María J. Cáceres

Long-time behavior for a nonlinear fourth-order parabolic equation

We study the asymptotic behavior of solutions of the initial-boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.

Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles

Abstract We study the long-time behavior of a linear inhomogeneous Boltzmann equation. The collision operator is modeled by a simple relaxation towards the Maxwellian distribution with zero mean and fixed lattice temperature. Particles are moving under the action of an external potential that confines particles, i.e., there exists a unique stationary probability density. Convergence rate towards global equilibrium is explicitly measured based on the entropy dissipation method and apriori time independent estimates on the solutions. We are able to prove that this convergence is faster than any algebraic time function, but we cannot achieve exponential convergence.

Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states

Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.AMS Subject Classification:35K60, 82C31, 92B20.

Nonlinear Stability in Lp for a Confined System of Charged Particles

We prove the nonlinear stability in Lp, with

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

1\le p\le 2

A deterministic solver for a hybrid quantum-classical transport model in nanoMOSFETs

, of particular steady solutions of the Vlasov--Poisson system for charged particles in the whole space R6 . Our main tool is a functional associated to the relative entropy or Casimir-energy functional.

A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics

Abstract 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.

Deterministic kinetic solvers for charged particle transport in semiconductor devices

We model a nanoMOSFET by a mesoscopic, time-dependent, coupled quantum-classical system based on a sub-band decomposition and a simple scattering operator. We first compute the sub-band decomposition and electrostatic force field described by a Schrodinger-Poisson coupled system solved by a Newton-Raphson iteration using the eigenvalue/eigenfunction decomposition. The transport in the classical direction for each sub-band modeled by semiclassical Boltzmann-type equations is solved by conservative semi-lagrangian characteristic-based methods. Numerical results are shown for both the thermodynamical equilibrium and time-dependent simulations in typical nowadays nanoMOSFETs.

Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activity

To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in 15,14], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in 15] to further simplify the kinetic theory.

Rate of convergence to self-similarity for the fragmentation equation in L1 spaces

Statistical models [F91], [L00], [MRS90], [To93] are used to describe electron transport in semiconductors at a mesoscopic level. The basic model is given by the Boltzmann transport equation (BTE) for semiconductors in the semiclassical approximation: