THE ASYMPTOTIC BEHAVIOR OF A TRANSPORT EQUATION

IN CELL POPULATION DYNAMICS WITH A NULL MATURATION VELOCITY

M. BOULANOUAR AND H. EMAMIRAD

Laboratoire de Modélisation Mécanique et de Mathématiques Appliquées. Université de Poitiers, Boulevard 3, Teleport 2, BP 179. 86960 FUTURO-SCOPE Cedex. France.

boulanou@l3ma.univ-poitiers.fr emamirad@l3ma.univ-poitiers.fr

Submitted by F. G. Webb In this paper we complete the study of Rotenberg’s model [10] describing the growth of a cell population and treated partially in [4]. In contrast with [4], here we impose the condition that the maturation velocity for any cell can become null. This consideration implies that the cell population never leaves completely its initial distribution, because at every time we can find some cells of initial cell population that are not divided. In this case, the generated semigroup is not compact. To surmount this difficulty, after studying the irreducibility of the generated semigroup, we calculate explicitly its essential type and we show the asymptotic convergence of the generated semigroup to a projection of rank one.

1. Introduction

This paper is the continuation of [4] in which we distinguish any cell by its degree of maturity μ ∈ I = (0, 1) and its maturation velocity v ∈ J = (0, b). By taking the degree of maturity between 0 and 1, we mean that the cells are born at μ = 0 and divide at μ = 1. We note by r(μ, v, v′) the transition rate at which cells change their velocities from v to v′. If we note by f = f(μ, v, t) the density of this cell population with respect to degree of maturity μ and maturation velocity v, then this density satisfied the following partial differential equation

\frac{∂f}{∂t} + v \frac{∂f}{∂μ} = − [∫_{0}^{b} r(μ, v′, v)dv′] f + ∫_{0}^{b} r(μ, v, v′)f(μ, v′, t)dv′.  (1.1)

This model is proposed by Rotenberg [10] and can be considered as one of the models of structured population dynamics with inherited properties. Inherited properties models do allow memory of generation time and among such models are the age-time and maturity-time models of proliferating cell populations with inherited cycle length of Lebowitz and Rubinow [7]. This model is based upon the assumption that the duration of the cycle from cell birth to cell division is determined at birth. Lebowitz and Rubinow proposed that the birth law can be considered as a boundary condition and has the form of an integral equation on the trace space. The transition probability kernel of this equation is determined by the correlation between generation times of mother-daughter pairs in proliferating cell populations. Although it is controversial, many researchers have claimed that such correlations do exist. The recent analysis of Sennerstam and Strömberg [11] shown even that how it might be possible for a near zero mother-daughter generation time correlation to accord with epigenetic and genetic inheritance transferred from mother to daughters. This remark justifies the main objective of this paper to put v near zero.

An important experimentally observed property of proliferating cell populations is their dispertion of various physical characteristics as time evolves. This property is known as asynchronous exponential growth. A rigorous mathematical treatment of Lebowitz-Rubinow model can be found in [15, 16]. In these papers Webb proved that asynchronous exponential growth occurs provided that the transition kernel satisfies certain smoothness and positivity conditions.

Let p ≥ 0 be the average number of viable daughters per mitosis and k(v, v′) the positive correlation between velocities of mother and daughter cells. In [10], Rotenberg proposed as in [7] that the reproduction rule is given by the following boundary condition

vf(0, v, t) = p ∫_{0}^{b} k(v, v′)v′f(1, v′, t)dv′.  (1.2)

To insure the current continuity he impose the normalization condition

∫_{0}^{b} k(v, v′)dv = 1.  (1.3)

We show that the Cauchy problem (1.1) and (1.2) with initial condition

f(μ, v, 0) = φ(μ, v)  (1.4)

is generated by a C_{0}-semigroup {e^{tT_{p}}}_{t≥0}. For studying the asymptotic behavior of this semigroup we will use an operator–theoretic formulation of asynchronous exponential growth of {e^{tT_{p}}}_{t≥0} which is given in [17]. This formulation is based on three facts about this C_{0}–semigroup. The semigroup should be positive, irreducible and its essential type ω_{ess}(T_{p}) should be strictly less than its exponential growth bound ω_{0}(T_{p}), where

ω_{ess}(A) = lim_{t⟶∞} t^{−1} ln α [e^{tA}]

and α [ ] is the measure-of-noncompactness (see [17]). In order to establish the main inequality

ω_{ess}(T_{p}) < ω_{0}(T_{p})

we have imposed in [4] that the minimum of maturation velocity of all cells is positive. That is J = (a, b) with a > 0. This implies that ω_{ess}(T_{p}) = −∞. Since this assertion is biologically unjustifiable (see [11]), our main aim in this paper is to remove this assertion and recover the asynchronous exponential growth of the generated semigroup with a new treatment. If we impose that the maturation velocity of every cell may vanish, then at any moment one can find some cells of the initial cell population that are not divided. So this cell population never goes out of the initial evolution phase and this may explain the non-compactness of the generated semigroup in this case.

