ON THE THEORY OF REMEDIABILITY

HASSAN EMAMIRAD

Abstract. Suppose {G_{1}(t)}_{t≥0} and {G_{2}(t)}_{t≥0} be two families of semigroups on a Banach space X (not necessary of class C_{0}), such that for some initial datum u_{0}, G_{1}(t)u_{0} tends towards an undesirable state u^{∗}. After remedying by the means of an operator ρ we continue the evolution of the state by applying G_{2}(t) and after time 2t we retrieve a prosperous state u given by u = G_{2}(t)ρG_{1}(t)u_{0}. Here we are concerned with the different properties of the semigroup G(t) : ρ ⟶ G_{2}(t)ρG_{1}(t). We define R(X) the space of remedial operators for G_{1}(t) and G_{2}(t), when the above application is well defined for all ρ ∈ R(X) and satisfies the properties of a uniformly bounded semigroup on R(X). In this paper we study some properties of the space R(X) and we prove that when A_{i} generate a regularized semigroup for i = 1, 2, then the operator Δ defined on L(X) by Δρ = A_{2}ρ + ρA_{1} generates a tensor product regularized semigroup. Finally, we give two examples of the remedial operators in radiotherapy and chemotherapy in the proliferation of the cancer cells.

1. Introduction

The theory of remediability describes mathematical models of biological states, which are represented by a dynamical system such that the solution of the models stemming from such states tend inexorably towards undesirable states. We suppose that an undesirable state is remediable whether in the past or in the future. Think of the extinction of species or environmental degradation in ecology or a cohort of carcinogenic cells in the theory of proliferation of cell population, etc ...

First let us suppose that such a phenomenon can be described by a linear or nonlinear dynamical system

(DS)  \begin{cases}\frac{d}{dt}u(t) = Au(t)  t ∈ ℝ \\ u(0) = u_{0} ∈ X, \end{cases}

which has a global solution given by a group of operators {G(t)}_{t∈ℝ}.

Suppose that at time t = t_{0} we have at our disposal the initial datum u_{0} and the trajectory coming from u_{0} tends towards a disastrous state u^{∗} which is to be avoided. A utopian strategy to alleviate the situation is to go back in time to instant t_{0} − t < 0, when a remedy is plausible. After remedying by the means of an operator ρ we go back to t_{0} and we retrieve a prosperous state u given by

u = G(t − t_{0})ρG(t_{0} − t)u_{0}  (1)

We are concerned more precisely with the different properties of the application

L(X) ∋ ρ ⟶ G(t)ρG(−t) ∈ L(X).

If {G(t)}_{t∈R} is a C_{0}-group on X, then this application defines a group G(t) on L(X) in the following sense

(1) G(0)ρ = ρ

(2) G(t)G(s)ρ = G(t)[G(s)ρG(−s)] = G(t)G(s)ρG(−s)G(−t)

= G(t + s)ρG(−t − s)
= G(t + s)ρ

(3) G(t)ρ converges strongly to ρ, for any ρ ∈ L(X).

In general the convergence is not in the sense of uniform topology of L(X) unless X is a finite dimensional space or G(t) acts on the subalgebra L_{c}(X), the ideal of compact operators on X.

In quantum mechanics we can find such formalism in which G(t) is called quantum dynamical semigroup in theory of open systems(see [7]). In this theory G(t) appears as a unitary group generated by a skew-adjoint operator −iH, where H is the Hamiltonian of the system. In this formalism the dynamical semigroup G(t) acts on the state space V, which is the trace class operators on a Hilbert space H and for the trace norm topology we have lim_{t→0} ∥G(t)ρ − ρ∥_{tr} = 0. However, the use of such theory comes up against few difficulties.

First, in the practical cases the evolution dynamics are in general irreversible in time. For example in population dynamics the individuals cannot become less mature in time. This irreversibility forces to redefine the generalized dynamical semigroup G(t) and to replace G(t)ρG(−t) by G(t)ρG(t) or more generally by G_{2}(t)ρG_{1}(t). This type of semigroup have been also studied in the literature (see [1], [5], [8], [12], [13], [14] and [15]). It is formally clear that if the C_{0}–semigroups G_{1}(t) and G_{2}(t) are generated by the operators A_{1} and A_{2} on the Banach spaces X, then the semigroup G(t) has Δ as its infinitesimal generator where

Δρ = A_{2}ρ + ρA_{1}.  (2)

This is strictly true when the generators A_{1} and A_{2} are bounded. That is why this semigroup intervenes for studying the operator equation A_{2}ρ + ρA_{1} = Q in the scattering theory (see [9] and [10]).

By applying G(t) = G_{2}(t) ⊗ G_{1}(t)^{∗} to operators of rank one, u ⊗ v^{∗} = ⟨., v^{∗}⟩u (u ∈ X, v^{∗} ∈ X^{∗}), it comes out that

[G_{2}(t) ⊗ G_{1}^{∗}(t)] (u ⊗ v^{∗}) = G_{2}(t)u ⊗ G_{1}^{∗}(t)v^{∗}
= ⟨·, G_{1}^{∗}(t)v^{∗}⟩G_{2}(t)u
= G_{2}(t)(u ⊗ v^{∗})G_{1}(t)

Hence, G(t) is an extension to L(X) of the tensor product G_{2}(t) ⊗ G_{1}(t)^{∗} acting on the space of operators of finite rank, X ⊗ X^{∗}. Therefore G(t) is referred to as a tensor product semigroup. In the following proposition we gather the essential properties of a tensor product semigroup G(t).

Proposition 1. (see [8])

(1) If G_{1}(t) and G_{2}(t) are two C_{0}–semigroups on a Banach space X, then for any ρ ∈ L(X), G(t)ρ = G_{2}(t)ρG_{1}(t) is continuous in t ∈ [0, ∞) relative to the strong operator topology;

(2) Let D(Δ) := {ρ ∈ L(X) such that ρD(A_{1}) ⊂ D(A_{2}) and the operator Δρ has a bounded extension on X defined by (2)}. Then Δ is closed for weak operator topology and

Δρ = s − lim_{t→0} \frac{G(t)ρ − ρ}{t}.

Furthermore, if G_{1}(t) and G_{2}(t) are defined on two different Banach spaces X and Y, then we can also define the family {G(t) : t ≥ 0} on the Banach algebra L(Y, X) (see [12]).

The second difficulty is related to the ill-posedness of the dynamics G_{1}(t). Even in the linear case, if we assume that X is the space of prosperious states and if for some initial datum u_{0} ∈ X, G_{1}(t)u_{0} tends towards an undesirable state u^{∗} ∉ X, such a semigroup cannot be strongly continuous and needs to be regularized or to be extended in a larger space. This situation requires the theory of C-regularized semigroups and C-existence families which is extensively studied in [6].

Definition 1. For a bounded injective operator C, the strongly continuous family {W (t)}_{t≥0} in L(X) is a C–regularized semigroup if

(1) W (0) = C;

(2) W (t)W (s) = CW (t + s) for all s, t ≥ 0. An operator A is the generator of the semigroup {W (t)}_{t≥0} if

Ax = C^{−1} [lim_{t→0} \frac{1}{t}(W (t)x − Cx)]

with D(A) := {x | the limit exists and is in Im(C)}.

Note that, when C = I, {W (t)}_{t≥0} is a C_{0}–semigroup generated by A. When A generates a C–regularized semigroup {W (t)}_{t≥0}, then we can define the semigroup {e^{tA}}_{t≥0} of (possibly unbounded) operators by

e^{tA} := C^{−1}W (t), t ≥ 0  (3)

with domain

D(e^{tA}) := {x ∈ X | W (t)x ∈ Im(C)}.

If the operator C do not satisfies the commutation relationship CG_{1}(t) = G_{1}(t)C, then one can even so use the theory of C–existence family in such a manner that W (t) = CG_{1}(t) becomes a linear bounded operator.

