Positive semigroups and perturbations of boundary conditions

We present a generation theorem for positive semigroups on an L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} space. It provides sufficient conditions for the existence of positive and integrable solutions of initial-boundary value problems. An application to a two-phase cell cycle model is given.


Introduction
We study well-posedness of linear evolution equations on L 1 of the form where Ψ 0 , Ψ are positive and possibly unbounded linear operators on L 1 , the linear operator A is such that Eq. (1) with Ψ = 0 generates a positive semigroup on L 1 , i.e., a C 0 -semigroup of positive operators on L 1 . We present sufficient conditions for the operators A, Ψ 0 , and Ψ under which there is a unique positive semigroup on L 1 providing solutions of the initial-boundary value problem (1). For a general theory of  [4,7,11,14,34]. An overview of different approaches used in studying initial-boundary value problems is presented in [13]. Our result is an extension of Greiner's [19] by considering unbounded Ψ and positive semigroups. Unbounded perturbations of the boundary conditions of a generator were studied recently in [1,2] by using extrapolated spaces and various admissibility conditions. In the proof of our perturbation theorem we apply a result about positive perturbations of resolvent positive operators [3] with non-dense domain in AL-spaces in the form given in [37,Theorem 1.4]. It is an extension of the well known perturbation result due to Desch [15] and by Voigt [41]. For positive perturbations of positive semigroups in the case when the space is not an AL-space we refer to [5,10]. We also present a result about stationary solutions of (1). We illustrate our general results with an age-size-dependent cell cycle model generalizing the discrete time model of [22,25,38]. This model can be described as a piecewise deterministic Markov process (see Sect. 5 and [34]). Our approach can also be used in transport equations [8,23].

General results
Let ∂ is positive and there exists ω ∈ R such that the operator Theorem 1 Assume conditions (i)-(iv). Then the operator (A Ψ , D(A Ψ )) defined by is the generator of a positive semigroup on L 1 . Moreover, the resolvent operator of A Ψ at λ > ω is given by Let I be the identity operator on Combining this with (ii) we conclude that I − B R(λ, A) is invertible with positive inverse (I − B R(λ, A)) −1 for all λ > ω. Hence, the spectral radius of the positive operator B R(λ, A) is strictly smaller than 1 for some λ > ω. It follows from [37,Theorem 1.4] that the part of (A is densely defined and generates a positive semigroup on L 1 . Finally, the operator (A + B, D(A)) is resolvent positive with resolvent given by for all nonnegative f ∈ D(A 0 ). If, additionally, is densely defined and resolvent positive, is the generator of a substochastic semigroup on L 1 , i.e., a positive semigroup of contractions on L 1 . This is a consequence of the Hille-Yosida theorem, see e.g. [34,Theorem 4.4]. Thus it is enough to assume condition (v) instead of (iii). Observe also that (iii) and (iv) imply that (0, ∞) ⊆ ρ(A 0 ).

Remark 4 Greiner [19, Theorem 2.1] establishes that (A Ψ , D(A Ψ ))
is the generator of a C 0 -semigroup for any bounded Ψ provided that conditions (a) and (b) hold true, (A 0 , D(A 0 )) is the generator of a C 0 -semigroup, and that there exist constants γ > 0 and λ 0 such that This is condition (2.1) of Greiner [19,Theorem 2.1]. Some extensions of this result are provided in [20,29] for unbounded Ψ , as well as in [1,2].

Remark 5
Recall that a positive operator on an AL-space defined everywhere is automatically bounded. Thus our assumption (i) implies that Ψ (λ) is bounded for each λ > 0. Moreover, its norm is determined through its values on the positive cone. From assumptions (i) and (iv) it follows that λ Ψ (λ) ≤ 1 for each λ > 0, as was shown in the proof of Theorem 1. Thus, for f = Ψ (λ) f ∂ , we get (9) with γ = 1. Now suppose, as in [19], that Ψ is bounded. Then Ψ Ψ (λ) ≤ Ψ /λ for all λ > 0. Hence, the operator (4), Consequently, if Ψ is bounded and positive, then we get the same result as in [19].
We now look at a simple example where Theorem 1 can be easily applied and it should be compared with [1,Corollary 25].  (1). Let the boundary operators Ψ 0 and Ψ be defined by where μ is a finite Borel measure. Note that for each λ > 0 and Hence condition (i) holds true. We have and the restriction A 0 of the operator A to is the generator of a positive semigroup. Thus conditions (iii) and (iv) hold true. If there exists λ > 0 such that then condition (ii) holds true and the operator is the generator of a positive semigroup, by Theorem 1. Now suppose that μ is a probability measure, so that μ([0, 1]) = 1. Then Consequently, if μ is a probability measure such that μ = δ 1 then (A Ψ , D(A Ψ )) is the generator of a positive semigroup.
It should be noted that in [34,Theorem 4.6] the assumption that the domain D(A Ψ ) of the operator A Ψ is dense is missing. Making use of Theorem 1, we get the following result.