In this paper we adopt the notations of [4] and we will show that in spite of noncompactness of the generated semigroup, the conclusion of [4] on the asymptotic behavior holds. In the next section we give a new expression of the resolvent of the unperturbed operator (i.e r = 0). This expression is used for proving that the consideration a = 0 does not spoil the generation theorem and positivity of the C_{0}– semigroup for this model. In section 3 we recover the irreducibility of the generated semigroup either by using the irreducibility of the boundary operator or by imposing an appropriate assumption on the kernel r. In the last section, the boundedness of the kernel k permits us to calculate explicitly the essential type of the semigroup generated without perturbation (i.e. r = 0). The essential type of the globally generated semigroup is also calculated explicitly under a assumption on the kernel r. Finally, we show that the generated semigroup converges asymptotically to a projection of the rank one in the operator norm topology. Some of these results are announced in [1]. Here we complete [1] by stating some new results and outlining the proofs. The case b = ∞ is discussed partially in [2].

2. A new formulation of the generated semigroup

Let Ω = (0, 1) × (0, b) = I × J with 0 < b < ∞. We begin by introducing the Lebesgue’s space L^{1}(Ω) with its natural norm denoted by ∥·∥_{1} and the partial Sobolev space W^{1}(Ω) = \{φ ∈ L^{1}(Ω) | v \frac{∂φ}{∂μ} ∈ L^{1}(Ω)\} and correspondence trace space L^{1}(J, vdv). W^{1}(Ω) and L^{1}(J, vdv) are the Banach spaces endowed with the norms

∥φ∥_{W^{1}(Ω)} = ∥φ∥_{1} + ∥v \frac{∂φ}{∂μ}∥_{1} and ∥g∥_{L^{1}(J,vdv)} = ∫_{0}^{b} v |g(v)| dv.

In [4, Theorem 3.1], we have shown that the trace mappings

\begin{matrix}γ_{0} : φ ⟼ φ(0, .) \\ γ_{1} : φ ⟼ φ(1, .) \end{matrix}

are linear continuous from W^{1}(Ω) on L^{1}(J, vdv). and we have characterized the boundary condition (1.2) by introducing the multiplying boundary operator K_{p} defined by

K_{p}ψ(v) ≡ \frac{p}{v} ∫_{0}^{b} k(v, v′)v′ψ(v′)dv′.

This operator is positive from the Banach lattice L^{1}(J, vdv) into itself. Furthermore the relation (1.3) implies that

∫_{0}^{b} vK_{p}ψ(v)dv = p ∫_{0}^{b} vψ(v)dv for any ψ ∈ L^{1}(J, vdv),

thus

∥K_{p}∥_{L(L^{1}(J,vdv))} = p.  (2.1)

If for any p ≥ 0, one defines the operator T_{p,0} by

T_{p,0}φ(μ, v) ≡ −v \frac{∂φ}{∂μ}(μ, v),

with domain

D(T_{p,0}) = {φ ∈ W^{1}(Ω) | γ_{0}φ = K_{p}γ_{1}φ},

the operator T_{0,0} on D(T_{0,0}) = {φ ∈ W^{1}(Ω) | γ_{0}φ = 0} generates a strongly continuous positive contraction semigroup given by

e^{tT_{0,0}}φ(μ, v) = χ(μ, v, t)φ(μ − tv, v),

where

χ(μ, v, t) = \begin{cases}1 & if μ ≥ tv; \\ 0 & if μ < tv, \end{cases}

and for λ > 0, we have

(λ − T_{0,0})^{−1}φ(μ, v) = ∫0^{\frac{μ}{v}} e^{−λt}φ(μ − tv, v)dt.

The operators (λ − T_{0,0})^{−1} and γ_{1}(λ − T_{0,0})^{−1} are obviously bounded and strictly positive. That is

(λ − T_{0,0})^{−1} ((L^{1}(Ω))_{+} \ {0}) ⊂ ((L^{1}(Ω))_{+} \ {0})  (2.2)

and

γ_{1}(λ − T_{0,0})^{−1} ((L^{1}(Ω))_{+} \ {0}) ⊂ ((L^{1}(J, vdv))_{+} \ {0}) .  (2.3)

Now, let us define the operator \overline{K}_{p,λ} by

\overline{K}_{p,λ} = (γ_{1}ϵ_{λ}) K_{p}

where

ϵ_{λ}(μ, v) = e^{−λ \frac{μ}{v}} .  (2.4)

Theorem 2.1. For any λ > max{0, b ln p}, the resolvent operator of T_{p,0} is a positive operator given by

(λ − T_{p,0})^{−1} = ϵ_{λ}K_{p}(I − \overline{K}_{p,λ})^{−1}γ_{1}(λ − T_{0,0})^{−1} + (λ − T_{0,0})^{−1}.  (2.5)

Proof. Let λ > max{0, b ln p}. We know from [4] that

(λ − T_{p,0})^{−1} = ϵ_{λ}(I − K_{p,λ})^{−1}K_{p}γ_{1}(λ − T_{0,0})^{−1} + (λ − T_{0,0})^{−1}  (2.6)

where the operator K_{p,λ} is given by

K_{p,λ}ψ = K_{p} (γ_{1}ϵ_{λ}) ψ,  (2.7)

for all ψ ∈ L^{1}(J, vdv). For obtaining the new expression (2.5) of (λ − T_{0,0})^{−1} it is suffices to show that

K_{p}(I − \overline{K}_{p,λ})^{−1} = (I − K_{p,λ})^{−1}K_{p}.