In the next section we will define the space of remedial operators ρ for which G(t)ρ := G_{2}(t)ρG_{1}(t) is well defined and we will indicate some properties of this space. In the third section we suppose that the operator A_{1} and A_{2} are the generators of the C_{1} and C_{2}–regularized semigroups we show that Δ is also the generator of some regularized semigroup. A similar result is already proved in [5], when A_{i} generates exponentially bounded n-times integrated semigroup, for i = 1, 2. In the last section we give two examples of the remedial operators in radiotherapy and chemotherapy in the proliferation of the cancer cells.

2. Space of the remedial operators

Definition 2. Let {G_{1}(t)}_{t≥0} and {G_{2}(t)}_{t≥0} be two families of semigroups in L(X) (not necessary of class C_{0}). We say that R(X) a subset of L(X) is the set of remedial operators for G_{1}(t) and G_{2}(t), if G(t)ρ := G_{2}(t)ρG_{1}(t) is well defined for all ρ ∈ R(X) and we have

(i) G(0)ρ = ρ for all ρ ∈ R(X); (ii) G(t)G(s)ρ = G(t + s)ρ for all ρ ∈ R(X) and all t, s ≥ 0; (iii) G(t)ρ converges strongly to ρ, for any ρ ∈ R(X), as t → 0; (iv) For any ρ ∈ R(X), there exists M > 0 such that sup_{t≥0} ∥G(t)ρ∥ ≤ M.

It is clear that R(X) is a subspace of L(X) and the condition (iv) implies that R(X) can be endowed with the norm

|||ρ||| := sup_{t≥0} ∥G(t)ρ∥.

Let us denote once more by R(X) the completion of R(X) for this norm.

Theorem 1. On the Banach space R(X), G(t) is a family of contractions in L(L(X)).

Proof. It is not hard to verify that all the conditions of Definition 2 are true (by passing into limit of the Cauchy sequences) for all ρ ∈ R(X). Hence G(t) defines a generalized tensor product semigroup on R(X) and we have

|||G(s)ρ||| = sup_{t≥0} ∥G(t + s)ρ∥ ≤ |||ρ|||.

This implies that

|||G(t)|||_{L(L(X))} := sup{|||G(t)ρ||| : |||ρ||| ≤ 1} ≤ 1.
□

Since, in general, G_{1}(t) and G_{2}(t) are not continuous in norm operator topology G(t) is not strongly continuous on R(X). However, in the next section we construct a C–regularized semigroup which is generated by an extension of some restriction the generator Δ defined by Δρ = A_{2}ρ + ρA_{1}.

3. Tensor product regularized semigroup

For i = 1, 2, let A_{i} generates a C_{i}–regularized semigroups W_{i}(t) and A_{1} densely defined. Let D(Δ) be domain of Δ defined as in Proposition 1 (2), with Δρ = A_{2}ρ + ρA_{1}. We suppose that ρ(A_{i}) is the resolvent set of A_{i} and ρ(A_{1}) ∩ ρ(A_{2}) is not empty. For r ∈ ρ(A_{1}) ∩ ρ(A_{2}), we define Λ_{r} by Λ_{r}ρ := (r − A_{2})^{−1}ρ(r − A_{1})^{−1} on L(X).

Lemma 1. Under the above assumptions Δ is a closed operator on L(X) and if we denote Δ̃ the restriction of Δ to

L_{A_{2},A_{1}}(X) := {ρ ∈ L(X) ; ρ(X) ⊂ D(A_{2}) and A_{2}ρA_{1} is bounded on D(A_{1})},

then we have

Im(Λ_{r}) = D(Δ̃) := D(Δ) ∩ L_{A_{2},A_{1}}(X).

Proof. Let ρ_{n} be a sequence in D(Δ) which converges to ρ and Δρ_{n} to B in L(X). Hence for any x ∈ D(A_{1}), ρ_{n}x converges to ρx and ρ_{n}A_{1}x converges to ρA_{1}x. This implies that A_{2}ρ_{n}x converges also to Bx − ρA_{1}x. Now, since any generator of a regularized semigroup is closed (see [6, Theorem 3.4]), thus A_{2} is closed and consequently ρx ∈ D(A_{2}) and A_{2}ρx = Bx − ρA_{1}x, which implies the closedness of Δ.

