (a ∧ n)f n -g n X 1 + [m(m -1) + 1]κ ≤ m sup l≥1 { (a ∧ l)f l X 1 } + (m 2 + 1)κ ≤ (m 2 + m + 1)κ ,
where the last inequality is due to (4.2). Thus, in all cases for m ≥ 1, we have shown that

∞ 0 (x + x m ) (a(x) ∧ n)|f n (x)| dx ≤ (m 2 + m + 2)κ .
Fixing N ≥ 1, we deduce from the previous estimate that, for all n ≥ N,

∞ 0 (x + x m ) (a(x) ∧ N)|f n (x)| dx ≤ ∞ 0 x m (a(x) ∧ n)|f n (x)| dx ≤ (m 2 + m + 2)κ .
We then let n → ∞ and infer from (4.1) that

∞ 0 (x + x m ) (a(x) ∧ N)|f (x)| dx ≤ (m 2 + m + 2)κ . Using Fatou's lemma to let N → ∞, we conclude that af ∈ X 1,m with af X 1,m ≤ (m 2 + m + 2)κ . (4.3)
Step 2. We next show that ((a ∧ n)f n ) n≥1 converges to af in X 1,m . Let χ ∈ C ∞ ((0, ∞)) be such that χ(x) = 1 for x > 2, χ(x) = 0 for x ∈ (0, 1), and

χ(x) ∈ [0, 1] for x ∈ [1, 2]. Introducing χ R (x) := χ(x/R) for x ∈ (0, ∞) and R > 1, we deduce from Lemma 2.3 (with ℓ(x) = x m χ R (x)) that ∞ 0 x m (a(x) ∧ n)χ R (x)|f n (x)| dx + ∞ 0 [x m χ ′ R (x) + mx m-1 χ R (x)]|f n | ′ (x) dx ≤ ∞ 0 xχ R (x)sign(f n (x))g n (x) dx ≤ ∞ R x |g n (x)| dx . Since ∞ 0 [x m χ ′ R (x) + mx m-1 χ R (x)]|f n | ′ (x) dx = - ∞ 0 [x m χ ′′ R (x) + 2mx m-1 χ ′ R (x) + m(m -1)x m-2 χ R (x)]|f n (x)| dx = - ∞ 0 x m |f n (x)| 1 R 2 χ ′′ x R + 2m Rx χ ′ x R + m(m -1) x 2 χ x R dx and χ ′′ (y) + 2m y χ ′ (y) + m(m -1) y 2 χ(y) ≤ 3m 2 χ ′′ L∞(0,∞) , y ∈ (0, ∞) ,
we further obtain

∞ 2R x m (a(x) ∧ n)|f n (x)| dx ≤ ∞ 0 x m (a(x) ∧ n)χ R (x)|f n (x)| dx ≤ sup l≥1 ∞ R x m |g l (x)| dx + 3m 2 R 2 χ ′′ L∞(0,∞) f n Xm ≤ sup l≥1 ∞ R x m |g l (x)| dx + 3m 2 κ R 2 χ ′′ L∞(0,∞) for n ≥ 1. Since lim R→∞ sup l≥1 ∞ R (x + x m ) |g l (x)| dx = 0 by (4.1), we conclude lim R→∞ sup n≥1 ∞ 2R (x + x m ) (a(x) ∧ n)|f n (x)| dx = 0 . (4.4) Now, let R > 1. Since a ∈ L ∞ (0, 2R) by (1.2), there is n R ≥ 1 such that a(x) ∧ n = a(x) for x ∈ (0, 2R) and n ≥ n R . Consequently, for n ≥ n R , (a ∧ n)f n -af X 1,m ≤ 2R 0 (x + x m ) a(x)|(f n -f )(x)| dx + ∞ 2R (x + x m ) (a(x) ∧ n)|f n (x)| dx + ∞ 2R (x + x m ) a(x)|f (x)| dx ≤ a L∞(0,2R) f n -f X 1,m + sup l≥1 ∞ 2R (x + x m ) (a(x) ∧ l)|f l (x)| dx + ∞ 2R (x + x m ) a(x)|f (x)| dx .
We then pass to the limit as n → ∞ and infer from (4.1) that lim sup

n→∞ (a ∧ n)f n -af X 1,m ≤ sup l≥1 ∞ 2R (x + x m ) (a(x) ∧ l)|f l (x)| dx + ∞ 2R (x + x m ) a(x)|f (x)| dx .
We finally let R → ∞ with the help of (4.3) and (4.4) and end up with

lim n→∞ (a ∧ n)f n -af X 1,m = 0 . (4.5)
Step 3. We finally show that f ∈ dom(A 0,m ). Indeed, it readily follows from (4.1) that (f ′′ n ) n≥1 converges to f ′′ in the sense of distributions, while (4.1), (4.3), and (4.5) guarantee that (f ′′ n ) n≥1 = ((a ∧ n)f n -g n ) n≥1 converges to af -g in X 1,m . Therefore, f ′′ belongs to X 1,m with f ′′ = af -g and (f ′′ n ) n≥1 converges to f ′′ in X 1,m . Since f n (0) = 0 for n ≥ 1, this convergence along with Lemma 2.1 ensures that f (0) = 0 and we have proved that f ∈ dom(A 0,m ).

For further use, we show that the graph norm of A a,m in X 1,m controls independently the diffusive and absorption terms in X 1,m .

Lemma 4.3. Assume (1.2). For f ∈ dom(A a,1 ), 1 3 ( A 0,1 f X 1 + af X 1 ) ≤ A a,1 f X 1 . (4.6) Let m > 1. For f ∈ dom(A a,m ), 1 4(m + 1) A 0,m f X 1,m + af X 1,m -m f X 1,m ≤ A a,m f X 1,m . (4.7) Proof. Let f ∈ dom(A a,1 ) and set g := -A a,1 f = -f ′′ + af . It follows from Lemma 2.3 (with ℓ(x) = x) that g X 1 ≥ ∞ 0 x sign(f (x))g(x) dx ≥ ∞ 0 |f | ′ (x) dx + af X 1 = af X 1 .
Consequently,

af X 1 ≤ A a,1 f X 1 and A 0,1 f X 1 = A a,1 f + af X 1 ≤ 2 A a,1 f X 1 , (4.8)
from which we deduce (4.6).

Next, let m > 1 and consider f ∈ dom(A a,m ). We set g := -A a,m f = -f ′′ + af and infer from Lemma 2.3 (with ℓ(x) = x m ) that

g Xm ≥ ∞ 0 x m sign(f (x))g(x) dx ≥ m ∞ 0 x m-1 |f | ′ (x) dx + af Xm = af Xm -m(m -1) f X m-2 .
Either m ≥ 3 and it follows from Young's inequality and the above inequality that

af Xm ≤ g Xm + m(m -1) m -3 m -1 f Xm + 2 m -1 f X 1 ≤ g Xm + m(m -3) f Xm + 2m f X 1 ≤ g Xm + m 2 f X 1,m . (4.9a)
Or m ∈ (1, 3) and we infer from (2.2) and (4.8) that

af Xm ≤ g Xm + m(m -1) 1 m -1 f ′′ X 1 + f X 1 ≤ g Xm + m A 0,1 f X 1 + m(m -1) f X 1 ≤ g Xm + 2m A a,1 f X 1 + m 2 f X 1 . (4.9b)
Collecting (4.8) and (4.9) leads us to

af X 1,m ≤ (1 + 2m) A a,m f X 1,m + m 2 f X 1,m , which in turn gives A 0,m f X 1,m ≤ A a,m f X 1,m + af X 1,m ≤ 2(1 + m) A a,m f X 1,m + m 2 f X 1,m .
Consequently,

1 4(1 + m) A 0,m f X 1,m + af X 1,m ≤ A a,m f X 1,m + m 2 2(m + 1) f X 1,m ,
from which (4.7) follows.

The Fragmentation-Diffusion Semigroup

We now consider the operator A m = A a,m + B m , where we recall that the nonlocal operator B m on X 1,m is defined by

dom(B m ) = {f ∈ X 1,m : af ∈ X 1,m } , B m f (x) = ∞ x a(y)b(x, y)f (y) dy , x ∈ (0, ∞) , f ∈ dom(B m ) .
We first, show that B m is A a,m -bounded in X 1,m . (5.1)

In addition, M 1 (B m f ) = M 1 (af ) , f ∈ dom(B m ) , (5.2) and B m is A a,m -bounded in X 1,m .
Proof. We infer from (1.3) and Fubini-Tonelli's theorem that Finally, let f ∈ dom(A a,m ) ⊂ dom(B m ). By (4.7) and (5.1),

B m f X 1,m ≤ ∞ 0 (x + x m ) ∞ x a(y)b(x, y)f (y) dy dx ≤ af X 1,m ≤ 4(m + 1) A a,m f X 1,m + 4m(m + 1) f X 1,m , so that B m is A a,m -bounded.
As already observed in the literature, see, e.g., [7, Theorem 5.1.47 (c)], the inequality (1.8) implies that, for each m > 1, there is δ m ∈ (0, 1) such that

(1 -δ m )y m ≥ y 0 x m b(x, y) dx , y ∈ (0, ∞) . (5.3)
An immediate consequence of (5.3) is a strict domination of af over B m f in X m .

