Mathematical Sciences: Solutions for Functional DifferentialEquations

数学科学：泛函微分方程的解

• 批准号: 9401823
• 负责人: Roger Nussbaum
• 金额: $ 7.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401823&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Solutions; Functional; DifferentialEquations

9401823 Nussbaum Many natural phenomena, for example, in biology, seem best described by nonlinear "functional differential equations" or "FDE's". Roughly speaking, FDE's are equations in which an unknown function of time t, x(t), appears and x'(t), the instantaneous rate of change of x(t) with time, depends in a specified way not only on x(t) but also on the past history of the function x. For example,in some models the rate of increase of a population of a class of mature red blood cells at time t may well depend on population levels of those same mature cells six to ten days earlier. A rigorous mathematical theory of of nonlinear FDE's poses serious challenges and various equations of interest in applications have been neglected. We propose to study some classes of examples which were, until quite recently, considered intractable, but for which it now seems possible to obtain a wide variety of surprisingly detailed theorems. The starting point of this proposal is the study of the equation (*), ax'(t) = f(x(t),x(t-r)), r = r(x(t)), where f and r are given functions and a0. Equation (*) and generalizations of equation (*) arise in a variety of applications. In joint work with John Mallet-Paret, the author has shown that, under natural assumptions on f and r and for all sufficiently small a, equation (*) has nonconstant periodic solutions. These periodic solutions often seem to have strong global stability properties. Furthermore, in many cases it has proved possible to determine the limiting profile of shape of such periodic solutions as a approaches zero. In this proposal we consider many questions about equation (*), and we discuss possible extensions of results for (*) to much more general classes of equations.

9401823 Nussbaum许多自然现象，例如生物学中的许多自然现象，似乎都可以用非线性“泛函微分方程”或“泛函微分方程”来描述。粗略地说，FDE是这样的方程，其中出现未知的时间t的函数x(T)，以及x(T)随时间的瞬时变化率x‘(T)不仅取决于x(T)，而且取决于函数x的过去历史。例如，在某些模型中，某类成熟红细胞在时间t的增长率很可能取决于6至10天前那些相同的成熟细胞的种群水平。严格的非线性偏微分方程组的数学理论提出了严重的挑战，各种有趣的方程在应用中被忽视了。我们建议研究一些直到最近还被认为是难以解决的例子，但现在似乎有可能获得各种各样令人惊讶的详细定理。这个建议的出发点是研究方程(*)，ax‘(T)=f(x(T)，x(t-r))，r=r(x(T))，其中f和r是给定的函数和a0。方程式(*)和方程式(*)的推广在各种应用中出现。在与John Mallet-Paret的合作中，作者证明了，在关于f和r的自然假设下，对所有足够小的a，方程(*)有非常数周期解。这些周期解往往具有很强的全局稳定性。此外，在许多情况下，已经证明可以确定当a趋近于零时这种周期解的形状的极限轮廓。在这个建议中，我们考虑了许多关于方程(*)的问题，并讨论了(*)的结果可能推广到更一般的方程类。