126 citations · 619 across the 30 of their papers we have counts for
31 papers
Subdiffusive fractional limit of a jump-renewal equation
Hugues Berry, Pierre Gabriel, Thomas Lepoutre +1
In this paper, we consider an age-structured jump model that arises as a description of continuous time random walks with infinite mean waiting time between jumps. We prove that un…
Spectral description of a cell growth and division equation
Pierre Gabriel, Bruce van Brunt, Graeme Charles Wake +1
We give a refined description of the dominant spectrum of a non-local operator that models growth and equal mitosis of cells. More precisely we look at the spectrum in half planes…
The principal eigenvalue problem for time-periodic nonlocal equations with drift
Bertrand Cloez, Adil El Abdouni, Pierre Gabriel
In this work, we consider a general time-periodic linear transport equation with integral source term. We prove the existence of a Floquet principal eigenvalue, namely a real numbe…
Constructive Krein-Rutman result for Kinetic Fokker-Planck equations in a domain
Kleber Carrapatoso, Pierre Gabriel, Richard Medina +1
We consider a general Kinetic Fokker-Planck (KFP) equation in a domain with Maxwell reflection condition on the boundary, not necessarily with conservation of mass. We establish th…
On the Krein-Rutman theorem and beyond
Claudia Fonte Sanchez, Pierre Gabriel, Stéphane Mischler
In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results wi…
Fast, slow convergence, and concentration in the house of cards replicator-mutator model
Bertrand Cloez, Pierre Gabriel
We propose a fine analysis of the various possible long time behaviours of the solutions of the replicator-mutator equation with so-called Kingman's house of cards mutations. In pa…
Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics
Matthieu Alfaro, Pierre Gabriel, Otared Kavian
We consider nonlinear mutation selection models, known as replicator-mutator equations in evolutionary biology. They involve a nonlocal mutation kernel and a confining fitness pote…
Spectral gap for the growth-fragmentation equation via Harris's Theorem
José A. Cañizo, Pierre Gabriel, Havva Yoldaş
We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics.…
Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation
Pierre Gabriel, Hugo Martin
We are interested in a non-local partial differential equation modeling equal mitosis. We prove that the solutions present persistent asymptoticoscillations and that the convergenc…
On an irreducibility type condition for the ergodicity of nonconservative semigroups
Bertrand Cloez, Pierre Gabriel
We propose a simple criterion, inspired from the irreducible aperiodic Markov chains, to derive the exponential convergence of general positive semi-groups. When not checkable on t…
A non-conservative Harris ergodic theorem
Vincent Bansaye, Bertrand Cloez, Pierre Gabriel +1
We consider non-conservative positive semigroups and obtain necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm. This ensures t…
Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate
Etienne Bernard, Pierre Gabriel
The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted spaces and under general assumptions on the coefficients.…
Steady distribution of the incremental model for bacteria proliferation
Pierre Gabriel, Hugo Martin
We study the mathematical properties of a model of cell division structured by two variables, the size and the size increment, in the case of a linear growth rate and a self-simila…
The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states
Grégory Dumont, Pierre Gabriel
Neural network dynamics emerge from the interaction of spiking cells. One way to formulate the problem is through a theoretical framework inspired by ideas coming from statistical…
Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions
Vincent Bansaye, Bertrand Cloez, Pierre Gabriel
We provide quantitative estimates in total variation distance for positive semi-groups, which can be non-conservative and non-homogeneous. The techniques relies on a family of cons…
Measure solutions to the conservative renewal equation
Pierre Gabriel
We prove the existence and uniqueness of measure solutions to the conservative renewal equation and analyze their long time behavior. The solutions are built by using a duality app…
Limiting Hamilton-Jacobi Equation for the Large Scale Asymptotics of a Subdiffusion Jump-Renewal Equation
Vincent Calvez, Pierre Gabriel, Álvaro Mateos González
Subdiffusive motion takes place at a much slower timescale than diffusive motion. As a preliminary step to studying reaction-subdiffusion pulled fronts, we consider here the hyperb…
Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
Etienne Bernard, Marie Doumic, Pierre Gabriel
We study the asymptotic behaviour of the following linear growth-fragmentation equation$$\dfrac{\partial}{\partial t} u(t,x) + \dfrac{\partial}{\partial x} \big(x u(t,x)\big) + B(x…
Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate
Etienne Bernard, Pierre Gabriel
We are interested in the large time behavior of the solutions to the growth-fragmentation equation. We work in the space of integrable functions weighted with the principal dual ei…
Global stability for the prion equation with general incidence
Pierre Gabriel
We consider the so-called prion equation with the general incidence term introduced in [Greer et al., 2007], and we investigate the stability of the steady states. The method is ba…
Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems
Vincent Calvez, Pierre Gabriel, Stéphane Gaubert
We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmenta…
Exponential relaxation to self-similarity for the superquadratic fragmentation equation
Pierre Gabriel, Francesco Salvarani
We consider the self-similar fragmentation equation with a superquadratic fragmentation rate and provide a quantitative estimate of the spectral gap.
Optimization of an Amplification Protocol for Misfolded Proteins by using Relaxed Control
Jean-Michel Coron, Pierre Gabriel, Peipei Shang
We investigate an optimal control problem which arises in the optimization of an amplification technique for misfolded proteins. The improvement of this technique may play a role i…
Optimal growth for linear processes with affine control
Vincent Calvez, Pierre Gabriel
We analyse an optimal control with the following features: the dynamical system is linear, and the dependence upon the control parameter is affine. More precisely we consider $\dot…