Lemma 5.2. Assume (1.2), (1.3), and (1.8). Let m > 1 and consider f ∈ dom(B m ). Then

B m f Xm ≤ (1 -δ m ) af Xm .
Proof. It readily follows from (5.3) and Fubini's theorem that

B m f Xm ≤ ∞ 0 x m ∞ x a(y)b(x, y)|f (y)| dydx = ∞ 0 a(y)|f (y)| y 0 x m b(x, y) dxdy ≤ (1 -δ m ) ∞ 0 y m a(y)|f (y)|dy = (1 -δ m ) af Xm .
We shall see next that the property (5.3) ensures that B m is a Miyadera perturbation of A a,m . Recall that a similar result is available for the fragmentation equation without diffusion [START_REF]Global solutions of continuous coagulation-fragmentation equations with unbounded coefficients[END_REF].

Proposition 5.3. Let m > 1 and assume (1.2), (1.3), and (1.8). Then there are q m ∈ (0, 1) and

t m > 0 such that tm 0 B m e sAa,m f X 1,m ds q m f X 1,m , f ∈ dom(A a,m ) .
In particular, B m is a Miyadera perturbation of A a,m .

Proof. Consider f ∈ dom(A a,m ) and set F (t) := e tAa,m f for t ≥ 0. Owing to Proposition 4.1, we have

F (t) X 1 ≤ f X 1 , F (t) X 1,m ≤ e ωmt f X 1,m , t ≥ 0 . (5.4) In addition, F is a classical solution to d dt F -A a,m F = 0 , t > 0 , F (0) = f ,
and we deduce from Lemma 2.3 (with

ℓ(x) = x m ) that d dt F Xm + aF Xm ≤ m(m -1) F X m-2 , t ≥ 0 .
Hence, after integration with respect to time,

t 0 aF (s) Xm ds ≤ f Xm + m(m -1) t 0 F (s) X m-2 ds , t ≥ 0 . (5.5)
Now, let t > 0. It follows from (1.3), (5.1) (with m = 1), and Lemma 5.2 that

t 0 B m F (s) X 1,m ds ≤ t 0 aF (s) X 1 ds + (1 -δ m ) t 0 aF (s) Xm ds . (5.6)
For R > 1, we infer from (1.2) and (5.4) that

t 0 aF (s) X 1 ds ≤ a L∞(0,R) t 0 R 0 x|F (s, x)| dxds + R 1-m t 0 ∞ R x m a(x)|F (s, x)| dxds ≤ a L∞(0,R) t 0 F (s) X 1 ds + R 1-m t 0 ∞ 0 x m a(x)|F (s, x)| dxds ≤ a L∞(0,R) f X 1 t + R 1-m t 0 aF (s) Xm ds .
Combining (5.5), (5.6), and the above estimate with

R = R m := (δ m /2) 1/(1-m) gives t 0 B m F (s) X 1,m ds ≤ a L∞(0,Rm) f X 1 t + 1 - δ m 2 t 0 aF (s) Xm ds ≤ a L∞(0,Rm) f X 1 t + 1 - δ m 2 f Xm + m(m -1) 1 - δ m 2 t 0 F (s) X m-2 ds ≤ a L∞(0,Rm) f X 1 t + 1 - δ m 2 f Xm + m(m -1) t 0 F (s) X m-2 ds . (5.7)
At this point, we handle the cases m ≥ 3 and m ∈ (1, 3) in a different way. We first consider m ≥ 3. use Young's inequality, along with (5.4), to obtain

m(m -1) t 0 F (s) X m-2 ds ≤ m(m -1) t 0 m -3 m -1 F (s) Xm + 2 m -1 F (s) X 1 ds ≤ m(m -3) 1 + ω m f X 1,m e (1+ωm)t -1 + 2m f X 1 t . (5.8)
Collecting (5.7) and (5.8) leads us to

t 0 B m F (s) X 1,m ds ≤ a L∞(0,Rm) t + m(m -3) 1 + ω m e (1+ωm)t -1 + 2mt f X 1 + 1 - δ m 2 + m(m -3) 1 + ω m e (1+ωm)t -1 f Xm .
We now pick t m > 0 such that a L∞(0,Rm) + 2m t m ≤ 1 -δ m 2 and m(m -3) 1 + ω m e (1+ωm)tm -1 ≤ δ m 4 and infer from the previous estimate (with

t = t m ) that tm 0 B m F (s) X 1,m ds ≤ 1 - δ m 4 f X 1,m .
Recalling that B m is A a,m -bounded by Lemma 5.1, we have thus established that B m is a Miyadera perturbation of A a,m for m ≥ 3.

Let us now consider m ∈ (1, 3). In that case, m -2 ∈ (-1, 1) and it follows from Lemma 2.1, Lemma 4.3, and (5.4) that, for s ∈ (0, t),

F (s) X m-2 ≤ 2(3 -m) (m-3)/2 m -1 F ′′ (s) (3-m)/2 X 1 F (s) (m-1)/2 X 1 ≤ 6(3 -m) (m-3)/2 m -1 A a,1 F (s) (3-m)/2 X 1 f (m-1)/2 X 1 .
Owing to the analyticity of e tAa,m t≥0 , see Proposition 4.1, we further infer from [26, Theorem 2.5.2] that there is C > 0 such that

A a,1 e sA a,1 L(X 1 ) ≤ C e s s ≤ C e t s , s ∈ (0, t) .
Combining the above two estimates gives

F (s) X m-2 ≤ C(m) f X 1 e (3-m)t/2 s (m-3)/2 , s ∈ (0, t) .
Hence, recalling (5.7),

t 0 B m F (s) X 1,m ds ≤ a L∞(0,Rm) t + C(m)e (3-m)t/2 t (m-1)/2 f X 1 + 1 - δ m 2 f Xm .
We now choose t m > 0 such that

a L∞(0,Rm) t m C(m)e (3-m)tm/2 t (m-1)/2 m ≤ 1 - δ m 2 and deduce from the previous inequality (with t = t m ) that tm 0 B m F (s) X 1,m ds ≤ 1 - δ m 2 f X 1,m .
Consequently, using again Lemma 5.1, B m is also a Miyadera perturbation of A a,m when m ∈

We are now in a position to prove the first two statements in Theorem 1.1 for the operator

A m = A a,m + B m :
Proof of Theorem 1.1 (a)-(b). We handle the cases m = 1 and m > 1 separately.

(a). If m = 1, then A a,1 ∈ G + (X 1 , 1, 0) by Proposition 4.1, so that it generates a substochastic semigroup in X 1 . Moreover, dom(A a,1 ) ⊂ dom(B 1 ) and B 1 is obviously positive due to the nonnegativity of a and b. Also, for f ∈ dom(A a,1 ),

M 1 (A a,1 f + B 1 f ) = - ∞ 0 f ′ (x) dx -M 1 (af ) + M 1 (B 1 f ) = 0
by Lemma 2.1, (5.2), and the Dirichlet boundary condition. Consequently, we infer from [START_REF] Voigt | On substochastic C 0 -semigroups and their generators[END_REF] and [7, Theorem 4.9.16] that there is an extension Ã1

∈ G + (X 1 , 1, 0) of A 1 . (b). Let m > 1. Since A a,m ∈ H(X 1,m ) by Proposition 4.1 and B m is a Miyadera perturbation of A a,m by Proposition 5.3, it follows from [12, Corollary III.3.16 & Exercise III.3.17] that A m = A a,m + B m ∈ H(X 1,m ) with dom(A m ) = dom(A a,m ). Note that D(A a,m ) and D(A m ) are both Banach spaces and that D(A a,m ) is continuously embedded in D(A m ), since B m is A a,m -bounded in X 1,m according to Lemma 5.1. Consequently, D(A m ) . = D(A a,m
) by the open mapping theorem. We now check the positivity of e tAm t≥0 , bearing in mind that we already know from Proposition 4.1 that A a,m is resolvent positive. Pick λ > 0 sufficiently large. Then λ -A m is invertible with inverse given by

(λ -A m ) -1 = (λ -A a,m -B m ) -1 = (λ -A a,m ) -1 1 -B m (λ -A a,m ) -1 -1 = (λ -A a,m ) -1 ∞ j=0 B m (λ -A a,m ) -1 j ,
where the Neumann series converges since B m is a Miyadera perturbation of A a,m , see the proof of [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]Theorem III.3.14]. Now, B m is obviously a positive operator on X 1,m due to the non-negativity of a and b, and the positivity of (λ -A m -1 directly follows from the above identity.

Finally, as in the proof of (a), we have M 1 (A m f ) = 0 for any f ∈ dom(A a,m ) by Lemma 2.1, (5.2), and the Dirichlet boundary condition, so that (1.9) immediately follows.

Proof of Proposition 1.2 (a). Let m ≥ 1. The operator A a,m belongs to G + (X 1,m ) ∩ H(X 1,m ) by Proposition 4.1. Since a ∈ L ∞ (0, ∞) and b satisfies (1.
3), the operator B m is a positive bounded operator on X 1,m . On the one hand, it now follows from well-known perturbation results that A m = A a,m + B m belongs to H(X 1,m ), see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Theorem 3.2.1]. On the other hand, the same argument as in the proof of Theorem 1.1 (b) ensures the positivity of e tAm t≥0 . Finally, for m = 1, it readily follows from [7, Proposition 4.9.16] that A 1 = Ã1 ∈ G + (X 1 , 1, 0), thereby completing the proof.

