Functionals-preserving cosine families generated by Laplace operators in C[0,1]

Bobrowski, Adam; Gregosiewicz, Adam; Murat, Małgorzata

doi:10.3934/dcdsb.2015.20.1877

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…(4.8). Continuing this procedure of repeated reflections (see [39, p. 341] or [20]), we construct a twice continuously differentiable function on the entire line such that (4.7) and (4.8) are true for all x ≥ 1 and all x ≥ 0, respectively. This allows defining the family (C r (t)) t∈R of operators in…”

Section: A Building Block: a Holomorphic Feller Semigroup In [0 1] mentioning

confidence: 99%

“…Having defined (an extension of) f on [−2, 2] we may extend its definition to [−2, 4] by formula (3.13), and then again to [−4, 4] by formula (3.14). Continuing this procedure of repeated reflections (see [29] p. 341 or [13]) we construct a twice continuously differentiable function on the entire line such that (3.13) and (3.14) are true for all x ≥ 1 and all x ≥ 0, respectively. This allows defining the family…”

Section: 2mentioning

confidence: 99%

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Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Bobrowski

2020

J. Evol. Equ.

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Using techniques of the theory of semigroups of linear operators, we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the layers converges to 0, the solutions, which by nature are functions of 3 variables, gradually lose dependence on the vertical variable and thus may be regarded as functions of 2 variables. The limit equation describes diffusion on the lower and upper sides of a two-dimensional surface (the membrane) with jumps from one side to the other. The latter possibility is expressed as an additional term in the generator of the limit semigroup, and this term is built from permeability coefficients of the membrane featuring in the transmission conditions of the approximating equations (i.e., in the description of the domains of the generators of the approximating semigroups). We prove this convergence result in the spaces of square integrable and continuous functions, and study the way the choice of transmission conditions influences the limit.

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Section: A Building Block: a Holomorphic Feller Semigroup In [0 1] mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Bobrowski

2020

J. Evol. Equ.

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“…converges to the sum of this series. Since D is closed, g n (given by (21)) belongs to D(D), i.e., it is absolutely continuous with g ′ n equal to the sum of the series (23). Combining…”

Section: Let D(d) ⊂ Lmentioning

confidence: 99%

Lord Kelvin and Andrey Andreyevich Markov in a Queue with Single Server

Bobrowski¹

2018

Bulletin of the SUSU. MMP

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“…(For detailed introduction to the method of images see [6] and references given there. More examples may be found in [5,7,8].) As a by-product we obtain a semi-explicit formula for the semigroup T = {T (t), t 0} related to the Rotenberg model.…”

Section: Introductionmentioning

confidence: 99%

Lord Kelvin's method of images approach to the Rotenberg model and its asymptotics

Kosmulski¹

2017

Preprint

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We study a mathematical model of cell populations dynamics proposed by M. Rotenberg [18] and investigated by M. Boulanouar [9]. Here, a cell is characterized by her maturity and speed of maturation. The growth of cell populations is described by a partial differential equation with a boundary condition. In the first part of the paper we exploit semigroup theory approach and apply Lord Kelvin's method of images in order to give a new proof that the model is well posed. Next, we use a semi-explicit formula for the semigroup related to the model obtained by the method of images in order to give growth estimates for the semigroup. The main part of the paper is devoted to the asymptotic behaviour of the semigroup. We formulate conditions for the asymptotic stability of the semigroup in the case in which the average number of viable daughters per mitosis equals one. To this end we use methods developed by K. Pichór and R. Rudnicki [17].

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Functionals-preserving cosine families generated by Laplace operators in C[0,1]

Cited by 5 publications

References 15 publications

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Lord Kelvin and Andrey Andreyevich Markov in a Queue with Single Server

Lord Kelvin's method of images approach to the Rotenberg model and its asymptotics

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