Since the operators K_{p,λ} and \overline{K}_{p,λ} are positive on L^{1}(J, vdv), from (2.1) we have

∥K_{p,λ}∥_{L(L^{1}(J,vdv))} < 1 and ∥\overline{K}_{p,λ}∥_{L(L^{1}(J,vdv))} < 1,

for all λ > max{0, b ln p}. On the other hand, for any n ∈ ℕ we have

K_{p}\overline{K}_{p,λ}^{n} = K_{p,λ}^{n}K_{p},

which implies that

K_{p}(I − \overline{K}_{p,λ})^{−1} = ∑_{n≥0} K_{p}\overline{K}_{p,λ}^{n}

= ∑_{n≥0} K_{p,λ}^{n}K_{p}

= (I − K_{p,λ})^{−1}K_{p}.

Te positivity of the operator (λ − T_{p,λ})^{−1} follows from the positivity of the operators (λ − T_{0,0})^{−1}, γ_{1}(λ − T_{0,0})^{−1} and K_{p}.

The coincidence between (2.5) and (2.6) infers that the Theorem 3.5 and 3.6 of [4] are valid, even if a = 0. As consequence we have

Lemma 2.1. (see [4]). The operator T_{p,0} generates, on L^{1}(Ω) a strongly continuous semigroup {e^{tT_{p,0}}}_{t≥0} satisfying

(1) e^{tT_{0,0}} ≤ e^{tT_{p,0}};

(2) e^{tT_{p,0}} ≤ qe^{tb ln q}e^{tT_{\frac{p}{q} ,0}} if p ≥ 1 and all q > p;

(3) ∥e^{tT_{p,0}} ∥_{1,1} ≤ 1  if  p < 1;
(4) ∥e^{tT_{p,0}} ∥_{1,1} = 1  if  p = 1;
(5) ∥e^{tT_{p,0}} ∥_{1,1} ≥ 1  if  p > 1,

(6) when p < 1, this semigroup is expressed by

e^{tT_{p,0}}φ(μ, v) = e^{tT_{0,0}}φ(μ, v) + A(t)φ(μ, v) t ≥ 0  (2.8)

with

A(t)φ(μ, v) = \overline{χ}(μ, v, t) [K_{p}γ_{1}e^{(t− \frac{μ}{v})tT_{p,0}}φ] (v)  (2.9)

and

\overline{χ} = 1 − χ,  (2.10)

where ∥ · ∥_{1,1} ≡ ∥ · ∥_{L(L^{1}(Ω))}.

When t ∈ (0, b^{−1}), we give the following expression of the semigroup {e^{tT_{p,0}}}_{t≥0} for all p ≥ 0.

Theorem 2.2. If t ∈ (0, b^{−1}), then for all p ≥ 0 we have

e^{tT_{p,0}}φ(μ, v) = e^{tT_{0,0}}φ(μ, v) + \overline{χ}(μ, v, t) [K_{p}γ_{1}e^{(t− \frac{μ}{v})T_{0,0}}φ] (v).

Proof. Let t ∈ (0, b^{−1}) and φ ∈ L^{1}(Ω). We define the following function

f(t)(x, y) = e^{tT_{0,0}}φ(μ, v) + \overline{χ}(μ, v, t) [K_{p}γ_{1}e^{(t− \frac{μ}{v})T_{0,0}}φ] (v).

If t > 0 is small then the same calculation of [4, Theorem 4.5] gives us

lim_{t→0} ∥\frac{f(t) − φ}{t} + v \frac{∂φ}{∂μ}∥_{1} = lim_{t→0} ∥\frac{e^{tT_{0,0}φ} − φ}{t} + v \frac{∂φ}{∂μ}∥_{1} = 0.

Furthermore, if φ ∈ D(T_{p,0}) then we have γ_{0}f(t) = K_{p}γ_{1}e^{tT_{0,0}} and γ_{1}f(t) = γ_{1}e^{tT_{0,0}}, because t ∈ (0, b^{−1}). Thus f(t) ∈ D(T_{p,0}). Consequently, the function f is a strong solution of the Cauchy problem

\{ \begin{matrix}\frac{df}{dt} = T_{p,0}f \\ f(0) = φ \end{matrix}

But e^{tT_{p,0}} is an other solution of the same Cauchy problem. By uniqueness we achieves the proof.

Next we define the absorption operator T_{p,1} and the transport operator T_{p} by

T_{p,1} = T_{p,0} + S and T_{p} = T_{p,0} + S + R = T_{p,1} + R

where the operators R and S are given by

Rφ(μ, v) = ∫_{0}^{b} r(μ, v, v′)φ(μ, v′)dv′

and

Sφ(μ, v) = − [∫_{0}^{b} r(μ, v′, v)dv′] φ(μ, v).

For the boundedness of these operators in L^{1}(Ω) we assume that

(H1)  \{ \begin{matrix}r measurable positive function; \\ M = ess sup_{(μ,v)∈Ω} ∫_{0}^{b} r(μ, v′, v)dv′ < ∞. \end{matrix}

In the following Lemma we announce some facts concerning the perturbed operator T_{p}, which is used in the sequel . This Lemma is proved in [4] for a > 0, and the same proof is valid for a = 0.

