Abstract

In this work, we investigate the problem of constructing asymptotic representations for weak solutions of a certain class of linear differential equations in the Banach space as an independent variable tends to infinity. We consider the class of equations that represent a perturbation of a linear autonomous equation, in general, with an unbounded operator. The perturbation takes the form of a family of bounded operators that, in a sense, oscillatorally decreases at infinity. It is assumed that the unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence of a center-like manifold (a critical manifold) for the initial equation. This manifold is positively invariant with respect to the initial equation and attracts all trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional system of ordinary differential equations. The asymptotics of the fundamental matrix of this system can be constructed by using the method developed by P.N. Nesterov for asymptotic integration of systems with oscillatory decreasing coefficients. We illustrate the proposed technique by constructing the asymptotic representations for solutions of the perturbed heat equation.

中文翻译：

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Banach空间中某些微分方程的渐近积分

摘要

在这项工作中，我们研究由于自变量趋于无穷大而在Banach空间中构造一类线性微分方程的弱解的渐近表示的问题。我们考虑通常用无界算子表示线性自治方程组摄动的方程组。摄动采取有界算子族的形式，从某种意义上说，其在无穷大处振荡减小。假定无扰动方程满足中心流形理论的标准要求。所提出的渐近积分方法的本质是证明初始方程存在中心样流形（临界流形）。该流形相对于初始方程为正不变的，并且吸引了弱解的所有轨迹。临界流形上初始方程的动力学由常微分方程的有限维系统描述。可以使用PN Nesterov开发的方法构造该系统基本矩阵的渐近性，该方法用于渐进积分具有振荡递减系数的系统。我们通过构造摄动热方程解的渐近表示来说明所提出的技术。可以使用PN Nesterov开发的方法构造该系统基本矩阵的渐近性，该方法用于渐进积分具有振荡递减系数的系统。我们通过构造摄动热方程解的渐近表示来说明所提出的技术。可以使用PN Nesterov开发的方法构造该系统基本矩阵的渐近性，该方法用于渐进积分具有振荡递减系数的系统。我们通过构造摄动热方程解的渐近表示来说明所提出的技术。