Growth-fragmentation Equations Research Articles

Growth-fragmentation equations with McKendrick–von Foerster boundary condition

The paper concerns the well-posedness and long-term asymptotics of growth–fragmentation equation with unbounded fragmentation rates and McKendrick–von Foerster boundary conditions. We provide three different methods of proving that there is a strongly continuous semigroup solution to the problem and show that it is a compact perturbation of the corresponding semigroup with a homogeneous boundary condition. This allows for transferring the results on the spectral gap available for the later semigroup to the one considered in the paper. We also provide sufficient and necessary conditions for the irreducibility of the semigroup needed to prove that it has asynchronous exponential growth. We conclude the paper by deriving an explicit solution to a special class of growth–fragmentation problems with McKendrick–von Foerster boundary conditions and by finding its Perron eigenpair that determines its long-term behaviour.

A growth-fragmentation model related to Ornstein–Uhlenbeck type processes

Growth-fragmentation processes describe systems of particles in which each particle may grow larger or smaller, and divide into smaller ones as time proceeds. Unlike previous studies, which have focused mainly on the self-similar case, we introduce a new type of growth-fragmentation which is closely related to L\'evy driven Ornstein-Uhlenbeck type processes. Our model can be viewed as a generalization of compensated fragmentation processes introduced by Bertoin, or the stochastic counterpart of a family of growth-fragmentation equations. We establish a convergence criterion for a sequence of such growth-fragmentations. We also prove that, under certain conditions, this system fulfills a law of large numbers.

A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual.

Probabilistic aspects of critical growth-fragmentation equations

AbstractThe self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered in Doumic and Escobedo (2015) for the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a probabilistic approach based on Lévy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the nonhomogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter (0,∞) continuously from either 0 or ∞, we exhibit unexpected spontaneous generation of mass in the solutions.

Time asymptotics for a critical case in fragmentation and growth-fragmentation equations

Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to the description of the long-time asymptotics of two critical cases of these equations, when the division rate is constant and the growth rate is linear or zero. The study of these cases may be reduced to the study of the following fragmentation equation:$$\frac{\partial }{\partial t} u(t,x) + u(t,x) = \int\limits_x^\infty k_0 (\frac{x}{y}) u(t,y) dy.$$Using the Mellin transform of the equation, we determine the long-time behavior of the solutions. Our results show in particular the strong dependence of this asymptotic behavior with respect to the initial data.

Spectral analysis of semigroups and growth-fragmentation equations

The aim of this paper is twofold:(1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl's Theorems and the Krein–Rutman Theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the Dyson–Phillips series in the spirit of those used in [96,87,51,86].(2) On the other hand, we present the semigroup spectral analysis for three important classes of “growth-fragmentation” equations, namely the cell division equation, the self-similar fragmentation equation and the McKendrick–Von Foerster age structured population equation. By showing that these models lie in the class of equations for which our general semigroup analysis theory applies, we prove the exponential rate of convergence of the solutions to the associated first eigenfunction or self-similar profile for a very large and natural class of fragmentation rates. Our results generalize similar estimates obtained in [103,73] for the cell division model with (almost) constant total fragmentation rate and in [19,18] for the self-similar fragmentation equation and the cell division equation restricted to smooth and positive fragmentation rate and total fragmentation rate which does not increase more rapidly than quadratically. It also improves the convergence results without rate obtained in [84,36] which have been established under similar assumptions to those made in the present work.

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

We consider the growth-fragmentation equation and we address the problem of estimating 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 spaces, as the measurement error goes to 0. This improves the 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.

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

We are concerned with the long-time behavior of the growth-frag-mentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to $0$ and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [1], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.

Long-time asymptotics for nonlinear growth-fragmentation equations

We are interested in the long-time asymptotic behavior of growth-fragmentation equations with a nonlinear growth term. We present examples for which we can prove either the convergence to a steady state or conversely the existence of periodic solutions. Thanks the General Relative Entropy method applied to well chosen self-similar solutions, we show that the equation can "asymptotically" be reduced to a system of ODEs. Then stability results are proved by using a Lyapunov functional, and existence of periodic solutions are proved thanks to the Poincar\'e-Bendixon theorem or by Hopf bifurcation.

Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

We study the asymptotic behavior of linear evolution equations of the type ∂ t g = D g + L g − λ g , where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator D g + L g . In the case D g = − ∂ x g , this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case D g = − ∂ x ( x g ) , it is known that λ = 1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂ t f = L f . By means of entropy–entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L 2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.