Lemma 2.2. Under the hypothesis (H1), −S and R are positive linear bounded operators on L^{1}(Ω). Furthermore, the operators T_{p} and T_{p,1} with the domain D(T_{p}) = D(T_{p,1}) ≡ D(T_{p,0}), generate on L^{1}(Ω) the positive C_{0}-semigroups {e^{tT_{p}}}_{t≥0} and {e^{tT_{p,1}}}_{t≥0} satisfying for all p ≥ 0 the following assertations

(1) e^{tT_{0,1}} ≤ e^{tT_{p,1}}, t ≥ 0;

(2) e^{−Mt}e^{tT_{p,0}} ≤ e^{tT_{p,1}} ≤ e^{tT_{p,0}}, t ≥ 0;

(3) e^{tT_{p,1}} ≤ e^{tT_{p}}, t ≥ 0;

(4) if p < 1, then ∥e^{tT_{p}} ∥_{1,1} ≤ 1, t ≥ 0;

(5) if p ≥ 1, then 1 ≤ ∥e^{tT_{p}} ∥_{1,1} ≤ pe^{tb ln p}, t ≥ 0.

3. Irreducibility

The main result of this section is to show that the semigroup semigroup {e^{tT_{p}}}_{t≥0} is irreducible. The proof of irreducibility given in [4, Lemma 6.1] is not valid here, because we cannot find a constant c > 0 such that the function ϵ_{λ} given by (2.4) satisfies γ_{1}ϵ_{λ}(v) ≥ c > 0, for all v ∈ J. Consequently, the inequality K_{p,λ} ≥ cK_{p}, is no longer valid in the present context and this forces us to choose a new treatment. For irreducibility of e^{tT_{p}}, we have two alternatives, we can either impose the appropriate conditions on the boundary operator K_{p}, or we can endow the function r with some conditions as it is defined in the sequel. First assume that the kernel of the operator K_{p} satisfies the following condition

(H2)  ∫_{0} ∫_{J\A} k(v, v′)dv′dv > 0,

where A is a measurable subset of J such that meas(A) > 0 and meas(J\A) > 0.

By the standard argument the assumption (H2) implies the irreducibility of the operator K_{p} on L^{1}(J, vdv) (see [18]).

Lemma 3.1. Under the assumption (H2) the operator \overline{K}_{p,λ} is irreducible for all λ > max{0, b ln p}.

Proof. Let λ > max{0, b ln p} and M a closed ideal of the trace space L^{1}(J, vdv) such that

\overline{K}_{p,λ}(M) ⊂ M  (3.1)

By the characterization of the closed ideal in L^{1}(J, vdv) (see [9]), there exists a measurable subset ω ⊂ J such that

M = {ψ ∈ L^{1}(J, vdv) | ψ(v) = 0 for a.e v ∈ ω} .  (3.2)

If g ∈ K_{p}(M), then there exists ψ ∈ M such that g = K_{p}ψ and (γ_{1}ϵ_{λ}) g = (γ_{1}ϵ_{λ})K_{p}ψ = \overline{K}_{p,λ}ψ. Thus by (3.1) we obtain (γ_{1}ϵ_{λ}) g ∈ M and (3.2) implies that K_{p}(M) ⊂ M. Finally, the irreducibility of the operator K_{p} implies that M = ∅ or M = L^{1}(J, vdv) and this achieves the proof.

Theorem 3.1. Under the assumption (H2), the semigroup {e^{tT_{p,0}}}_{t≥0} is irreducible.

Proof. Let λ > max{0, b ln p} and g be a strictly positive function in L^{1}(Ω). According to (2.3) the function γ_{1}(λ − T_{0,0})^{−1}g ∈ L^{1}(J, vdv) is also strictly positive and from the irreducibility of the operator \overline{K}_{p,λ}, it follows that there exists an integer m > 0 such that

\overline{K}_{p,λ}^{m}γ_{1}(λ − T_{0,0})^{−1}g > 0 a.e. in J.

The positivity of the operator (λ − T_{0,0})^{−1} and the inequality K_{p} ≥ \overline{K}_{p,λ} imply that

\begin{aligned}(λ − T_{p,0})^{−1}g & ≥ ϵ_{λ}K_{p}(I − \overline{K}_{p,λ})^{−1}γ_{1}(λ − T_{0,0})^{−1}g \\ & = ϵ_{λ}K_{p} ∑_{n≥0} \overline{K}_{p,λ}^{n}γ_{1}(λ − T_{0,0})^{−1}g \\ & ≥ ϵ_{λ}\overline{K}_{p,λ}\overline{K}_{p,λ}^{m−1}γ_{1}(λ − T_{0,0})^{−1}g \\ & = ϵ_{λ}\overline{K}_{p,λ}^{m}γ_{1}(λ − T_{0,0})^{−1}g. \end{aligned}  (3.3)

Thus, the resolvent operator of T_{p,0} is irreducible which is equivalent to irreducibility of the semigroup {e^{tT_{p,0}}}_{t≥0} (see [5] or [4, Lemma 2.1].

Theorem 3.2. Suppose that the hypothesis (H1) and (H2) hold. Then the semigroup {e^{tT_{p}}}_{t≥0} is irreducible.

Proof. By using the assertions 2 and 3 of Lemma 2.2, we obtain

e^{tT_{p}} ≥ e^{−Mt}e^{tT_{p,0}}, t ≥ 0.