Immediate Compactness of the Semigroup

We now turn to compactness properties of the semigroup (e tAm ) t≥0 for m > 1 as stated in Theorem 1.1 (c). To avoid loss of compactness for large sizes, we further require a to diverge to infinity for large sizes, thus excluding bounded overall fragmentation rates.

(x)| + x|f ′ n (x)|} ≤ C , n ≥ 1 , (6.1a)
f n X 1,m + f ′′ n X 1 + af n Xm ≤ C , n ≥ 1 . (6.1b)
On the one hand, we infer from (6.1a) and Arzelà-Ascoli's theorem that (f

n ) n≥1 is relatively compact in C([1/R, R]) for each R > 1.
There are thus a subsequence (f n j ) j≥1 and f ∈ C((0, ∞)) such that

lim j→∞ f n j (x) = f (x) , x ∈ (0, ∞) . (6.2)
On the other hand, it follows from (6.1) that, if R > 1 and E is a measurable subset of (0, ∞), then, for n ≥ 1,

E (x + x m )|f n (x)| dx ≤ E (x + x m )1 (0,R) (x)|f n (x)| dx + ∞ R (x + x m )|f n (x)| dx ≤ (R + R m ) f n L∞(0,∞) |E ∩ (0, R)| + 2 inf x≥R {a(x)} ∞ R x m a(x)|f n (x)| dx ≤ 2CR m |E ∩ (0, R)| + 2C inf x≥R {a(x)} . ( 6.3)
A first consequence of (6.3) with We next infer from (6.3) that, for δ > 0,

E = (R, ∞) is that sup n≥1 ∞ R (x + x m )|f n (x)| dx ≤ 2C inf x≥R {a(x)} ,
η(δ) := sup E (x + x m )|f n (x)| dx : n ≥ 1 , E ∈ B((0, ∞)) , |E| ≤ δ satisfies η(δ) ≤ 2CR m δ + 2C inf x≥R {a(x)}
for all R > 1. We first let δ → 0 and then R → ∞ in the above estimate and use once more (1.10) to conclude that lim δ→0 η(δ) = 0 . (6.5)

Gathering (6.4) and (6.5) implies that the sequence (f n ) n≥1 is uniformly integrable in X 1,m and thus weakly compact in X 1,m by Dunford-Pettis' theorem. This just established weak compactness in X 1,m , along with the pointwise convergence (6.2) and Vitali's theorem, entails that (f n j ) j≥1 converges to f in X 1,m , thereby completing the proof.

We are now in a position to finish off the proof of Theorem 1.1.

Proof of Theorem 1.1 (c). Let m > 1. By Lemma 6.1, (λ -A m ) -1 is compact for λ > 0 large enough and, since m > 1, the analyticity of e tAm t≥0 implies that it is continuous with respect to the operator norm for positive times [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Lemma 2.4.2]. We now may apply [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Theorem 2.3.3] to conclude that e tAm t≥0 is immediately compact.

The compactness result of Lemma 6.1 is not valid under the sole assumption (1.2) on a. In particular, we show that it fails when a is bounded. Lemma 6.2. Let m ≥ 1 and assume that a ∈ L ∞ (0, ∞). Then the embedding of [START_REF] Amann | Ordinary differential equations[END_REF], and ϕ L 1 (R) = 1. We fix m ≥ 1 and set

D(A a,m ) . = D(A m ) in X 1,m is not compact. Proof. Let ϕ ∈ C ∞ c (R) be such that 0 ≤ ϕ ≤ 1, supp ϕ ⊂ [-1,
ϕ n (x) := 1 n + n m ϕ(x -n) , x ∈ (0, ∞) , n ≥ 1 .
Straightforward computations show that

ϕ n X 1,m + aϕ n X 1,m 1 + a L∞(0,∞) ϕ n X 1,m ≤ 1 + a L∞(0,∞) 1 + 1 -1 1 + m2 m n m-1 n + n m ϕ(y) dy ≤ (2 + m2 m ) 1 + a L∞(0,∞)
and

ϕ ′′ n X 1,m ≤ n + 1 + (n + 1) m n + n m 1 -1 |ϕ ′′ (y)| dy ≤ 2 m ϕ ′′ L 1 (R) ,
as well as lim

n→∞ ϕ n X 1,m = 1 , lim n→∞ ϕ n L∞(0,∞) = 0 . (6.6)
Therefore, the sequence (ϕ n ) n≥1 is bounded in D(A a,m ) but cannot converge in X 1,m due to (6.6).

Proof of Proposition 1.2 (b). Let m ≥ 1 and a ∈ L ∞ (0, ∞). Since D(A m ) is not compactly embedded in X 1,m by Lemma 6.2, the resolvent (λ -A m ) -1 is not compact. Hence, [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Theorem 2.3.3] implies that the semigroup (e tAm ) t≥0 is not compact.

Steady States and Convergence

Throughout this section, we assume that a and b satisfy (1.2), (1.3), (1.8), and (1.10), and that a > 0 and b > 0.

We begin with the construction of stationary solutions with the help of Schauder's fixed point theorem.

Lemma 7.1. There is a unique nonnegative

ψ 1 ∈ r≥1 dom(A r )
such that M 1 (ψ 1 ) = 1 and ker(A m ) = ker(A 2 m ) = Rψ 1 for all m ≥ 1. Proof. We split the proof into three steps.

Step 1. The uniqueness of a solution ψ ∈ dom(A 1 ) to A 1 ψ = 0 satisfying M 1 (ψ) = 1 relies on the dissipativity properties of A 1 in X 1 and can be shown exactly as in the proofs of [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]Lemma 3.5] and [START_REF] Ph | Steady states for a fragmentation equation with size diffusion[END_REF]Proposition 3], to which we refer.

Step 2. We now turn to the existence part. Let m ≥ 3 and consider f ∈ X + 1,m satisfying M 1 (f ) = 1. Setting F (t) := e tAm f for t ≥ 0, it readily follows from Theorem 1.1 (b) that

F (t) ≥ 0 and M 1 (F (t)) = 1 , t ≥ 0 . (7.1)
Next, by (5.3) and Fubini's theorem,

d dt M m (F (t)) = -m ∞ 0 x m-1 ∂ x F (t, x) dx - ∞ 0 x m a(x)F (t, x) dx + ∞ 0 a(y)F (t, y) y 0 x m b(x, y) dxdy ≤ m(m -1)M m-2 (F (t)) -δ m M m (aF (t)) .
Owing to (1.10), there is x * > 0 that a(x) ≥ 1 for x ≥ x * . Consequently, using (7.1),

d dt M m (F (t)) + δ m M m (F (t)) ≤ d dt M m (F (t)) + δ m ∞ x * x m F (t, x) dx + δ m x * 0 x m F (t, x) dx ≤ d dt M m (F (t)) + δ m M m (aF (t)) + δ m x m-1 * x * 0 xF (t, x) dx ≤ m(m -1)M m-2 (F (t)) + δ m x m-1 * .
Since m ≥ 3, we now deduce from Young's inequality that

d dt M m (F (t)) + δ m M m (F (t)) ≤ δ m 2 M m (F (t)) + 2m 2m(m -3) δ m (m-3)/2 M 1 (F (t)) + δ m x m-1 * .
Hence, by (7.1),

d dt M m (F (t)) + δ m 2 M m (F (t)) ≤ 2m 2m(m -3) δ m (m-3)/2 + δ m x m-1 * =: δ m µ m 2 , t ≥ 0 .
After integration with respect to time, we conclude that

M m (F (t)) ≤ max {M m (f ), µ m } , t ≥ 0 . (7.2) Now, introducing C m := {f ∈ X + 1,m : M 1 (f ) = 1 , M m (f ) ≤ µ m } , which is a closed convex subset of X 1,m ,
an immediate consequence of (7.1) and (7.2) is that

e tAm C m ⊂ C m , t ≥ 0 .
Owing to the compactness of e tAm in X 1,m for all t > 0, see Theorem 1.1 (c), we argue as in the proofs of [START_REF] Amann | Ordinary differential equations[END_REF]Theorem 22.13] and [17, Theorem 5.2] to deduce from Schauder's fixed point theorem that there is ψ m ∈ C m such that e tAm ψ m = ψ m , t ≥ 0 . Equivalently, A m ψ m = 0 and we have thus shown the existence of a stationary solution to (1.1a) for m ≥ 3. Obviously, ψ 3 also belongs to dom(A m ) and satisfies A m ψ 3 = 0 for any m ∈ [1, 3). Thus, there is at least one stationary solution ψ m to (1.1) for any m ≥ 1. Obviously, ψ m ∈ dom(A 1 ) solves A 1 ψ m = 0 for every m ≥ 1, and we infer from Step 1 that ψ m = ψ 1 for every m ≥ 1.

Step 3. We finally identify ker(A 2 m ). To this end, let f ∈ ker(A 2 m ). Then A m f belongs to ker(A m ), so that Step 2 implies that there is µ ∈ C such that A m f = µψ 1 . Therefore,

µ = µM 1 (ψ 1 ) = M 1 (A m f ) = 0 .
Hence, f ∈ ker(A ).

