Fine
  asymptotics of profiles and relaxation to equilibrium for
  growth-fragmentation equations with variable drift rates

Balagué, D.; Cañizo, José A.; Gabriel, Patrick

doi:10.3934/krm.2013.6.219

Cited by 36 publications

(97 citation statements)

References 11 publications

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“…Usually, even when it is possible to establish (5), it is difficult to find an explicit expression for the asymptotic profile. In [2], the authors provided fine estimates on the principal eigenfunctions, giving their first order behaviour close to 0, and +∞ (see also [7]). In this work, we have been able to characterize the asymptotic profile in a fairly explicit way using special properties of refracted Lévy processes.…”

Section: Ifmentioning

confidence: 99%

See 1 more Smart Citation

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

Cavalli

2019

Acta Appl Math

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The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo (2016) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namelywith 0 < a + < a − . We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted Lévy processes that arise naturally in this setting.

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Section: Ifmentioning

confidence: 99%

“…By definition of Φ − , for a solution to (18) to exist, one must have that Ψ ′ − (−1) ≥ 0. First of all, Ψ ′ − (−1) is well-defined and finite thanks to (2). Moreover, by (13), Ψ ′ − (−1) ≥ 0 is equivalent to − 1 0 log(s) ρ(s)ds ≤ a − and in this case the solution to (18) is given by…”

Section: Preliminaries and General Strategymentioning

confidence: 99%

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

Cavalli

2019

Acta Appl Math

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“…We notice that B only appears multiplied by N in Equation (2), so that it cannot be accurately estimated from this equation where N vanishes (near 0 and near +∞). Hence, we denote H = BN and in this article we focus on estimating H rather than B (the interested reader may find in [12] a fully rigorous estimate of B, obtained after a truncated division by N, i.e.…”

Section: Introductionmentioning

confidence: 98%

“…Inverse Problem (IP): Given a measure N ε of the solution N of Equation (2), if N ε satisfies Estimate (3), how can we get an approximation H ε of H solution of Equation (4), and estimate the approximation error ∥H − H ε ∥ L 2 (x q dx) in terms of ε, for small enough values of q and for large enough values of q?…”

Section: Introductionmentioning

confidence: 99%

Estimating the division rate of the growth-fragmentation equation with a self-similar kernel

2014

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We consider the growth-fragmentation equation and we address the problem of finding the division rate from the stable size distribution of the population, which is easily measured, but non-smooth. We propose a method based on the Mellin transform for growth-fragmentation equations with self-similar kernels. We build a sequence of functions which converges to the density of the population in division, simultaneously in several weighted L 2 spaces, as the measurement error goes to 0. This improves previous results for self-similar kernels and allows us to understand the partial results for general fragmentation kernels. Numerical simulations confirm the theoretical results. Moreover, our numerical method is tested on real biological data, arising from a bacteria growth and fission experiment.

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“…This is true not only for BeckerDöring equation [5], but also for growth fragmentation equations [15,4], for stochastic models [16], etc. This is one of the reasons that makes size distribution analysis so important: if taken at an instant where steady state or steady growth is observed, it is possible to interpret it as solution of a time-independent equation and thus to estimate kinetic coefficients from its shape.…”

Section: Statistical Analysis Of the Datamentioning

confidence: 99%

Size Distribution of Amyloid Fibrils. Mathematical Models and Experimental Data

Prigent¹,

Haffaf²,

Banks³

et al. 2014

Int. J. of Pure and Appl. Math.

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More than twenty types of proteins can adopt misfolded conformations, which can co-aggregate into amyloid fibrils, and are related to pathologies such as Alzheimer's disease. This article surveys mathematical models for aggregation chain reactions, and discuss the ability to use them to understand amyloid distributions. Numerous reactions have been proposed to play a role in their aggregation kinetics, though the relative importance of each reaction in vivo is unclear: these include activation steps, with nucleation compared to initiation, disaggregation steps, with depolymerization compared to fragmentation, and additional processes such as filament coalescence or secondary nucleation. We have statistically analysed the shape of the size distribution of prion fibrils, with the specific example of truncated data due to the experimental technique (electron microscopy). A model of polymerization and depolymerization succeeds in explaining this distribution. It is a very plausible scheme though, as evidenced in the review of other mathematical models, other types of reactions could also give rise to the same type of distributions.

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Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates

Cited by 36 publications

References 11 publications

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

Estimating the division rate of the growth-fragmentation equation with a self-similar kernel

Size Distribution of Amyloid Fibrils. Mathematical Models and Experimental Data

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