Now the irreducibility of the semigroup {e^{tT_{p}}}_{t≥0} follows immediately from the previous Theorem.

When K_{p} is not irreducible, we impose the following hypothesis

(H3) \{ \begin{matrix}There exists a′ and b′ (0 ≤ a′ < b′ ≤ b) such that \\ r(μ, v, v′) > 0 a.e. on ((0, 1) × J × (a′, b′)) ⋃ ((0, 1) × (a′, b′) × J) . \end{matrix}

The proof of [4, Theorem 6.4], is valid and stablishes the following Lemma.

Lemma 3.2. Under the hypothesis (H1) and (H3), the C_{0}-semigroup {e^{tT_{p}}}_{t≥0} is irreducible.

4. Asymptotic behavior of the semigroup {e^{tT_{p}}}_{t≥0}

In this section we describe the asymptotic behavior of the semigroup {e^{tT_{p}}}_{t≥0} under some hypotheses on the functions k and r. We remark that from the points (3), (2) of the Lemma 2.2 and the point (1) of the Lemma 2.1 we get

e^{−Mt}e^{tT_{0,0}} ≤ e^{−Mt}e^{tT_{p,0}} ≤ e^{tT_{p,1}} ≤ e^{tT_{p}}, t ≥ 0

Since the semigroup {e^{tT_{0,0}}}_{t≥0} is not compact, the semigroup {e^{tT_{p}}}_{t≥0} is neither (see [6, Proposition 2.1]). Consequently, we cannot use the method in [4] that ω_{ess}(T_{p}) = −∞. For obtaining this result we apply a new treatment, assuming that

(H4)  k ∈ L^{∞}(J × J).

This assumption permits us to calculate explicitly the essential type of the semigroup {e^{tT_{p,1}}}_{t≥0}.

Lemma 4.1. Suppose that the assumption (H4) holds, then for all 0 < t < b^{−1} and s ≥ 0, the operator

L(t, s) ≡ [e^{tT_{p,0}} − e^{tT_{0,0}}] e^{sT_{0,0}} [e^{tT_{p,0}} − e^{tT_{0,0}}]

is weakly compact on L^{1}(Ω).

Proof. Let φ ∈ (L^{1}(Ω))_{+}. If 0 < t < b^{−1}, it follows from Theorem 2.2 and the hypothesis (H4) that

\begin{matrix}[e^{tT_{p,0}} − e^{tT_{0,0}}] φ(μ, v) & ≤ p ∥k∥_{∞} \frac{\overline{χ}(μ, v, t)}{v} ∫_{0}^{b} v′γ_{1} [e^{(t− \frac{μ}{v})T_{0,0}}φ] (v′)dv′ \\ = & p ∥k∥_{∞} \frac{\overline{χ}(μ, v, t)}{v} ∫_{0}^{b} v′χ(1, v′, t − \frac{μ}{v})φ(1 − (t − \frac{μ}{v})v′, v′)dv′ \end{matrix}

Let s ≥ 0, then

L(t, s)φ(μ, v) ≤ p ∥k∥_{∞} \frac{\overline{χ}(μ, v, t)}{v} ∫_{0}^{b} v′γ_{1} [e^{(t+s− \frac{μ}{v})T_{0,0}} \{e^{tT_{p,0}} − e^{tT_{0,0}}\} φ] (v′)dv′

= p ∥k∥_{∞} \frac{\overline{χ}(μ, v, t)}{v} ∫_{0}^{b} v′χ(1, v′, t + s − \frac{μ}{v})

[e^{tT_{p,0}} − e^{tT_{0,0}}] φ(1 − (t + s − \frac{μ}{v})v′, v′)dv′

≤p^{2} ∥k∥_{∞}^{2} \frac{\overline{χ}(μ, v, t)}{v}∫_{0}^{b}∫_{0}^{b}v”χ(1, v′, t + s − \frac{μ}{v})\overline{χ}(1 − (t + s − \frac{μ}{v})v′, v′, t)

χ(1, v”, 2t + s − \frac{μ}{v} − \frac{1}{v′})φ(1 − (2t + s − \frac{μ}{v} − \frac{1}{v′})v”, v”)dv”dv′.

The following change of variables

\begin{cases}μ′ = 1 − (2t + s − \frac{μ}{v} − \frac{1}{v′})v” \\ dv′ = −\frac{u^{2}}{v”}dμ′ where u = u(t, s, μ, v, μ′, v”) = [2t + s − μv^{−1} + (μ′ − 1)\frac{1}{v”}]^{−1} \end{cases}

implies that

\begin{matrix}L(t, s) & φ(μ, v) ≤ p^{2} ∥k∥_{∞}^{2} \frac{\overline{χ}(μ, v, t)}{v} ∫_{0}^{b}∫_{1−(2t+s− \frac{μ}{v} − \frac{1}{b})v”}^{∞} χ(1, v”, 2t + s − \frac{μ}{v} − \frac{1}{u}) \\ u^{2}χ(1, u, t + s − μv^{−1})\overline{χ}(1 − (t + s − \frac{μ}{v})u, u, t)φ(μ′, v”)dμ′dv”. \end{matrix} (4.1)