We now supply refined information on the spectrum of A m for m > 1. Since zero belongs to the spectrum of A m by Lemma 7.1, we deduce from (7.5) that s(A m ) = 0 is a pole of the resolvent of A m . Recalling that (e tAm ) t≥0 is a positive semigroup on the Banach lattice X 1,m , it follows from [START_REF] Clément | One-parameter semigroups[END_REF]Theorem 8.14] that σ(A m )∩iR is either reduced to {0} or contains infinitely many elements. The latter being ruled out by (7.4), we conclude that σ(A m ) ∩ iR = {0}. Since all eigenvalues are isolated, this last property ensures that there is ε m > 0 such that (7.3) holds true. Finally, since ker(A m ) = ker(A 2 m ) = Rψ 1 by Lemma 7.1, zero is a simple eigenvalue of A m according to [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF]Section IV.1.17].

Proof of Theorem 1.5. Let m > 1. From Lemma 7.1 we obtain the existence of a unique nonnegative

ψ 1 ∈ r≥1 dom(A r )
such that M 1 (ψ 1 ) = 1 and ker(A m ) = Rψ 1 . We next infer from Lemma 7.2 that zero is a dominant eigenvalue of A m and a first-order pole of its resolvent with residue P , where P ∈ L(X 1,m ) denotes the spectral projection onto ker(A m ) and is given by It only remains to identify the spectral projection P . Introducing g λ := λ(λ-A m ) -1 f for f ∈ X 1,m , we λf = λg λ -A m g λ , from which we readily deduce that M 1 (f ) = M 1 (g λ ). Therefore, (7.6) implies M 1 (P f ) = M 1 (f ). Since P f ∈ Rψ 1 and M 1 (ψ 1 ) = 1, we conclude that

P f = lim λ→0 λ(λ -A m ) -1 f , f ∈ X 1,m , (7.6
P f = M 1 (f )ψ 1 , f ∈ X 1,m .
Recalling (7.7), the above identity completes the proof of Theorem 1.5.

Stationary Solutions Revisited

We now prove the existence of a stationary solution to (1.1) when the overall fragmentation rate a may be bounded for large sizes but does not decay to zero. Specifically, we assume that a satisfies (1.12); that is,

α := 1 2 lim inf x→∞ a(x) ∈ (0, ∞) .
Proof of Proposition 1.6. As in Lemma 7.1, the proof of the uniqueness assertion in Proposition 1.6 relies on the dissipativity properties of A 1 in X 1 and can be shown exactly as in the proofs of [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]Lemma 3.5] and [START_REF] Ph | Steady states for a fragmentation equation with size diffusion[END_REF]Proposition 3], to which we refer.

As for the existence assertion, we employ a compactness method. Let n ≥ 1. We set a n (x) := a(x) + x/n for x > 0 and, for m ≥ 1, we denote the operators B m and A m with a n instead of a by B m,n and A m,n , respectively. Since a n (x) → ∞ as x → ∞, we infer from Lemma 7.1 that there is a unique nonnegative ψ 1,n ∈ r≥1 dom(A r,n ) such that M 1 (ψ 1,n ) = 1 and ker(A m,n ) = Rψ 1,n for all m ≥ 1. In particular, given m > 3, the function ψ 1,n belongs to dom(A 0,m ) with a n ψ 1,n ∈ X 1,m and solves for n ≥ 1. Moreover, a straightforward consequence of (8.5), (8.6), and Hölder's inequality is that (8.5) is also true for m ∈ (1, 3] with a suitable constant c 2 (m). We next claim that (ψ 1,n ) n≥1 is relatively compact in X 1,m for any m ≥ 1. To this end, we note that, thanks to (8.5), (8.6), Lemma 2.1, and Fubini's theorem,

ψ 1,n X 0 ≤ 1 √ 2 ψ ′′ 1,n 1/2 X 1 ψ 1,n 1/2 X 1 ≤ c 3 2 and ψ 1,n X 1 + ψ ′ 1,n X 1 ≤ 1 + ∞ 0 x ∞ x |ψ ′′ 1,n (y)| dydx ≤ 1 + ψ ′′ 1,n X 2 2 ≤ 1 + c 2 (2)
for n ≥ 1. Now, let m ≥ 2. In view of (8.5) and the above estimates, (ψ 1,n ) n≥1 is a bounded sequence in X 0 ∩ X m ∩ W 1 1 ((0, ∞), xdx) and it follows from [8, Proposition 7.2.2] that (ψ 1,n ) n≥1 is relatively compact in X r for any r ∈ (0, m). As m ≥ 2 is arbitrary, we conclude that there are ψ 1 ∈ r>0 X r and a subsequence (ψ 1,n j ) j≥1 of (ψ 1,n ) n≥1 such that lim j→∞ ψ 1,n j -ψ 1 Xr = 0 for all r > 0 .

(8.7)

An immediate consequence of (8.7) and the properties of (ψ 1,n ) n≥1 is that It readily follows from (8.5) that

ψ 1 ∈ X +
R n X 1,m ≤ 2 n ψ 1,n X 2 + ψ 1,n X m+1 ≤ 2 n (c 2 (2) + c 2 (m + 1)) , so that lim n→∞ R n X 1,m = 0 . (8.9)
In view of (8.8) and (8.9), the sequence (A m ψ 1,n j ) j≥1 converges to zero in X 1,m as j → ∞ and, since A m is closed on X 1,m , we readily deduce from (8.7) that ψ 1 ∈ dom(A m ) and A m ψ 1 = 0.

Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F-31062 Toulouse Cedex 9, France Email address: laurenco@math.univ-toulouse.fr

Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, D-30167 Hannover, Germany

Email address: walker@ifam.uni-hannover.de

  )as well as the nonlocal operatorB m on X 1,m by dom(B m ) := {f ∈ X 1,m : af ∈ X 1,m } ⊂ dom(A a,m ) , B m f (x) := ∞ x a(y)b(x, y)f (y) dy , x ∈ (0, ∞) , f ∈ dom(B m ) .(1.5)Owing to (1.3) the operator B m turns out to be well-defined, see Lemma 5.1. SettingA m := A a,m + B m with dom(A m ) := dom(A a,m ) ,(1.6)Equation (1.1) can be equivalently formulated as the Cauchy problem d dt φ = A m φ , t > 0 , φ(0) = f ,

Lemma 4 . 2 .

 42 Assume that a satisfies (1.2) and let m ≥ 1. Then dom( Âa,m ) = dom(A a,m ).

Lemma 5 . 1 .

 51 Assume (1.2) and (1.3). Let m ≥ 1 and consider a measurable function f on (0, ∞) such that af ∈ X m . Then )b(x, y)f (y) dy dx ≤ af Xm .

  )b(x, y)|f (y)| dydx = ∞ 0 a(y)|f (y)| y 0 x m b(x, y) dxdy ≤ ∞ 0 y m-1 a(y)|f (y)| y 0 xb(x, y) dxdy = af Xm , from (5.1) readily follows. Next, (5.2) is a straightforward consequence of (1.3) and Fubini's theorem.

Lemma 6 . 1 .

 61 Let m ≥ 1 and assume that a satisfies (1.2) and (1.10). Then D(A a,m ) . = D(A m ) is compactly embedded in X 1,m . Proof. Recall that the relation D(A a,m ) . = D(A m ) is established in the proof of Theorem 1.1 (b). Let (f n ) n≥1 be a bounded sequence in D(A a,m ). According to Lemma 2.1 and Lemma 4.3, there is C > 0 such that sup x≥0 {|f n

from which we deduce by ( 1

 1 m )|f n (x)| dx = 0 . (6.4)

Lemma 7 . 2 .

 72 Let m > 1. The spectrum σ(A m ) of A m only consists of isolated eigenvalues and satisfiesσ(A m ) ⊂ {0} ∪ {λ ∈ C : Re λ < -ε m } (7.3)for some ε m > 0. Moreover, s(A m ) = 0 is a simple eigenvalue of A m .Proof. Owing to the immediate compactness of (e tAm ) t≥0 , see Theorem 1.1 (c), and [12, Corollary V.3.2], the spectrum σ(A m only consists of isolated eigenvalues which are poles of the resolvent with finite algebraic multiplicity. Moreover, for any r ∈ R,#{λ ∈ σ(A m ) : Re λ ≥ r} < ∞ . (7.4)We next claim that s(A m ) = 0. Indeed, since A m ⊂ A 1 ⊂ Ã1 and Ã1 ∈ G(X 1 , 1, 0) by Theorem 1.1, any eigenvalue of A m is also an eigenvalue of Ã1 and it follows from[START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] Corollary 1.3.6] that {λ ∈ C : Re λ > 0} ⊂ ρ( Ã1 ) .Consequently, any eigenvalue of A m has a non-positive real part. Thus, σ(A m ) ⊂ {λ ∈ C : Re λ ≤ 0} .(7.5)