If ρ ∈ Im(Λ_{r}), then there exists σ ∈ L(X) such that ρ = (r − A_{2})^{−1}σ(r − A_{1})^{−1}. Hence on D(A_{1}) we have

Δρ = (r(r − A_{2})^{−1} − I)σ(r − A_{1})^{−1} + (r − A_{2})^{−1}σ(r(r − A_{1})^{−1} − I),

which has a bounded extension

B = −[σ(r − A_{1})^{−1} + (r − A_{2})^{−1}σ] + 2rρ

that implies Im(Λ_{r}) ⊂ D(Δ). Furthermore, for any x ∈ X, ρx ∈ D(A_{2}) and for σ = (r − A_{2})ρ(r − A_{1}) on D(A_{1}), σ − r^{2}ρ + rB is a bounded extension of A_{2}ρA_{1} from D(A_{1}). This implies also that Im(Λ_{r}) ⊂ L_{A_{2},A_{1}}(X).

Conversely if ρ ∈ D(Δ̃), then there exists a bounded operator B such that for any x ∈ D(A_{1}), (A_{2}ρ + ρA_{1})x = Bx. With the bounded operator σ = rρ − rB + A_{2}ρA_{1} we retrieve ρ = (r − A_{2})^{−1}σ(r − A_{1})^{−1} and D(Δ̃) ⊂ Im(Λ_{r}). □

Let us note by by Γ the operator Γρ = C_{2}ρC_{1}, on L(X).

Theorem 2. Under the above assumptions, suppose there exists a real r such that for any t ≥ 0, W_{i}(t) commutes with (r − A_{i})^{−1}, for i=1,2. Then an extension of Δ̃ generates a Λ_{r}Γ–regularized semigroup W(t), which is continuous in the uniform operator topology of L(L(X)), that leaves D(Δ̃) invariant.

Proof. Define W(t) on L(X) by

W(t)ρ := (r − A_{2})^{−1}W_{2}(t)ρW_{1}(t)(r − A_{1})^{−1}.

For any x ∈ X and i = 1, 2,

\frac{d}{dτ} W_{i}(τ)(r − A_{i})^{−1}x = W_{i}(τ)A_{i}(r − A_{i})^{−1}x.

Since W_{i}(t) is a strongly continuous family in L(X), W_{i}(τ)A_{i}(r − A_{i})^{−1}x is uniformly bounded on [s, t] by mean value theorem in the Banach space X, there exists τ_{0} ∈ [s, t], such that

(W_{i}(t) − W_{i}(s))(r − A_{i})^{−1}x = (t − s)W_{i}(τ_{0})A_{i}(r − A_{i})^{−1}x

and by uniform boundedness principle ∥W_{i}(τ)A_{i}(r − A_{i})^{−1}∥ is bounded in a closed neighborhood of τ_{0}. Thus, W_{i}(t)(r − A_{i})^{−1} is continuous in operator norm topology of L(X). Since

∥W(t)ρ − Λ_{r}Γρ∥ = ∥(r − A_{2})^{−1}(W_{2}(t) − C_{2})ρW_{1}(t)(r − A_{1})^{−1}
+ (r − A_{2})^{−1}C_{2}ρ(W_{1}(t) − C_{1})(r − A_{1})^{−1}∥
≤ (∥(r − A_{2})^{−1}W_{2}(t) − (r − A_{2})^{−1}C_{2}∥.∥W_{1}(t)(r − A_{1})^{−1}∥
+ ∥(r − A_{2})^{−1}C_{2}∥.∥(W_{1}(t)(r − A_{1})^{−1} − C_{1}(r − A_{1})^{−1}∥)∥ρ∥

and W_{i}(t)(r − A_{i})^{−1} = (r − A_{i})^{−1}W_{i}(t) converges uniformly to C_{i}(r − A_{i})^{−1} = (r − A_{i})^{−1}C_{i}, as t → 0, this implies the continuity of W(t) in the uniform operator topology of L(L(X)).