Replacing u by its value, a simple calculation gives that

χ(1, u, t + s − μv^{−1})\overline{χ}(1 − (t + s − μv^{−1})u, u, t) = \begin{cases}1 if 1 − tv” < μ′ ≤ 1 \\ 0 elsewhere, \end{cases}

and 0 ≤ u ≤ b and in any case χ(1, v′, 2t + s − μv^{−1} − u^{−1}) = 1. Thus the relation (4.1) becomes

L(t, s)φ(μ, v) ≤ p^{2}b^{2} ∥k∥_{∞}^{2} \frac{\overline{χ}(μ, v, t)}{v} ∫_{0}^{b}∫_{0}^{1}φ(μ′, v′)dμ′dv′ ≡ H_{t}φ(μ, v)

Since ∫_{Ω} \frac{\overline{χ}(μ,v,t)}{v}dμdv < ∞, the operator H_{t} is of rank one, and consequently weakly compact on L^{1}(Ω). Further, so is the operator L(t, s) which is dominated by H_{t} (see [6, Proposition 2.1]).

Lemma 4.2. Suppose that the assumption (H4) holds. Then the operator

[e^{tT_{p,1}} − e^{tT_{0,1}}] e^{sT_{0,1}} [e^{tT_{p,1}} − e^{tT_{0,1}}]

is weakly compact on L^{1}(Ω) for all 0 < t < b^{−1} and s ≥ 0

Proof. To show this Lemma, it suffices to remark that the operator T_{p,1} (resp. T_{0,1}) is a perturbation of the operator T_{p,0} (resp. T_{0,0}) by the bounded operator S. The Duhamel’s formula gives us

e^{tT_{p,1}} = e^{tT_{p,0}} + ∫_{0}^{t} e^{(t−s)T_{p,0}}Se^{sT_{p,1}}ds

and

e^{tT_{0,1}} = e^{tT_{0,0}} + ∫_{0}^{t} e^{(t−s)T_{0,0}}Se^{sT_{0,1}}ds

The positivity of the operator −S and (1) of Lemma 2.1 implies that

e^{tT_{p,1}} − e^{tT_{0,1}} = e^{tT_{p,0}} − e^{tT_{0,0}} + ∫_{0}^{t} e^{(t−s)T_{p,0}}Se^{sT_{p,1}}ds − ∫_{0}^{t} e^{(t−s)T_{0,0}}Se^{sT_{0,1}}ds
≤ e^{tT_{p,0}} − e^{tT_{0,0}} + ∫_{0}^{t} e^{(t−s)T_{0,0}}S [e^{sT_{p,1}} − e^{sT_{0,1}}] ds
≤ e^{tT_{p,0}} − e^{tT_{0,0}}.

Consequently,

[e^{tT_{p,1}} − e^{tT_{0,1}}] e^{sT_{0,1}} [e^{tT_{p,1}} − e^{tT_{0,1}}] ≤ [e^{tT_{p,0}} − e^{tT_{0,0}}] e^{sT_{0,0}} [e^{tT_{p,0}} − e^{tT_{0,0}}]

Once more, the fact that the operator [e^{tT_{p,1}} − e^{tT_{0,1}}] e^{sT_{0,1}} [e^{tT_{p,1}} − e^{tT_{0,1}}] is dominated by a weakly compact operator in L^{1}(Ω), implies that it is weakly compact.

Lemma 4.3. If the assumption (H4) holds then ω_{ess}(T_{p,1}) = ω_{ess}(T_{0,1}).

Proof. Let 0 < t < b^{−1} . Then by the previous Lemma the operator

2^{−n} [e^{tT_{p,1}} − e^{tT_{0,1}}] e^{ntT_{0,1}} [e^{tT_{p,1}} − e^{tT_{0,1}}]

is weakly compact for all n ∈ N. Consequently, the following operator

2^{−1} [e^{tT_{p,1}} − e^{tT_{0,1}}] [∑_{n=0}^{m} 2^{−n}e^{ntT_{0,1}}] [e^{tT_{p,1}} − e^{tT_{0,1}}]

is also weakly compact for any m ∈ N. Since {e^{tT_{0,1}}}_{t≥0} is a contraction semigroup, then the last sum converges in L(L^{1}(Ω)) to the operator

[e^{tT_{p,1}} − e^{tT_{0,1}}] (2 − e^{tT_{0,1}})^{−1} [e^{tT_{p,1}} − e^{tT_{0,1}}] .

which becomes weakly compact. The Dunford-Pettis property of L^{1}(Ω) (see [8]), implies the compactness of

[(e^{tT_{p,1}} − e^{tT_{0,1}}) (2 − e^{tT_{0,1}})^{−1}]^{4}

According to [12, Corollary 1.4], this implies that the spectral radius r_{ess}(e^{tT_{0,1}}) of e^{tT_{0,1}} and r_{ess}(e^{tT_{p,1}}) are the same. We achieves the proof of the Lemma by recalling that for all t ≥ 0 we have r_{ess}(e^{tT_{p,1}}) = e^{tω_{ess}(T_{p,1})} and r_{ess}(e^{tT_{0,1}}) = e^{tω_{ess}(T_{0,1})} (see [?]).

In order to calculate the essential type of the semigroup {e^{tT_{p}}}_{t≥0}, we have to assume that

(H5)  r_{1} = r_{1}(μ, v, v′) = \frac{r(μ, v, v′)}{μ} ∈ L^{∞}(I × J^{2}).