-ψ ′′ 1 , 2 . 2 . 1 0 4 ∞ 1 x 4

 1221414 n + a n ψ 1,n = B m,n ψ 1,n in (0, ∞) , ψ 1,n (0) = 0 . (8.1)It follows from (8.1), Lemma 5.2, and Young's inequality that, for ε > 0,M m (a n ψ 1,n ) = M m (B m,n ψ 1,n ) -m ∞ 0 x m-1 ψ ′ 1,n (x) dx ≤ (1 -δ m )M m (a n ψ 1,n ) + m(m -1)M m-2 (ψ 1,n ) ≤ (1 -δ m )M m (a n ψ 1,n ) + m(m -3)εM m (ψ 1,n ) + 2mε (3-m)/2 M 1 (ψ 1,n ) .Hence,δ m M m (a n ψ 1,n ) ≤ m(m -3)εM m (ψ 1,n ) + 2mε (3-m)/2 . (8.2)Owing to (1.12), there isx * > 0 such that a(x) ≥ α , x ≥ x * . (8.3)In view of (8.2) and (8.3), we obtainαδ m M m (ψ 1,n ) ≤ αδ m x m-1 * x * 0 xψ 1,n (x) dx + δ m ∞ x * x m a n (x)ψ 1,n (x) dx ≤ αδ m x m-1 * M 1 (ψ ) + m(m -3)εM m (ψ 1,n ) + 2mε (3-m)/Choosing ε = αδ m /2m(m -3) in the above inequality givesαδ m 2 M m (ψ 1,n ) ≤ αδ m x m-1 *Therefore, there is a positive constant c 1 (m) depending only on a and m such thatM m (ψ 1,n ) ≤ c 1 (m) , n ≥ 1 . (8.4)Several additional estimates can now be derived from(8.4). Indeed, it readily follows from (8.1), (8.2)(with ε = 1), and Lemma 5.2 that, for n ≥ 1,ψ ′′ 1,n Xm + a n ψ 1,n Xm + B m,n ψ 1,n Xm ≤ 2 ( a n ψ 1,n Xm + B m,n ψ 1,n Xm ) ≤ 2(2 -δ m ) a n ψ 1,n Xm ≤ 4M m (a n ψ 1,n ) ≤ 4m(m -3)c 1 (m) + 8m δ m =: c 2 (m) .(8.5)Similarly, by (1.2), (8.1), (8.4) (with m = 4), and Lemma 5.1,ψ ′′ 1,n X 1 + a n ψ 1,n X 1 + B m,n ψ 1,n X 1 ≤ 2 ( a n ψ 1,n X 1 + B m,n ψ 1,n X 1 ) ≤ 4 a n ψ 1,n Xm ≤ 4 1 + a L∞(0,1) xψ 1,n (x) dx + a n (x)ψ 1,n (x) dx ≤ 4 1 + a L∞(0,1) + 4c 2 (4) =: c 3 (8.6)

1 and M 1 (ψ 1 ) = 1 .Finally, let m ≥ 1 .

 11111 We observe that dom(A m,n ) ⊂ dom(A m ) for n ≥ 1 (as a n ≥ a) and that (8.1) also readsA m ψ 1,n = R n ,(8.8) whereR n (x) := -x n ψ 1,n (x) + 1 n ∞ x yb(x, y)ψ 1,n(y) dy , x > 0 .

  ) see, e.g.,[START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF] Section IV.1.17]. It then follows from[START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF] Corollary V.3.3] that there are N m ≥ 1 and ν m > 0 such that e tAm -P L(X 1,m ) ≤ N m e -νmt , t ≥ 0 . (7.7)

We actually compute explicit stationary solutions to (1.1) when a(x) = x γ and b(x, y) = (ν +

2)x ν y -ν-1 for γ ≥ 0, ν ∈ (-2, 0] in[22].

Acknowledgments

This work was done while PhL enjoyed the kind hospitality of the Institut für Angewandte Mathematik, Leibniz Universität Hannover.

Partially supported by Deutscher Akademischer Austauschdienst funding programme Research Stays for University Academics and Scientists, 2021 (57552334).


Sergiy Zhuk

Andrey Polyakov

Homogeneous Observer Design for Finite-dimensional projections of Homogeneous PDEs

Keywords: nonlinear filtering, uniform observability, Lyapunov exponents, bilinear systems, Riccati equations, fixedtime convergence

Sufficient conditions for existence and uniqueness of solutions for a coupled system of homogeneous equations defining dynamics of the gain and observer for ODEs obtained as a Galerkin projection of homogeneous PDEs are proposed. The conditions rely upon fundamental concept of uniform complete observability which is also used to design an exponentially convergent observer. Convergence of the observer is confirmed by numerical experiments with ODEs obtained from a hyperbolic PDE in 1D (Burgers-Hopf equation).

I. INTRODUCTION

Stochastic filtering is fundamental in diverse fields including synchronization in complex networks, data assimilation and control engineering to name just a few. Theoretically, the optimal solution of stochastic filtering problem for Markov diffusions is given by the so-called Kushner-Stratonovich (KS) equation [START_REF] Stratonovich | On the theory of optimal non-linear filtering of random functions[END_REF], a stochastic Partial Differential Equation (PDE) which describes evolution of the conditional density of the states of the underlying diffusion process. For linear systems, KS equation is equivalent to the Kalman-Bucy Filter (KF) equations.

In contrast, deterministic state estimators (observers), including the algorithm presented in this paper, assume that errors have bounded energy and belong to a given bounding set. The state estimate is then defined as the minimax center of the reachability set, and temporal dynamics of the minimax center is described by a minimax filter (or minimum energy estimate) [START_REF] Baras | Dynamic observers as asymptotic limits of recursive filters: Special cases[END_REF]. There is a fundamental connection between stochastic filters and observers: namely, an observer can be obtained as asymptotic limit of KFs associated with the "noisy" version of the state equation and observations; when "noise" dissapears (e.g. noise variance approaches zero) the mean-squared error between stochastic and deterministic state estimates goes to zero, and, in fact, the conditional measure associated with the stochastic filter converges to a degenerate measure concentrated at the observer [START_REF] Baras | Dynamic observers as asymptotic limits of recursive filters: Special cases[END_REF]. Having this in mind we consider the case of exact measurements and zero disturbances. In addition, we do not assume any form of diffusion (or damping): in fact, bilinear ODEs studied below can be obtained as Galerkin projection of nonlinear hyperbolic PDEs including Euler equation in two spatial dimensions [START_REF] Zhuk | Exponentially convergent data assimilation algorithm for navier-stokes equations[END_REF], [START_REF] Zhuk | Parameter estimation for euler equations with uncertain inputs[END_REF], or a finite-difference discretization of the Sergiy Zhuk is with IBM Research, IBM Tech. campus, Damastown, Dublin, D15 HN66, Ireland sergiy.zhuk@ie.ibm.com

Andrey

Polyakov is with Univ. Lille, Inria, CNRS, UMR 9189 CRIStAL, Centralle Lille, F-59000 Lille, France andrey.polyakov@inria.fr Burgers equation in 1D (as discussed in the next section). Such systems have a number of conserved quantities, e.g. energy and volume, and so all the solutions evolve on a given bounded manifold (e.g. a sphere of a certain radius) and never approach zero. Hence, even the case of exact observations is very challenging for dynamical systems of this class.

Our key contribution is in using intrinsic symmetries provided by homogeneity of the state equation in combination with rather mild uniform observability assumptions along observer's trajectory to prove existence and uniqueness of solutions for the system of non-linear differential equations, which describe dynamics of the observer and its gain. The latter coincides with the classical Extended KF locally, on the basin of attraction. Our design relies upon Lyapunov methods like many other works on nonlinear observer design, however we do not require rather strict unifrom observability assumptions on the Jacobian which are imposed to prove existence and uniqueness of the observer and the gain (e.g. [START_REF] Baras | Dynamic observers as asymptotic limits of recursive filters: Special cases[END_REF]), neither do we make an extra assumption of existense of the gain [START_REF] Hinrichsen | Modelling, State Space Analysis, Stability and Robustness[END_REF]. Yet another contribution is the description of the basin of attraction, which can be used in numerical computations as it is demonstrated below, and, more importantly, the description of the deformations of the basin in responce to rescaling of the set of "the true intial conditions". In addition, we quantify the effect of rescaling onto the estimation error convergence speed.

State estimation for bilinear systems is a challenging problem especially in the case of high dimension of the state vector. Indeed, in the latter case it is hard to implement the differential algebraic approach conventional for nonlinear systems [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], [START_REF] Isidori | Nonlinear Control Systems Design[END_REF], [START_REF] Hermann | Nonlinear controllability and observability[END_REF]. Our design however is minimal in that one cannot make use of Lyapunov subspace reduction (e.g. [START_REF] Frank | A detectability criterion and data assimilation for nonlinear differential equations[END_REF], [START_REF] Tranninger | Detectability analysis and observer design for linear time varying systems[END_REF]) and restrict the observer's gain corrections just to the stable Lyapunov subspace due to the following fact: all the Lyapunov exponents of the considered dynamical system equal to zero. Nevertheless, unlike [START_REF] Frank | A detectability criterion and data assimilation for nonlinear differential equations[END_REF] we do not require to observe all "unstable directions" instead requiring the uniform complit observability along the observer's trajectory.

II. MATHEMATICAL PRELIMINARIES

In this section we introduce notations and collect all the required mathematical notions and results. a) Notation: R n -Euclidean space of n-dimensional real vectors with orthonormal basis {ψ 1 . . . ψ n }, inner product (x, y) = Ṗ or ẋ denotes the time derivative of the matrix-or vectorvalued function of time; S n -Hilbert space of symmetric non-negative definite n × n-matrices. λ min (P ) and λ max (P ) denote minimal and maximal eigen-values of a matrix P ; Tr(P ) -trace of P ; I n ∈ S n -the n × n-identity matrix; P = {P ij } n i,j=1 -stands for a matrix P ∈ S n with components P ij .

b) Trace Inequalities: If A is symmetric and B is skewsymmetric then Tr(AB + B A) = 0.

Lemma 1:

If 0 < A ∈ S n then A ∞ = max ij |a ij | ≤ Tr A ≤ √ n Tr A 2 .