It is clear that W(0) equals Λ_{r}Γ and

W(t)W(s)ρ = (r − A_{2})^{−1}W_{2}(t)(r − A_{2})^{−1}W_{2}(s)ρW_{1}(s)(r − A_{1})^{−1}W_{1}(t)(r − A_{1})^{−1}
= (r − A_{2})^{−2}C_{2}W_{2}(t + s)ρW_{1}(t + s)C_{1}(r − A_{1})^{−2}
= Λ_{r}ΓW(t + s)ρ.

Hence, W(t) is a Λ_{r}Γ–regularized semigroup.

Now, by taking ρ ∈ D(Δ̃) = Im(Λ_{r}), we can write W(t) as

W(t)ρ = (r − A_{2})^{−2}W_{2}(t)(Λ_{r}^{−1}ρ)W_{1}(t)(r − A_{1})^{−2}.

Again using the mean value theorem, one may show that (r − A_{i})^{−2}W_{i}(t) is a differentiable function of t, in operator norm topology of L(X), for i = 1, 2. Thus W(t)ρ is differentiable, with

\frac{d}{dt}W(t)ρ = [\frac{d}{dt}(r − A_{2})^{−2}W_{2}(t)](Λ_{r}^{−1}ρ)W_{1}(t)(r − A_{1})^{−2}

+ (r − A_{2})^{−2}W_{2}(t)(Λ_{r}^{−1}ρ)[\frac{d}{dt}(r − A_{1})^{−2}W_{1}(t)]

= (r − A_{2})^{−1}W_{2}(t)A_{2}(r − A_{2})^{−1}(Λ_{r}^{−1}ρ)(r − A_{1})^{−1}W_{1}(t)(r − A_{1})^{−1}

+ (r − A_{2})^{−1}W_{2}(t)(r − A_{2})^{−1}(Λ_{r}^{−1}ρ)(r − A_{1})^{−1}A_{1}W_{1}(t)(r − A_{1})^{−1}

= (r − A_{2})^{−1}W_{2}(t)(A_{2}ρ)W_{1}(t)(r − A_{1})^{−1}

+ (r − A_{2})^{−1}W_{2}(t)(ρA_{1})W_{1}(t)(r − A_{1})^{−1}

= W(t)Δρ.

This implies that an extension of Δ̃ generates W(t). Since W(t) commutes with Λ_{r}, W(t) leaves D(Δ̃) = Im(Λ_{r}) invariant. □

The following definition is a natural consequence of the preceding theorem.

Definition 3. Whenever A_{1} and A_{2} generate two regularized semigroups, then the regularized semigroup generated by an extension of the operator Δ̃ on L(X) is called tensor product regularized semigroup.

Corollary 1. Let W_{1}(t) be an exponentially bounded C-regularized semigroup of the type ω generated by A_{1} and let G_{2}(t) be a C_{0}–semigroup of the type −ω generated by A_{2}. Then all the elements of

I_{C} := {ρ ∈ L(X) ; ρ = BC with B ∈ L(X)}

the right ideal generated by C are the remedial operators for e^{tA_{1}} = C^{−1}W_{1}(t) and G_{2}(t).

Proof. Since G_{2}(t)BCe^{tA_{1}} = G_{2}(t)BW_{1}(t) all the properties of Definition 2 are readily satisfied for ρ ∈ I_{C}. □

In the following section we give an application of the above Corollary.