Lemma 4.4. If (H1), (H4) and (H5) hold, then ω_{ess}(T_{p}) = ω_{ess}(T_{p,1}).

Proof. Let t > 0. If p < 1, then for any function φ ∈ (L^{1}(Ω))_{+} the relation (2.8) gives us

Re^{tT_{p,0}}Rφ = Re^{tT_{0,0}}Rφ + RA(t)Rφ,

where the operator A(t) is defined in (2.9). As the hypothesis (H5) holds, then the kernel r is bounded and by a simple calculation we obtain

Re^{tT_{0,0}}Rφ ≤ \frac{∥r∥_{∞}^{2}}{t} ∫_{Ω} φ(μ′, v′)dμ′dv′1  (4.2)

where 1 ∈ L^{1}(Ω) is the function equal to 1 on Ω. By using the assumptions (H4) and (H5) we obtain

RA(t)Rφ(μ, v) ≤ ∥k∥_{∞}∥r_{1}∥_{∞} ∫_{0}^{b} ∫_{0}^{b} μ\overline{χ}(μ, v′, t)e^{(t−μv′^{−1})T_{p,0}}Rφ(1, v′′)v′′dv′dv′′.

The change of variables s = t − μv′^{−1} implies that

\begin{aligned}RA(t)Rφ(μ, v) & ≤ ∥k∥_{∞}∥r_{1}∥_{∞} ∫_{0}^{b} ∫_{−∞}^{t− \frac{μ}{b}} \overline{χ}(μ, \frac{μ}{t − s}, t) \\ & e^{sT_{p,0}}Rφ(1, v′′)v′′ [\frac{μ}{t − s}]^{2} dsdv′′ \end{aligned}

Since \overline{χ}(μ, \frac{μ}{t−s}, t) = 0 for s < 0 then

RA(t)Rφ(μ, v) ≤ b^{2}∥k∥_{∞}∥r_{1}∥_{∞} ∫_{0}^{b} ∫_{0}^{t} e^{sT_{p,0}}Rφ(1, v′′)v′′dsdv′′.

As there exist an integer n such that n ≤ bt < n + 1, then

RA(t)Rφ(μ, v) ≤ b^{2}∥k∥_{∞}∥r_{1}∥_{∞} ∑_{i=0}^{n} ∫_{0}^{b} ∫ \frac{i+1}{b} ; \frac{i}{b} e^{sT_{p,0}}Rφ(1, v′′)dsdv′′

= b^{2}∥k∥_{∞}∥r_{1}∥_{∞} ∑_{i=0}^{n} ∫_{0}^{b} ∫0^{\frac{1}{b}} e^{(s+ \frac{i}{b})T_{p,0}}Rφ(1, v′′)dsdv′′

= b^{2}∥k∥_{∞}∥r_{1}∥_{∞} ∑_{i=0}^{n} ∫_{0}^{b} ∫0^{\frac{1}{b}} e^{sT_{p,0}}e^{\frac{i}{b} T_{p,0}}Rφ(1, v′′)dsdv′′.

When 0 ≤ s ≤ b^{−1}, by Theorem 2.2 we obtain e^{sT_{p,0}}φ(1, ·) = e^{sT_{0,0}}φ(1, ·) and

RA(t)Rφ(μ, v) ≤ b^{2}∥k∥_{∞}∥r_{1}∥_{∞} ∑_{i=0}^{n} ∫_{0}^{b} ∫0^{\frac{1}{b}} e^{sT_{0,0}}e^{\frac{i}{b} T_{p,0}}Rφ(1, v′′)dsdv′′.

Using the definition of semigroup {e^{tT_{0,0}}}_{t≥0} we get

\begin{aligned}RA(t)Rφ(μ, v) & ≤ b^{2}∥k∥_{∞}∥r_{1}∥_{∞} ∑_{i=0}^{n} ∫_{0}^{b} ∫0^{\frac{1}{b}} χ(1, v′′, s)e^{\frac{i}{b} T_{p,0}}Rφ(1 − sv′′, v′′)dsdv′′ \\ & ≤ b^{2}∥k∥_{∞}∥r_{1}∥_{∞} ∑_{i=0}^{n} ∫_{0}^{b} ∫_{0}^{1} e^{\frac{i}{b} T_{p,0}}Rφ(μ′, v′′)dμ′dv′′. \end{aligned}

The contraction of the linear operator e^{\frac{i}{b} T_{p,0}}, i = 0, ⋯, n − 1, (see Lemma 2.1) and the boundedness of the positive operator R yield that

RA(t)Rφ(μ, v) ≤ (n + 1)b^{2}∥k∥_{∞}∥r_{1}∥_{∞}∥R∥_{1,1} ∫_{Ω} φ(μ′, v′′)dμ′dv′′

Thus

RA(t)Rφ(μ, v) ≤ (bt + 1)b^{2}∥k∥_{∞}∥r_{1}∥_{∞}∥R∥_{1,1} ∫_{Ω} φ(μ′, v′)dμ′dv′1.  (4.3)

Now the relation (4.2) and (4.3) gives us

Re^{tT_{p,0}}Rφ ≤ [\frac{∥r∥_{∞}^{2}}{t} + (bt + 1)b^{2}∥k∥_{∞}∥r_{1}∥_{∞}∥R∥_{1,1}] ∫_{Ω} φ(μ′, v′)dμ′dv′1.

As the operator φ ∈ L^{1}(Ω) ↦ ∫_{Ω} φ(μ′, v′′)dμ′dv′′1 ∈ L^{1}(Ω) is one rank, then by [6, Proposition 2.1], we obtain the weak compactness of Re^{tT_{p,0}}R for all t > 0. Finally, the same result holds for p ≥ 1 (see Lemma 2.1). We achieve this proof by applying the main result of [14] (see also [4]) .

To describe the asymptotic behavior of the semigroup {e^{tT_{p}}}_{t≥0} we introduce the following assumption

(H6)  m = ess inf_{(μ,v)∈Ω} ∫_{Ω} r(μ, v′, v)dv′ > 0

The following Theorem is a simple application of Lemma 2.3 in [4] and describes the asymptotic behavior of the semigroup {e^{tT_{p}}}_{t≥0} for p > 1. In fact only in this case does the density of cells increase (see Lemma 2.2), which is the biologically meaningful case.

Theorem 4.1. Suppose that the hypotheses (H1), (H4), (H5), (H6) and (H2) or (H1), (H4), (H5), (H6) and (H3) hold. Then there exists ε > 0 such that for all η ∈ (0, ε), there exists M(η) ≥ 1 such that for all φ ∈ L^{1}(Ω) and t ≥ 0 we have

∥e^{−ω(T_{p})t}e^{tT_{p}}φ− < φ, φ_{0}^{∗} > φ_{0}∥_{1} ≤ M(η)e^{−ηt}∥φ∥_{1}.

Proof. By using the hypotheses (H1),(H4),(H5) and (H6), the Lemma 4.3 and Lemma 4.4 we get

ω_{ess}(T_{p}) = ω_{ess}(T_{p,1}) = ω_{ess}(T_{0,1}) ≤ ω_{0}(T_{0,1}).

Following the arguments given in [13] the hypothesis (H6) yields that

ω_{0}(T_{0,1}) ≤ −m < 0,

thus

ω_{ess}(T_{0,1}) ≤ −m < 0.

The claim (5) of Lemma 2.2 gives us

ω_{0}(T_{p}) ≥ 0,

and thus from previous relations we infer that

ω_{ess}(T_{p}) < ω_{0}(T_{p})

Now either one of the hypotheses (H2) or (H3) implies the irreducibility of the semigroup generated by T_{p} and this achieves the proof by using the main theorem of [17].

References

[1] M. Boulanouar, Un modèle de Rontenberg avec des vitesses de maturité nulles C. R. Acad. Sci. Paris Série I 326 (1998), 443–446.

[2] M. Boulanouar, Un modèle de Rontenberg avec la loi à mémoire parfaite,C. R. Acad. Sci. Paris Série I 327 (1998) 965–968.

[3] M. Boulanouar and L. Leboucher, Une équation de transport dans la dynamique des populations cellulaires,C. R. Acad. Sci. Paris Série I 321 (1995) 305–308.

[4] M. Boulanouar and H. Emamirad, A transport equation in cell population dynamics,J. Diff. Integ. Equ., (to appear).

[5] Ph. Clément and al., ”One-Parameter Semigroups” North-Holland. Amerterdam, New York.1987.

[6] G. Greiner, Spectral properties and asymptotic behaviour of linear transport equation,Math. Z. 185 (1984) 167-177.

[7] J. L. Lebowitz and S. I. Rubinow, A theory for the age and generation time distribution of a microbial population, J. Math. Bio. 1 (1974), 17–36.

[8] P. Meyer-Nieberg, ”Banach Lattices,” Springer–Verlag. Berlin Heidelberg New York. 1991.

[9] R. Nagel ed., ”One-Parameter Semigroups of Positive operators,” Springer–Verlag. L.N.M. 1184. Berlin, New York. 1986.

[10] M. Rotenberg, Transport theory for growing cell populations, J. theor. Biol.103(1983) 181–199.

[11] R. Sennerstam and J.-O. Strömberg, Sister, mother-daughter and cousin cell generation time correlations and cell cycle progression, in ”Mathematical Models in Medical and Health Sciences,” (M. Horn, G. Simonette and G. Webb, Eds.), pp. 337–350, Vanderbilt Univ. Press, 1998.

[12] J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups,Mh. Math.90(1980) 153–161.

[13] J. Voigt, Positivity in time dependent linear transport theory,Acta. Appl. Math. 2(1984) 311-331.

[14] J. Voigt, Stability of the essential type of strongly continuous semigroups,Trudy Mat. Inst. Steklov, Izbran. Voprosy Mat. Fiz. I Anal.203 (1994) 469–477.

[15] G.F. Webb, A model of proliferating cell populations with inherited cycle lenght, J. Math. Bio. 23 (1986) 269–282.

[16] G.F. Webb, Dynamics of structured populations with inherited properties, Inter. J. Comput. Math. Appl. 18 (1987) 749–757.

[17] G.F. Webb, An operator-theoretic formulation of asynchronous exponential growth,Trans. Amer. Math. Soc.303 (1987) 751–763.

[18] M. Zerner, Quelques propriétés spectrales des opérateurs positifs, J. Funct. Anal.72(1987) 381–