A. Motivating examples: Finite dimensional projections

As noted in Section I the class of dynamical systems studied here includes finite dimensional projections of some important PDEs including Burgers(-Hopf) equation (in 1D) and Euler equations (in 2D).

Following [START_REF] Frank | A detectability criterion and data assimilation for nonlinear differential equations[END_REF] we consider the ODE obtained by discretizing the Burgers(-Hopf) equation u t = -

∂ ξ u 2 2
on (0, 1) with periodic boundary conditions by using the finite difference scheme:

ui = - n 6 u i (u i+1 -u i-1 ) + (u 2 i+1 -u 2 i-1 ) (1)
taken on a periodic lattice (i = 1 . . . n, u -1 = u n , u n+1 = u 1 ) which has the properties that

• the quadratic energy i u 2 i is conserved, implying that every sphere in R n is invariant under the motion of the system and u is constant,

• the trace of the Jacobian of the r.h.s. of (1) is zero, implying that the flow conserves the volume of the phase element. Denoting x = (u 1 . . . u n ) and setting D = {D ij } n i,j=1

with D ji = -D ij for j ≤ i, and D ii+1 = 1, D ij = 0 for j > i + 1 except for D 1n = -1, we can rewrite (1) as follows: ẋ = B(x)x with

B(x) = - n 6 (diag(x)D + D diag(x)).
Clearly,

B (x) = - n 6 (D diag(x) + diag(x)D ) = -B(x) since D = -D. Euler 1 equations in 2D ∂ t ω + u • ∇ω = 0 , -∆ψ = ω , u = ū + ∇ ⊥ ψ , ω(0) = curl( u 0 ) , (2)
with periodic boundary conditions at the boundary of a rectangular domain can be approximated by an ODE which has similar properties. Namely, assuming periodic boundary conditions, and applying Fourier-Galerkin (FG) approximation one can project Euler equation onto a 2N + 1-dimensional subspace generated by {e ikx e isy } |k|,|s|≤ N 2 and obtain an ODE for the projection coefficients in the form ẋ = B(x)x with skew-symmetric linear (in x) matrix B. In this case the 1 More specifically, vorticity-streamfunction formulation of Euler equation components of x will represent projection coefficents of the solution. Analogously, for homogeneous Dirichlet conditions one can take sin-basis. We refer the reader to [START_REF] Zhuk | Exponentially convergent data assimilation algorithm for navier-stokes equations[END_REF] for further details.

III. MAIN RESULTS

A. Problem statement

Let x(t) ∈ R n and y(t) ∈ R m denote the state vector and output of the following system:

ẋ(t) = B(x(t)) x(t), x(t 0 ) = x 0 , (3) y
(t) = C(t)x(t) (4)
provided B(x) = -B (x) and x → B(x) is a linear mapping, and C(t) ∈ R m×n is a given measurable matrixvalued function such that

λ max (C (t)C(t)) ≤ c < +∞ .
Let z(t 0 ) = z 0 ∈ R n and P (t 0 ) = P 0 ∈ S n and consider the following system of equations:

ż = B(z)z + P C R(y -Cz), (5)
Ṗ = B(z)P + P B (z) -P C RCP + Q, P (t 0 ) = P 0 (6)
where R ∈ S n and Q ∈ S n are given continuous matrixvalued functions such that

0 < r ≤ λ min (R(t)) ≤ λ max (R(t)) ≤ r < +∞, (7) 0 < q ≤ λ min (Q(t)) ≤ λ max (Q(t)) ≤ q < +∞, (8)
B(z) = B(z) + B 1 (z), B 1 (z) = B(ψ1)z ... B(ψn)z (9)
In the forthcoming sections we report the following results:

• existence and uniqueness of the unique bounded and continuous solution for the system ( 5)-( 6) on any finite interval [t 0 , T ], i.e. that neither z nor P solving (5)-( 6) "blow-up" in finite time, see Section III-B, Theorem 1) • z converges to x exponentially fast provided z 0 belongs to a certain vicinity of x 0 , see Section III-D, Theorem 3 • construction of a homogeneous dilation which appropriately rescales ( 5)-( 6) provided x 0 is rescaled as follows:

x 0 → λx 0 , see Section III-E

B. Existence and uniqueness

Given ρ > 0 let φ ρ be defined as follows

φ ρ (r) = r if r ≤ ρ, ρ if r > ρ.
Theorem 1: Let P 0 ∈ S n and z 0 ∈ R n and x 0 ∈ R n . If P 0 , Q, R > 0 and x(t 0 ) = x 0 , z(t 0 ) = z 0 and P (t 0 ) = P 0 then for any fixed ρ > 0 the system of differential equations composed of (3) and

ż = B(z)z + P C R(y -Cz), (10)
Ṗ = Bρ (z)P +P B ρ (z)-P C RCP +Q, (11)
Bρ (z) = B(z) + φ ρ ( z )B 1 z z (12)
has the unique solution (x, z, P ) such that

(i) x, y ∈ C(t 0 , +∞, R n ) and x(t) = x 0 , ∀t ≥ t 0 (ii) z ∈ C(t 0 , T, R n
) and P ∈ C(t 0 , T, S n ) for any T < +∞ (iii) P (t) > p 1 I for some p 1 > 0.

Proof: Since B(x) = -B (x) it follows that (x, ẋ) = (x, B(x)x) = 0, i.e. x(t) = x(t 0 ) = x 0 < +∞ for all t ≥ t 0 . Hence x ∈ C(t 0 , +∞, R n ) and so is y as c < +∞. This demonstrates (i).

Denote by F (z, P, t) a mapping from R n × S n × R + to R n × S n which is defined by the right hand side of eqs. [START_REF] Tranninger | Detectability analysis and observer design for linear time varying systems[END_REF] and [START_REF] Anderson | Stability properties of kalman-bucy filters[END_REF]. Clearly, F is continuously differentiable w.r.t. z and P everywhere on R n ×S n , hence F (•, •, t) is locally Lipschitz continuous w.r.t. z and P for all t (y(t) is bounded!). By using a fixed-point iteration (the standard argument based on Picard theorem) we conclude that eqs. ( 10) and ( 11) have the unique continuous solution (z, P ) defined on a maximal 2 segment [t 0 , T ), T > t 0 , and that one of the following cases holds:

1) T = +∞ 2) T < +∞, sup t∈[t0,T ) z(t) < +∞ and P (t) → +∞ as t → T 3) T < +∞, z(t) → +∞ as t → T and sup t∈[t0,T ) P (t) < +∞ 4) T < +∞, z(t) → +∞ and P (t) → +∞ as t → T 5) T < +∞, sup t∈[t0,T ) z(t) < +∞, sup t∈[t0,T ) P (t) < +∞ and P (T ) is not positive definite.
To conclude the proof let us show that the case 1) above is the only possible one, i.e. that for any T < +∞ there exists the unique solution of eqs. [START_REF] Tranninger | Detectability analysis and observer design for linear time varying systems[END_REF] and [START_REF] Anderson | Stability properties of kalman-bucy filters[END_REF] such that z ∈ C(t 0 , T, R n ) and P ∈ C(t 0 , T, S n ) (claim (ii)), and that P (t) > 0, P -1 < +∞ for all t ∈ [t 0 , T ] provided P 0 , Q, R > 0 (claim (iii)).

Indeed, by contradiction, assume that either 2) or 4) holds true. Note that Tr Bρ P +P B ρ = φ ρ Tr B 1 ( z z )P +P B 1 ( z z ) as Tr(B(z)P ) = 0. Recalling that Tr(B A) is the inner product of matrices B, A we apply Schwartz inequality:

Tr P B 1 ( z z ) ≤ Tr 1 2 (P 2 ) Tr 1 2 B 1 z z B 1 z z ≤ λ 1 2 max ( B1 ) Tr(P ), B1 = {Tr(B 1 (ψ i )B 1 (ψ j )} n i,j=1
Applying Tr to both sides of eq. ( 11) and recalling that φ ρ ( z ) = ρ if z > ρ we get:

Tr( Ṗ ) = Tr Bρ P +P B ρ + Tr(Q) -Tr(P C RCP ) ≤ Tr(Q) + 2λ max 1 2 (
B1 )ρ Tr(P ), 2 Maximal means that the solution cannot be extended beyond [t 0 , T )

By Bellman-Gronwall lemma it now follows that Tr(P (t)) ≤ e 2ρ(t-t0)λ 1 2 max ( B1) Tr(P (t 0 ))

+ Tr(Q) e 2ρ(t-t0)λ 1 2 max ( B1) -1 2ρλ 1 2 max ( B1 ) (13)
hence λ max (P (t)) ≤ Tr(P (t)) < +∞ on [t 0 , T ] which contradicts both 2) and 4). Assume that 3) holds true. Since P (t 0 ) > 0 it follows that P (t) > 0 for all t ∈ [t 0 , T ) (see [START_REF] Anderson | Stability properties of kalman-bucy filters[END_REF]), and hence P -1 (t) is well defined on t ∈ [t 0 , T ), and it is bounded from below on t ∈ [t 0 , T ) as P (t) is uniformly bounded from above on [t 0 , T ) by 3).