4. Application to transport theory for growing cell populations.

One of the most appealing application of this theory can be found in the modelling of a carcinogenic cell population. Since this kind of cells has a high proliferating rate, we can associate them an abnormally large maturation velocity. In 1983, Rotenberg [11] presented a model for growing cell populations. In his model cells are distinguished by their degree of maturity μ ∈ I = (0, 1) and their maturation velocity v(= \frac{dμ}{dt}) which is considered as an independent variable within J = (a, b) (0 ≤ a < b ≤ ∞). The positivity of velocities comes from the fact that a cell may not become less mature with time and adds to the irreversible character to the problem. The density of the population f = f(μ, v, t) is described by the following partial differential equation of transport type

\begin{aligned}\frac{∂f}{∂t} & = −v \frac{∂f}{∂μ} − σ(μ, v)f + ∫_{a}^{b} r(μ, v, v′)f(μ, v′, t)dv′ \\ & = T f, \end{aligned}  (4)

where σ(μ, v) = ∫_{a}^{b} s(μ, v′, v)dv′ and s(μ, v′, v) is the sinking rate at which cells drop out of the cohort characterized by v to join those of v′. The kernel r(μ, v, v′) is the transition rate at which cells change their velocities from v′ to v. The reproduction rule are given by the following boundary condition, so called Lebowitz–Rubinow boundary condition :

vf(0, v, t) = p ∫_{a}^{b} k(v, v′)v′f(1, v′, t)dv′  (5)

where p ≥ 0 is the average number of viable daughters per mitosis. This rule allows a choice in the degree of positive correlation k between velocities of mother and daughter cells.

For a mathematical study of this problem, let us denote Ω = (0, 1) × (a, b) = I × J, L^{1}(Ω) the Lebesgue’s space with its natural norm ∥·∥_{1} and the partial Sobolev space W^{1}(Ω) = \{φ ∈ L^{1}(Ω) | v \frac{∂φ}{∂μ} ∈ L^{1}(Ω)\} endowed with the norm ∥ φ ∥_{W^{1}(Ω)}=∥ φ ∥_{1} + ∥

v \frac{∂φ}{∂μ} ∥_{1}. In [3], we studied the well-posedness of this model under the condition that the maturation velocity v belong to J = (a, b) with (0 < a < b < ∞). and we proved that the operator A_{2} with domain D(T) := {f ∈ W^{1}(Ω) | such that (5) holds } generates a C_{0}–semigroup. We have also showed the positivity, irreducibility of the generated semigroup which converges asymptotically to a projection of the rank one. In [4], in contrast with [3], we allow 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, in spite of the fact that we lose the weak compact character of the generated semigroup, we have calculated explicitly its essential type and we have showed the asymptotic convergence of the generated semigroup to a projection of rank one. Finally, the case 0 ≤ a < b = ∞ is a pathological case. In fact, M. Boulanouar in [2] showed that even with a simplest reproduction rule, which is called by M. Rotenberg [11] a perfect memory reproduction rule, if b = ∞ and p > 1 then the problem (4) is ill-posed and he introduced a regularized operator C with which A_{2} generates a Cregularized semigroup in the sense of deLaubenfels [6]. This perfect memory kind of reproduction rule is one in which the maturation velocity of a parent cell at mitosis is transmitted to the daughter cells. This gives the following boundary condition

f(0, v, t) = pf(1, v, t).  (6)

This identity asserts that the last velocity the parent happened to have just before mitosis would be the initial velocity assumed by the daughters. Hence, this kind of reproduction rule produces a population at the maximum rate and can be simulated as a cohort of the cancer cells. If we take σ = r = 0 in (4), then A_{2} with boundary condition (6) generates a semigroup T (t) which has an explicit form

T (t)φ(μ, v) = p^{[tv−μ]}φ(μ + [tv − μ] − tv, v).  (7)

where [tv − μ] = n + 1 if n < tv − μ ≤ n + 1. This semigroup is not strongly continuous for p > 1. In fact

∥T (t)f∥ = ∫_{0}^{∞} ∫_{0}^{1} |T (t)f(μ, v)|dμdv

= ∑_{n≥0} ∫ \frac{n+1}{t} ; \frac{n}{t} [ ∫_{0}^{tv−n} p^{n+1}|f(μ + n + 1 − tv, v)|dμ

+ ∫_{tv−n}^{1} p^{n}|f(μ + n − tv, v)|dμ]dv.

Hence, the series diverges for p > 1 and for some f ∈ L^{1}((0, 1) × (a, ∞)). This implies that T (t) is an unbounded semigroup and consequently, is not strongly continuous.

Here, we are going to give two concrete examples of remedial operators in this situation.

A remedial operator in radiotherapy

Current radiotherapy techniques fire radiation beams into the body to kill cancer cells. This treatment reduces spontaneously the amount of cancer cell population and can be modelled by a multiplicative operator. In [2], Boulanouar has taken ρf(μ, v) := p^{−v^{2}}f(μ, v) for regularizing the ill-posed semigroup. In spite of the fact that this operator reduces considerably those cells which have a large maturation velocities, but

the resulting ρ-regularized semigroup W (t) = ρT (t) has a behavior as p^{\frac{t^{2}}{4} +1} (see [2, Lemme 2.3]). So it seems hard to find a C_{0}-semigroup S(t) such that S(t)ρT (t) remains bounded for all t > 0, unless taking S(t) as a superstable semigroup, that become zero after finite time. This choice of semigroup would be another strategy of remedying, which consiste of shifting the emphasis from the remedial operator ρ to the semigroup S(t). This sort of strategy is far from our primary objective.

Thus, such a remedial operator does not belongs to any space R(X) for any C_{0}– semigroup S(t). Experimentally one knows that after radiotherapy remediation the residual cells can generate an amount of the cancer cells of exponential growth.

A remedial operator in chemotherapy

During chemotherapy treatment a hostile biophysical environment is created in order to stop the proliferation of the cancer cells. This treatement reduce the population distribution by a kind of two compartmental actions; annihilation of the cohort with the large maturation velocities, while maintaining the colony with small maturation velocities, for a long period to render them inoffensive. This can be done be introducing a time dependent integral operator [ρ(t)f](μ, v) = ∫_{0}^{∞} r(v, v′, t)f(μ, v′)dv′, with a bounded positive kernel r(v, v′, t) such that the support of r is in [0, α]×[0, β]×[0, ∞) with α and β relatively small. Such integral operator annihilates the cells which have a large maturation velocity and also any function in its image has a small maturation velocities. For simplicity we take a time independent kernel r(v, v′), with sup_{(v,v′)∈[0,α]×[0,β]} |r(v, v′)| ≤ M. From (7) it follows that for any f ∈ L^{1}(Ω), if N − 1 < tβ ≤ N, the ρ-regularized semigroup W (t) = ρT (t) can be estimated by

∥W (t)f∥ = ∫_{0}^{∞} ∫_{0}^{1} ∫_{0}^{∞} r(v, v′)|T (t)f(μ, v′)|dv′dμdv

≤ Mα ∫_{0}^{β} ∫_{0}^{1} |T (t)f(μ, v)|dμdv

≤ Mα ∑_{0≤n≤N} ∫ \frac{n+1}{t} ; \frac{n}{t} [ ∫_{0}^{tv−n} p^{n+1}|f(μ + n + 1 − tv, v)|dμ+

∫_{tv−n}^{1} p^{n}|f(μ + n − tv, v)|dμ]dv

\begin{matrix}= Mα ∑_{0≤n≤N} ∫ \frac{n+1}{t} ; \frac{n}{t} [∫_{0}^{n+1−tv} p^{n+1}|f(μ, v)|dμ + ∫_{0}^{n+1−tv} p^{n}|f(μ, v)|dμ] dv \\ ≤ Mαp^{tβ+1}∥f∥. \end{matrix}

Thus, by taking a C_{0}-semigroup S(t) of type ω = −β ln p, ∥S(t)ρT (t)∥ would be bounded and ρ is a remedial operator in the sense of Definition 2.

Acknowledgement. I am extremely indebted to Professor Sen-Yen Shaw and the referee who read the first version of the paper with care for detail and giving many valuable suggestions.

References

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Laboratoire de Modélisation Mécanique et de Mathématiques Appliquées, Université de Poitiers, Boulevard 3, Teleport 2, BP 179, 86960 FU-TUROSCOPE, Cedex, France (e-mail: Hassan.Emamirad@univpoitiers.fr)