Theorem 2 Assume conditions (i)-(iv). If B is a bounded positive operator such that
We conclude this section with a result concerning the existence of steady states of the positive semigroup from Theorem 1. Note that given any λ, exists and Ψ (0) f ∂ is nonnegative.

Theorem 3 Assume conditions (i)-(iv).
Let Ψ (0) be as in (11). If a nonnegative The operators Ψ and Ψ 0 are positive and we have , ω} and completes the proof.

A model of a two phase cell cycle in a single cell line
The cell cycle is the period from cell birth to its division into daughter cells. It contains four major phases: G 1 phase (cell growth before DNA replicates), S phase (DNA synthesis and replication), G 2 phase (post DNA replication growth period), and M (mitotic) phase (period of cell division). The Smith-Martin model [36] divides the cell cycle into two phases: A and B. The A phase corresponds to all or part of G 1 phase of the cell cycle and has a variable duration, while the B phase covers the rest of the cell cycle. The cell enters the phase A after birth and waits for some random time T A until a critical event occurs that is necessary for cell division. Then the cell enters the phase B which lasts for a finite fixed time T B . At the end of the B-phase the cell splits into two daughter cells. We assume that individual states of the cell are characterized by age a ≥ 0 in each phase and by size x > 0, which can be volume, mass, DNA content or any quantity conserved trough division. We assume that individual cells of size x increase their size over time in the same way, with growth rate g(x) so that dx/dt = g(x), and all cells age over time with unitary velocity so that da/dt = 1. We assume that the probability that a cell is still being in the phase A at age a is equal to H (a), so the rate of exit from the phase A at age a is ρ(a) given by where h is a probability density function defined on [0, ∞), describing the distribution of the time T A , the duration of the phase A. We make the following assumptions: (I) The function h in (12) is a probability density function so that h : [0, ∞) → [0, ∞) is Borel measurable and the function H in (12) satisfies: The Smith and Martin hypothesis [36] states that h is exponentially distributed with parameter p > 0, so that ρ(a) = p for all a > 0. However, this does not agree with experimental data, see e.g. [18,43] for recent results. The generation time of a cell, i.e. the time from birth to division, can be written as T = T A + T B . Thus the distribution of the generation time has a probability density of the form Cell generation times can have lognormal or bimodal distribution (see [35]), exponentially modified Gaussian [17], or tempered stable distributions [30].
To describe the growth of cells we define wherex > 0 orx = 0, if the integral is finite. The value Q(x) has a simple biological interpretation. Ifx is the size of a cell, then Q(x) is the time it takes the cell to reach the size x. It follows from assumption (II) that the function Q is strictly increasing and continuous. We denote by Q −1 the inverse of Q. Define for t ≥ 0 and x 0 > 0. Then π t x 0 satisfies the initial value problem If Q(0) = −∞ then Q −1 is defined on R. Hence, formula (14) extends to all t ∈ R and x 0 > 0. We also set π t 0 = 0 for t > 0 in this case. If Q(0) = 0 then Q −1 is defined only on (0, ∞) and we set π t 0 = Q −1 (t) for t > 0. We can extend formula (14) to all negative t satisfying Q(x 0 ) + t > 0; otherwise we set π t x 0 = 0. Note that at time t = T , the generation time, a "mother cell" of size π T x 0 divides into two daughter cells of equal size 1 2 π T x 0 . In the probabilistic model of [22,25,38,39] a sequence of consecutive descendants of a single cell was studied. Let f be the probability density function of the size distribution at birth at time t 0 of mother cells and let t 1 > t 0 be a random time of birth of daughter cells. Then the probability density function of the size distribution of daughter cells is given by [25,38] where The iterates P 2 f , P 3 f , . . . denote densities of the size distribution of consecutive descendants born at random times t 2 , t 3 , . . .. The operator P defined by (15) is a positive contraction on L 1 (0, ∞), the space of Borel measurable functions defined on (0, ∞) and integrable with respect to the Lebesgue measure. Here we extend the probabilistic model to a continuous time situation by examining what happens at all times t and not only at t 0 , t 1 , t 2 , . . .. We denote by p 1 (t, a, x) and p 2 (t, a, x) the densities of the age and size distribution of cell in the A-phase and in the B-phase at time t, age a, and size x, respectively. Neglecting cell deaths the equations can be written as with boundary and initial conditions In this model, cells in the A-phase enter the B-phase at rate ρ. This is taken into account by the boundary condition (18). All cells stay in the B-phase until they reach the age T B . Then they divide their size into half (17). The model is complemented with initial conditions (19). The model we propose is different as compared to mass/maturity structured models [16,21,31,40] where a cell leaves the phase A with intensity being dependent on maturity, not age. In the case of T B = 0 there is only one phase present; a maturity structured model being a continuous time extension of [24] is studied in [27], while age and volume/maturity structured population models of growth and division were studied extensively since the seminal work of [12,26,33]. We refer the reader to [28] for historical remarks concerning modeling of age structured populations and to [35,42] for recent reviews. We look for positive solutions of (16)- (19) in the space and m is the product of the two-dimensional Lebesgue measure and the counting measure on {1, 2}, and E is the σ -algebra of all Borel subsets of E. We identify L 1 = L 1 (E, E, m) with the product of the spaces L 1 (E 1 ) and L 1 (E 2 ) of functions defined on the sets E 1 and E 2 , respectively, and being integrable with respect to the two-dimensional Lebesgue measure. We say that the operator P has a steady state in L 1 (0, ∞) if there exists a probability density function f such that P f = f . Similarly, a semigroup {S(t)} t≥0 has a steady state in L 1 if there exists a nonnegative f ∈ L 1 such that S(t) f = f for all t > 0 and f 1 = 1 where · 1 is the norm in L 1 . We give the proof of Theorem 4 in the next section. Theorem 4 combined with [9] implies the following sufficient conditions for the existence of steady states of (16)- (19).
If the cell growth is exponential so that we have g(x) = kx for all x > 0, where k is a positive constant, then it is known [22,38,39] that the operator P has no steady state. We now consider a linear cell growth and assume that g(x) = k for all x > 0. We see that Q(x) = x/k, the operator P is of the form (see [39] or the last section)