Define V 0 = z P -1 z, set e = x -z. If V 0 (z(T )) < +∞ then z(T ) < +∞. Recalling that B(z)z = B 1 (z)z consider V0 = -R 1 2 Ce 2 + R 1 2 Cx 2 -Q 1 2 P -1 z 2 -2φ ρ (P -1 z, B( z z )z) ≤ R 1 2 Cx 2 + φ 2 ρ Q -1 2 B( z z ) -Q 1 2 P -1 z + φ ρ Q -1 2 B( z z ) 2 ≤ R 1 2 Cx 2 + ρ 2 λV 0 (14)
with λ = λ max (P

1 2 B( z z )Q -1 B( z z )P 1
2 ). As P is uniformly bounded on [t 0 , T ) by assumption it follows that λ < +∞, t ∈ [t 0 , T ]. Thus ( 14) implies (e.g. using Bellman-Gronwall lemma) that V 0 (z(T )) < +∞, which contradicts 3).

Finally, assume that 5) holds true. By assumption, P (t) > 0 on [t 0 , T ) and P (T )x = 0 for some x = 0. Hence, W = P -1 (t) is defined for every t ∈ [t 0 , T ), and lim t↑T W (t) = +∞. In this case, Tr W (t) → ∞ as t → T due to Lemma 1. For any t ∈ [t 0 , T ) we have:

Ẇ = -W Ṗ W = -B ρ W -W Bρ -W QW + C RC .
Since Tr(A) = n j=1 ψ i Aψ i and

Tr(W QW ) = n j=1 ψ i (W QW )ψ i ≥ λ min (Q) n j=1 ψ i W 2 ψ i then Tr Ẇ = -Tr( B ρ W + W Bρ + Tr(C RC) -Tr(W QW ) ≤ Tr(C RC) - q n Tr 2 (W ) + 2λ 1 2 max ( B1 )ρ Tr(W ).
which shows that W (T ) < +∞. This completes the proof.

Remark 1: Note that the theorem also holds for the case of ρ = 0 which was considered in [START_REF] Zhuk | On practical fixed-time convergence for differential riccati equations[END_REF].

C. Uniform bounds for DRE

Definition 1: Let z solve [START_REF] Tranninger | Detectability analysis and observer design for linear time varying systems[END_REF] and let Ẋ = Bρ (z)X, X(0) = X 0 and set Φ(t, s) = X(t)X -1 (s). A symmetric matrix N ρ (t, t -σ; R, z) is said to be a gramian along the trajectory z of (10

) if N ρ (t, t -σ; R, z) = t t-σ Φ (s, t)C RCΦ(s, t)ds.
Obviously, N ∞ is a gramian for z being a solution of [START_REF] Hinrichsen | Modelling, State Space Analysis, Stability and Robustness[END_REF].

It is well-known that for LTV systems invertibility of the gramian allows to prove that the observer z(t) converges to the true state x(t) for the case of exact observations [START_REF] Bucy | The riccati equation and its bounds[END_REF], [START_REF] Anderson | Stability properties of kalman-bucy filters[END_REF]. In what follows we generalize this result to systems of class (3). To that end note tha eqs. ( 3), [START_REF] Tranninger | Detectability analysis and observer design for linear time varying systems[END_REF] and [START_REF] Anderson | Stability properties of kalman-bucy filters[END_REF] has the unique solution which is uniquely defined by the initial conditions x 0 , z 0 , P 0 .

Definition 2: Given ρ ≥ 0 we say that the system eqs. ( 3), ( 10) and ( 11) is uniformly completely observable from x 0 , z 0 , P 0 if there exist ν > 0 and α, β > 0 such that αI < N ρ (t, t -σ; R, z) < βI, ∀t > t 0 + σ . Theorem 2: Take ρ > 0. If the system eqs. ( 3), [START_REF] Tranninger | Detectability analysis and observer design for linear time varying systems[END_REF] and ( 11) is uniformly completely observable from x 0 , z 0 , P 0 then

λ max (P (t)) ≤ 1 rα + t t-σ λ max (Q)e ρ λ(t-s) ds ≤ 1 rα + q ρ λ (e ρ λσ -1) , t ≥ t 0 + σ, (15)
where λ2 = n i=1 λ 2 max (B 1 (ψ i ) + B 1 (ψ i )). Proof: Assume that t < +∞ and consider the standard LQR design problem for the system q = -B ρ (z)q + C u, q(t) = h with the cost

J(u) = P 1 2 (t 0 )q(t 0 ) 2 + t t0 R -1 2 u 2 + Q 1 2 q 2 ds
It is known that the feedback û = RCP q minimizes J and J(û) = (P (t)h, h). Hence for any other control u we have that J(û) = (P (t)h, h) ≤ J(u). Let us select u as follows: set u(s) = RC(s)Φ(s, t)N -1 ρ (t, t-σ; R, z)h for s ∈ [t-σ, t] and set u(s) = 0 for s ∈ [t 0 , t -σ). Let us show that for this u one gets: q(s) = 0 for all s ∈ [t 0 , t -σ]. To this end recall that

q(s) = Φ (t, s)h - t s Φ (τ, s)C (τ )u(τ )dτ and so q(t -σ) = 0 if 3 Φ (t, t -σ)h = Φ (t, t -σ) t t-σ Φ (τ, t)C (τ )u(τ )dτ
Clearly, for the above choice of u we get that q(s) = 0 for all s ∈ [t 0 , t -σ]. Hence, for this u we get that

J(û) = (P (t)h, h) ≤ t t-σ (R -1 u, u) + (Qq, q)ds = (N -1 ρ (t, t -σ; R, z)h, h) + t t-σ (Qq, q)ds ≤ h 2 rα + t t-σ (Qq, q)ds (16)
since N ρ (t, t -σ; R, z) ≥ rN ρ (t, t -σ; I, z) ≥ rα. To compute t t-σ q Qqds we first note that q(s) = Φ (t, s)(I -N ρ (t, s; R, z)N -1 ρ (t, t -σ; R, z))h

3 We used the obvious equality

Φ(τ, t -σ) = Φ(τ, t)Φ(t, t -σ) = Φ (t, s)N ρ (s, t -σ; R, z)N -1 ρ (t, t -σ; R, z)h for s ≥ t -σ and so t t-σ (Qq, q)ds = W 1 2 (t, t -σ)N -1 (t, t -σ; R, z)h 2 with W (t, τ ) = t τ N (s, τ ; R, z)Φ(t, s)QΦ (t, s)N (s, τ ; R, z)ds
Now, recall Vazhevskii estimate from [14, p. 110]: Φ(t, s)q(s) ≤ q(s) e

λ max ( Bρ (z)+ B ρ (z)) ≤ ρ z n i=1 z i λ max (B 1 (ψ i )+B 1 (ψ i )) ≤ ρ λ Hence Φ(t, s) ≤ e 1 2 ρ λ(t-s) . Now (w, Φ(t, s)QΦ (t, s)w) ≤ λ max (Q)λ max (Φ(t, s)Φ (t, s)) w 2 = λ max (Q)λ max (Φ (t, s)Φ(t, s)) w 2 = λ max (Q) Φ(t, s) 2 w 2
so that:

W (t, t -σ) ≤ t t-σ λ max (Q)e ρ λ(t-s) N 2 ρ (s, t -σ; R, z)ds.
Finally, we note that N 2 ρ (s, t -σ; R, z) ≤ N 2 ρ (t, t -σ; R, z) hence W (t, t -σ) ≤ N 2 ρ (t, tσ; R, z) t t-σ λ max (Q)e ρ λ(t-s) ds and so t t-σ z Qzds ≤ h 2 t t-σ λ max (Q)e ρ λ(t-s) ds. This and (16) completes the proof.

Corollary 1:

If λ = 0 then λ max (P (t)) ≤ 1 rα + t t-σ λ max (Q)ds ≤ 1 rα + qσ, t ≥ t 0 + σ (17)
Remark 2: Note that eq. ( 17) coincides with the upper bound derived in [START_REF] Zhuk | On practical fixed-time convergence for differential riccati equations[END_REF].

D. Convergence

Let us introduce the following definitions:

β 2 = i B(ψ i ) 2 and p = max e 2ρσλ 1 2 max ( B1) Tr(P (t 0 )) + Tr(Q) e 2ρσλ 1 2 max ( B1) -1 2ρλ 1 2 max ( B1 ) , 1 rα + q ρ λ (e ρ λσ -1) , (18)
Theorem 3:

Set ρ = x(t 0 ) + q 2β Tr(P (t0)) , V 0 = q 2β p 3 2
and let z ρ , P ρ solve eqs. ( 10) and [START_REF] Anderson | Stability properties of kalman-bucy filters[END_REF]. Assume that system eqs. ( 3), ( 10) and ( 11) is uniformly completely observable from x(t 0 ) = x 0 , z(t 0 ) = z 0 and P (t 0 ) = P 0 > 0. If

V (e, t) = (P -1 e, e) and V 2 (e(t 0 ), t 0 ) ≤ κV 0 for 0 < κ < 1 then • z ρ (t) ≤ ρ for all t ≥ t 0 and z ρ , P ρ coincide with the unique solution z, P of ( 5)-( 6)

• N ρ (t, t -σ; I, z ρ ) = N (t, t -σ; I, z); • V (e(t), t) ≤ e -(1-κ) q(t-s) p
V (e(s), s) for all t ≥ s. Proof: Note that since λ max (P (t 0 )) ≤ Tr(P (t

