John Appleby (Dublin City University, Dublin, Ireland)

报告1

报告题目: Characterisation of explosion and non-explosion in autonomous functional differential equations

报告摘要: In this talk, we describe recent results which characterise whether solutions of autonomous scalar functional differential equations can be continued for all time, or have a finite interval of existence. We consider two classes of equations: Volterra differential equations without a damping term, and finite distributed delay functional differential equations which are instantaneously damped. In the case of the Volterra equation, when the kernel is positive close to zero, we give necessary and sufficient conditions for explosion or growth to infinity which depend on the nonlinearity. For the finite memory FDEs, we again give necessary and sufficient conditions on the nonlinearity in the integral and the damping term which give continuable or non-continuable solutions, or bounded or unbounded solutions. The results in the talk can be adapted considerable to capture precise asymptotic rates of growth in many instances. Most results are obtained by carefully constructing upper or lower solutions to the given functional equation.  

报告时间：2018年7月3日(星期二)上午9:00-10:00

报告地点：科技楼602

报告2.

报告题目: Preservation and suppression of explosive and non-explosive growth in solutions of general functional differential equations by noise perturbations

报告摘要: In this talk, we add to the rapidly growing literature on the role of noise to control or preserve dynamics in an underlying functional differential equation. We concentrate on very general scalar functional equations which have positive and monotone solutions in the absence of noise perturbation, and whose solutions are guaranteed to grow to infinity, or to explode in finite time. We then perturb the equation by a stochastic term that can be split into a purely instantaneous term (which we make linear close to zero, and a square root for large arguments) and a functional term $g$, resulting in an SFDE with strictly positive solutions on its maximal interval of existence. Our results have the following character: if the drift functional is $f$, and $\lambda:=f/g^2$ is a functional always bounded below by 1/2, then the solutions of the SFDE still grow to infinity. If however, $\lambda$ is always bounded above by 1/2, the solutions are either recurrent on their interval of existence, or tend to zero, so the growth of the original equation is broken. We also show, in the small noise case, that the type of growth of solution (explosive or non-explosive) is generally preserved, in the sense that if certain deterministic perturbations of the original FDE have a given growth property, that property is also enjoyed by solutions of the SFDE. The results employ comparison and random time change arguments, as well as careful study of the asymptotic behaviour of Bessel-type processes.    

报告时间：2018年7月4日(星期三)上午9:00-10:00

报告地点：科技楼602