Asymptotic behaviour of some nonlocal equations in mathematical biology and kinetic theory
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We study the long-time behaviour of solutions to some partial differential equations arising in modeling of several biological and physical phenomena. In this work, the type of the equations we consider is mainly nonlocal, in the sense that they involve integral operators. %In order to check if the equation is valid at one point, we need an information about the region far from that point. Moreover, the equations we consider describe the time evolution of either some populations structured by several traits like age, elapsed-time and size or the distribution of the dynamical states of a single particle, depending on time, space and velocity. In the latter case, they are called kinetic equations. We are interested in showing quantitatively the asynchronous behaviour of interacting neuron populations which are composed of large and fully connected networks. Neurons undergo a charging period followed by a sudden discharge in the form of firing a spike. We consider two nonlinear models structured by the time elapsed since the last discharge and nonlinearity comes from the dependence of firing rate on the total neural activity at a time. In the second model, there is an addition of a fragmentation term to include the effect of the past activity of neurons by displaying adaptation and fatigue. With this addition, the equation shares many common properties with another class of integro-partial differential equations called the growth-fragmentation equation. This is the second type of equation we look at the convergence rate to a universal profile in a quantitative way. The growth-fragmentation equation describes a system of growing and dividing particles which may be used as a model for many processes in ecology, neuroscience, telecommunications and cell biology. We consider two types of fragmentation processes, namely mitosis and constant fragmentation and include nonconservative cases where eigenelements cannot be computed explicitly. We present quantitative exponential convergence speeds in the weighted total variation norm. Furthermore, we also study hypocoercivity of some space inhomogeneous linear kinetic equations including linear relaxation Boltzmann(linear BGK) and linear Boltzmann equations which are posed either on the torus or on the whole space with a confining potential. We prove exponential convergence in the torus or on the whole space with a potential growing quadratically at infinity. Moreover, for the weaker confining potentials (subquadratic) we present subgeometric convergence rates quantitatively. The physiologically structured population models and the space inhomogeneous linear kinetic equations we deal with in this work are well-studied from various aspects in the already-existing literature. We provide the references later. What differs from the past plentiful studies on the asymptotic behaviour of these equations is the techniques we use here. We consider a probabilistic approach which is first developed for studying ergodic properties of discrete-time Markov processes. The method is due to Doeblin and Harris; based on establishing a combination of a minorisation (irreducibility) and a geometric drift (Lyapunov) conditions for a Markovian process. This method gives a quantitative convergence speed and existence of a unique steady state even without having to calculate it explicitly. Application of Harris's Theorem into the aforementioned partial differential equations to study the long-time behaviour of solutions is the core of this thesis.