0 )) ≤ p then p-1 2 x(t 0 ) -z(t 0 ) ≤ V 1 2 (e(t 0 ), t 0 ) < V 0 implies that z(t 0 ) ≤ e(t 0 ) + x(t 0 ) < V 0 p 1 2 + x(t 0 ) = q 2β p + x(t 0 ) ≤ ρ (19)
and so z(t) < ρ on a small interval [t 0 , t 1 ]. Then, on [t 0 , t 1 ] we have that φ ρ ( z ) B( z z ) = B(z) + B 1 (z) and thus z ρ , P ρ also solve ( 5) and ( 6) on [t 0 , t 1 ]. Thus, by direct calculation one finds that the estimation error equation takes the following form: ė = B(z)e + B 1 (x)e -P C RCe, e(t 0 ) = e 0 . Hence, to complete the proof it is suffies to demonstrate that z does not grow over time. To this end consider the following Lyapunov function V (e, t) = (P -1 e, e). Compute V (e, t) = 2(P -1 e, B(e)e) -(RCe, Ce) -q P -1 e 2 . It is not hard to see that V (e, t) ≤ P -1 e e 2 B(e) -q P -1 e e -(RCe, Ce)

Note that λ min (P -1 ) ≤ P -1 e e and λ min (P -1 ) = λ -1 max (P ) ≥ p-1 , and B(e) ≤ e β. Also λ min (P -1 )(e, e) ≤ (P -1 e, e) hence e ≤ λ 1 2 max (P ) P -1 2 e ≤ p 1 2 V 1 2 (e, t). Hence 2 B(e) -q P -1 e e ≤ 2β e -q λ max (P )

≤ 2β p 1 2 V 1 2 - q p
and so 2 B(e) -q P -1 e e < 0 provided V 1 2 (e(t), t) <

q 2β p 3 2 = V 0 . Assume that V 1
2 (e(t 0 ), t 0 ) < κV 0 . The latter also holds for all t ∈ [t 0 , t 1 -ε] for some ε > 0, and on [t 0 , t 1 -ε] we get:

V (e, t) ≤ -P -1 e e q p -2β p

1 2 V 1 2 ≤ -V q p -2β p 1 2 κV 0 (20)
as 0 ≤ V (e(t), t) ≤ P -1 e e and so -V ≥ -P -1 e e . Since q p -2β p 1 2 κV 0 = (1-κ)q p > 0 it follows that V (e(t), t) < V (e(t 0 ), t 0 ) < κV 0 on [t 0 , t 1 -ε]. This and (20) implies that V (e(t), t) ≤ e -(1-κ) q(t-s) p V (e(s), s) ≤ κV 0 for all t ≥ t 0 . Hence, as above p- 1 2 x(t) -z(t) ≤ V 2 (e(t), t) < κV 0 . The latter and (19) shows that z(t) < ρ for t ≥ t 0 .

E. Homogeneity: dilation symmetry

It is well known that the homogeneity (dilation symmetry) of a vector field is inherited by its flow. Indeed, if

f ∈ C 1 (R m , R m ) ξ = f (ξ), f (λξ) = λ 2 f (ξ), ∀λ > 0, ∀ξ ∈ R m then ξ(t, λξ 0 ) = λξ(λt, ξ 0 ), (21)
where ξ(•, ξ) is a solution of the ODE with the initial condition ξ(0) = ξ. The obtained symmetry of solutions means that any solution with a scaled initial data λξ 0 can be obtained using a solution with ξ(0) = ξ 0 by means of the scaling of both time and state vector.

Let us rewrite the system (6) in the form

Ṗ = BP +P B -P C RCP + Ξ 2 , P (t 0 ) = P 0 (22) Ξ = 0, Ξ(t 0 ) = Ξ 0 , (23)
where Ξ 0 is an arbitrary symmmtric positive definite matrix. Assuming that t 0 = 0, C(t) ≡ C, R(t) ≡ R and Q(t) ≡ Q we derive that the system (3),( 5), ( 22), ( 23) can be rewritten as follows ξ = F (ξ), where ξ(t) = (x(t), z(t), P (t), Ξ(t)).

It is easy to see that F (λξ) = λ 2 F (ξ) and the identity (21) holds for ξ 0 = (x 0 , z 0 , P 0 , Ξ 0 ). Therefore, twe have proven the following claim. Corollary 2: Let t 0 = 0, C(t) = const, R(t) = const and Q(t) = const and z(t) → x(t) as t → +∞ for some x 0 ∈ R n , z 0 ∈ R n and P 0 ∈ S n then the observer ( 5), (6) will converge to the state of the system (3) with the scaled initial condition x 0 → λx 0 , λ > 0 provided that initial conditions of the observer are scaled z 0 → λz 0 , P 0 → λP 0 as well and the matrix Q is scaled quadratically with respect to λ, i.e. Q → λ 2 Q.

IV. EXPERIMENTS

We perform a number of numerical experiments for the ODE obtained by discretizing the Burgers(-Hopf) equation

u t = - ∂ ξ u 2 2
on (0, 1) with periodic boundary conditions by using the finite difference scheme eq. ( 1). Recall that this ODE conserves energy and multidimensional volume:

x(t) = x 0 and the Lebesgue measure of the set of initial conditions is preserved by ODE's semigroup.

We set n = 8 and select C so that Cx = (x 2 , x 4 x 6 ) , i.e. 3 out of 8 components of x were measured. We stress that the estimate of p given in eq. ( 18) is very conservative from the practical standpoint. To circumvent this we take the smallest possible p but setting ρ = 0 so that p = max{Tr(P (t

0 )) + Tr(Q)σ, 1 rα + qσ}
To estimate α and σ numerically we discretized eqs. (3), ( 5) and ( 6) together with the following Lyapunov equation

Ṅ (t, t 0 ) = -A N (t, t 0 ) -N (t, t 0 )A + C (t)C(t),
with condition N (t 0 , t 0 ) = 0 for the gramian N , A = B(z) + B 1 (z) in time by using RK method of 4th order with ∆t = 1.3 × 10 -4 on the time interval [0, T ] with T = 40.

N was computed on a set of intervals (∆ti, ∆ti + σ) with σ = 8, and i = 0, 1, 2, . . . N , N is such that N ∆t + σ ≤ T . At the beginning of each interval N (∆t(i -1), ∆t(i -1)) was set to 0I 8 , and λ min (N (∆ti+σ, ∆ti)) was computed to get an estimate for α. In all experiments we observed that for σ = 8 we always had λ min (N (∆ti + σ, ∆ti)) > 1 × 10 -3 . Based on this we assumed that α > 1 × 10 -3 and σ = 8. We also computed that β = 7.54.

To compute the basin of attraction, V with p 0 = 1, q = 0.25, r = 1 × 10 2 . In this case we must select z 0 so that x 0 -z 0 ≤ θ 0 = . Note that if we rescale p 0 (λ) = λp 0 , q(λ) = λ 2 q and x 0 (λ) = λx 0 , z 0 (λ) = λz 0 then p(λ) = λp then the convergence radius rescales as follows: θ(λ) = λθ 0 .

Finally we set κ = 0.9 and generated x 0 randomly, projected it onto the unit ball and then performed 4 experiments for different values of λ:

• λ = 1: pick z 0 so that x 0 -z 0 ≤ θ(λ) = 1.27 × 10 -4 • λ = 10: pick z 0 so that x 0 -z 0 ≤ θ(λ) = 0.0013 • λ = 10: pick z 0 so that x 0 -z 0 ≤ θ(λ) = 0.013 • λ = 125: pick z 0 so that x 0 -z 0 ≤ θ(λ) = 0.015 For each λ we rescaled parameters and the convergence radius as follows: p 0 (λ) = λp 0 , q(λ) = λ 2 q and x 0 (λ) = λx 0 , θ(λ) = λθ 0 . The results are presented in 2 we started z 0 outside of the estimated basin of attraction, yet after a transient period when V is increasing, the decay of V returns to monotone.

V. CONCLUSIONS

This contribution provides sufficient conditions for existense and uniquness of the system of non-linear differential equations eqs. ( 5) and [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], which describe dynamics of the observer and its gain. Basin of attraction for the "true solution" which evolves on a sphere and may have other invariants (e.g. conservation of volume) is described as a ball of certain radius. The radius is expressend in terms of upper bounds on the gain and parameters of the observer. It is also shown that the radius scales linearly if so are "true intial condition" together with observer's and gain's intial conditions, and if the constant term in the gain's equation scales quadratically.

t s 1 2e t s 1 2

 11 λmax( Bρ(z(τ ))+ B ρ (z(τ ))dτ so that Φ(t, s) ≤ λmax( Bρ(z(τ ))+ B ρ (z(τ ))dτ , and the latter estimate is exact provided B = B. Clearly

1

 1

1

 1

Fig. 1 :

 1 Fig. 1: Convergence of the estimation error and Lyapunov function for an ensemble of 4 members with rescaled initial conditions and parameters (λ = 1, 10, 100, 125, log-scale).

1 2p 3 2

 13 (e(t 0 ), t 0 ) ≤ κV 0 with V 0 = q 2β we took P (0) = p 0 I 8 , Q = qI 8 , R = rI 3

  Fig. fig. 1. Clearly, the decay of the Lyapunov function V is monotone (unlike the estimation error!), and rescaling the parameters speeds up the convergence. In Fig. fig.

Fig. 2 :

 2 Fig. 2: Convergence of the estimation error and Lyapunov function for z 0 which is outside of the estimated basin of attraction.