Proof of Theorem 4
We will show that Theorem 4 can be deduced from Theorems 1 and 3. To this end, we introduce some notation. Let us define where π t is given by (14). Then t → π(t, a 0 , x 0 ) solves the system of equations a (t) = 1 and x (t) = g(x(t)) with initial condition a(0) = a 0 and x(0) = x 0 . Recall that E 1 is an open set. For any x 0 , a 0 ∈ E 1 we define and the incoming part of the boundary ∂ E 1 Observe that t − (a 0 , x 0 ) = a 0 for all (a 0 , x 0 ) ∈ E 1 and that Γ − 1 = {0} × (0, ∞). We consider on Γ − 1 the Borel measure m − 1 being the product of the point measure δ 0 at 0 and the Lebesgue measure on (0, ∞). We define the operator T max on L 1 (E 1 ) by [6] where the differentiation is understood in the sense of distributions. Then it follows from [6] that for f ∈ D(T max ) the following limit exists for almost every z ∈ Γ − 1 with respect to the measure m − 1 on Γ − 1 . According to [6,Theorem 4.4] the operator T 0 = T max with domain is the generator of a substochastic semigroup on L 1 (E 1 ) given by By [6, Proposition 5.1], the operator (T, D(T)) defined by is the generator of a substochastic semigroup on L 1 (E 1 ) of the from Note that we can identify the space L 1 (E 2 ) with the subspace We also define the exit time from the set E 2 by and the outgoing part of the boundary ∂ E 2 We define the Borel measure m − 2 on Γ − 2 as the measure m − 1 and the m + 2 on Γ + 2 as the product of the point measure at T B and the one dimensional Lebesgue measure. Since U 0 (t)(L 1 (E 2 )) ⊆ L 1 (E 2 ), the part of the operator (T 0 , D(T 0 )) in L 1 (E 2 ) is the generator of a substochastic semigroup {U 2 (t)} t≥0 in L 1 (E 2 ). Moreover, the following pointwise limits exist for almost every z ∈ Γ ± 2 with respect to the Borel measure m ± 2 on Γ ± 2 .
, E ∂ be the σ -algebra of Borel subsets of E ∂ and m ∂ be the product of the Lebesgue measure on the line {0} × (0, ∞) and the counting measure on {1, 2}. To simplify the notation we identify L 1 1 (0, ∞). We define operators A 1 and A 2 by We set

