On a mathematical model of age–cycle length structured cell population with non‐compact boundary conditions (II)

Boulanouar, Mohamed

doi:10.1002/mma.3606

Cited by 3 publications

(20 citation statements)

References 3 publications

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“…Let t > 0 and let ϕ ∈ L 1 . Since , Theorem 2.3] we readily get that

double-struckT α, β (t) ϕ (a, l) = double-struckT 0, 0 (t) ϕ (a, l) + ξ (a, t) K α, β γ 1 (double-struckT α, β (t - a) ϕ) (l)

for almost all ( a , l )∈Ω, and by the explicit form ,

\begin{matrix} double-struckT α, β (t) ϕ (a, l) & = double-struckT 0, 0 (t) ϕ (a, l) + αξ (a, t) γ 1 (double-struckT α, β (t - a) ϕ) (l) \\ + βξ (a, t) \int_{0}^{\infty} k (l, l^{'}) γ 1 (double-struckT α, β (t - a) ϕ) (l^{'}) d l^{'} . \end{matrix}

Firstly, if α = 0 then becomes

\begin{matrix} T 0, β (t) ϕ (a, l) & = T 0, 0 (t) ϕ (a, l) + β ξ (a, t) \int \end{matrix}

…”

Section: Explicit Form Of Unperturbed Semigroupmentioning

confidence: 87%

“…In , we have proved that this model is governed by a strongly continuous semigroup, U α , β =( U α , β ( t )) t ≥0 , which was the first novelty. The second one follows from . Indeed, we have estimated the type ω 0 ( U α , β ), of the full semigroup U α , β =( U α , β ( t )) t ≥0 , by proving

μ (a, l) < ω 0 U α, β

For more explanation, we refer to and the references therein.…”

Section: Introductionmentioning

confidence: 85%

“…As we have pointed out in the introduction of , the proof of needs to define an adequate strategy whose first part ( i.e. the proof of ) has been successfully completed in . We are therefore tempted to prove

ω ess (U α, β) \leq - (a, l) \in normalΩ μ (a, l)

which is exactly the second part of our strategy.…”

Section: Introductionmentioning

confidence: 99%

“…The full model – has recently been the subject of a mathematical study ( and ). In , we have proved that this model is governed by a strongly continuous semigroup, U α , β =( U α , β ( t )) t ≥0 , which was the first novelty.…”

Section: Introductionmentioning

confidence: 99%

“…As we have pointed out in the introduction of , the proof of needs to define an adequate strategy whose first part ( i.e. the proof of ) has been successfully completed in .…”

Section: Introductionmentioning

confidence: 99%

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On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

Boulanouar¹

2015

Math Methods in App Sciences

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This work is a natural continuation of two other works in which a mathematical model has been studied. This model is based on age‐cycle length structured cell population. The cellular mitosis is mathematically described by a non‐compact boundary condition. We investigate the asymptotic behavior of the generated semigroup, and we prove that the cell population possesses Asynchronous Growth Property. Copyright © 2015 John Wiley & Sons, Ltd.

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“…Let t > 0 and let ϕ ∈ L 1 . Since , Theorem 2.3] we readily get that

double-struckT α, β (t) ϕ (a, l) = double-struckT 0, 0 (t) ϕ (a, l) + ξ (a, t) K α, β γ 1 (double-struckT α, β (t - a) ϕ) (l)

for almost all ( a , l )∈Ω, and by the explicit form ,

\begin{matrix} double-struckT α, β (t) ϕ (a, l) & = double-struckT 0, 0 (t) ϕ (a, l) + αξ (a, t) γ 1 (double-struckT α, β (t - a) ϕ) (l) \\ + βξ (a, t) \int_{0}^{\infty} k (l, l^{'}) γ 1 (double-struckT α, β (t - a) ϕ) (l^{'}) d l^{'} . \end{matrix}

Firstly, if α = 0 then becomes

\begin{matrix} T 0, β (t) ϕ (a, l) & = T 0, 0 (t) ϕ (a, l) + β ξ (a, t) \int \end{matrix}

…”

Section: Explicit Form Of Unperturbed Semigroupmentioning

confidence: 87%

μ (a, l) < ω 0 U α, β

For more explanation, we refer to and the references therein.…”

Section: Introductionmentioning

confidence: 85%

ω ess (U α, β) \leq - (a, l) \in normalΩ μ (a, l)

which is exactly the second part of our strategy.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…As we have pointed out in the introduction of , the proof of needs to define an adequate strategy whose first part ( i.e. the proof of ) has been successfully completed in .…”

Section: Introductionmentioning

confidence: 99%

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On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

Boulanouar¹

2015

Math Methods in App Sciences

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An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated to a Regular Field: Part II

Arlotti

Lods

2019

Mediterr. J. Math.

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An $L^{p}$--approach to the well-posedness of transport equations associated to a regular field

Arlotti,

Lods

2018

Preprint

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We investigate transport equations associated to a Lipschitz field F on some subspace of R N endowed with some general space measure µ in some L p -spaces 1 < p < ∞, extending the results obtained in two previous contributions [2,3] in the L 1 -context. We notably prove the well-posedness of boundary-value transport problems with a large variety of boundary conditions. New explicit formula for the transport semigroup are in particular given. CONTENTS 1. Introduction 2. Preliminary results 2.1. Integration along characteristic curves 2.2. The maximal transport operator and trace results 2.3. Fundamental representation formula: mild formulation 2.4. Additional Properties 3. Well-posedness for initial and boundary-value problems 3.1. Absorption semigroup 3.2. Some useful operators. 3.3. Generalized Cessenat's theorems 3.4. Boundary-value problem 3.5. Additional properties of the traces 4. Generation properties for unbounded boundary operators 4.1. The transport operator associated to bounded boundary operators 5. Explicit transport semigroup for bounded boundary operators 5.1. Boundary Dyson-Phillips iterated operators 5.2. Generation Theorem Appendix A. Proof of Theorem 2.6 Appendix B. Additional properties of T max, p . Appendix C. On the family (U k (t)) t 0

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On a mathematical model of age–cycle length structured cell population with non‐compact boundary conditions (II)

Cited by 3 publications

References 3 publications

On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated to a Regular Field: Part II

An $L^{p}$--approach to the well-posedness of transport equations associated to a regular field

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