and we define the operator A on D by setting
We take operators Ψ 0 , Ψ : D → L 1 ∂ of the form and To this end, we check that assumptions (i)-(iv) of Theorem 1 from Sect. 2 are satisfied.
We first show that conditions (iii) and (iv) hold. The operator A restricted to with the corresponding generators. The semigroup {S 0 (t)} t≥0 is substochastic. For all By [6,Proposition 4.6], this reduces to implying that condition (iv) holds.
For 1 (a, x)) − λ f 1 (a, x), Hence, we see that the right inverse of Ψ 0 when restricted to the nullspace of λI − A is given by for (a, x) ∈ E 1 and Thus f 1 ∈ D 1 . Similarly, f 2 ∈ D 2 . Hence, condition (i) holds.
Thus the assertion follows from Theorem 3.

Final remarks
Our model can be described as a piecewise deterministic Markov process {X (t)} t≥0 . We considered three variables (a, x, i), where i = 1 if a cell is in the phase A, i = 2 if it is in the phase B, the variable x describes the cell size, and a describes the time which elapsed since the cell entered the ith phase. Let t 0 = 0. If we observe consecutive descendants of a given cell and the nth generation time is denoted by t n , then t n+1 = s n + T B where s n is the time when the cell from the nth generation enters the phase B, n ≥ 0. A newborn cell at time t n is with age a(t n ) = 0 and with initial size equal to is the size of its mother cell. The cell ages with velocity 1 and its size grows according to the equations x (t) = g(x(t)) for t ∈ (t n , s n ). If the cell enters the phase B then its age is reset to 0 and its size still grows according to and at the end of the second phase the cell divides into two cells, so that we have Thus the process X (t) = (a(t), x(t), i(t)) satisfies the following system of ordinary differential equations between consecutive times t 0 , s 0 , t 1 , s 1 , . . ., called jump times. At jump times the process is given by (30) and (31). If the distribution of X (0) has a density f then X (t) has a density S(t) f , i.e., 1 is the density of the size distribution at time t 0 = 0 and f ∂,2 is the density of the distribution of size at time s 1 , then the distribution of size at time t 1 is given by whereπ is the density of the size x(a) of the cell at time a, if x(0) has a density f ∂, 1 . Thus the density of the mass x(t 1 ) is given by for Lebesgue almost every x ∈ (0, ∞), where P is as in (15). Now, if the operator P has a steady state f ∂,1 ∈ L 1 (0, ∞) so that f ∂,1 satisfies (29) and if f ∂,2 is as in (32), then f * = ( f * 1 , f * 2 ) given by is the steady state for the semigroup {S(t)} t≥0 existing by Theorem 4. Moreover, it is unique if P has a unique steady state.

Remark 6
It should be noted that in the two-phase cell cycle model in [31] the rate of exit from the phase A depends on x, not on a, and that there is no such equivalence between the existence of steady states as presented in Theorem 4. Our results remain true if we assume as in [31] that division into unequal parts takes place. Methods as in [31,34] can also be used in our model to study asymptotic behaviour of the semigroup {S(t)} t≥0 . For a different approach to study positivity and asymptotic behaviour of solutions of population equations in L 1 we refer to [32].
We conclude this section with an extension of the age-size dependent model from [12] to a model with two phases. Let p i (t, a, x) be the function representing the distribution of cells over all individual states a and x at time t in the phase A for i = 1 or B for i = 2, i.e., a 2 a 1 x 2 x 1 p i (t, a, x)dadx is the number of cells with age between a 1 and a 2 and size between x 1 and x 2 at time t in the given phase. Then p 1 and p 2 satisfy Eqs. (16), (18), (19) while the boundary condition (17) takes the form since a mother cell at the moment of division T B has size 2x and gives birth to two daughters of size x entering the phase A at age 0. This follows from Theorem 1 in the same way as Theorem 4, where now to check condition (ii) we note that h(a)e −λa da f ∂ for all f ∂ ∈ L 1 ∂ and λ > 0, implying that Ψ Ψ (λ) < 1 for all λ > ω with ω = log 2/T B .
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