Time-periodic solutions with internal and boundary layers to singularly perturbed parabolic problems: Existence, approximation and domain of attraction

本项目研究一维奇异摄动时间周期反应扩散对流方程边值问题，我们将发展具有内层和边界层的上下解序列的解析构造算法.通常，内部层的位置将随时间周期性地移动。下解和上解将用于证明具有内部和边界层的精确解的存在性，用于它们的一致逼近，用于验证它们的稳定性并用于估计它们的吸引域。此外，上下解对于创建分层时间周期解的数值算法也是有用的。主要的技术工具是形式渐近展开和使用拉伸变量接近层。渐近展开式中的项以及创建上下解的附加修改项可以通过求解半轴上的线性非齐次时间周期抛物问题来计算。操作员在半轴上。逆线性抛物型偏微分算子必须是保序的，并且它们应该将指数衰减函数映射成指数衰减函数。为了证明稳定性，我们使用Krein-Rutman定理。

The project concerns boundary value problems for singularly perturbed time-periodic reaction-diffusion-advection equations in 1D.We will develop algorithms of analytic construction of sequences of lower and upper solutions with internal and boundary layers. The positions of the internal layers will periodically move in time, in general. The lower and upper solutions will be used for proving existence of exact solutions with internal and boundary layers, for uniform approximation of them, for verifying their stability and for estimation of their domain of attraction. Moreover, the lower and upper solutions are useful also for creating numerical algorithms for layered time-periodic solutions. The main technical tools are formal asymptotic expansions and using of stretched variables close to the layers. The terms in the asymptotic expansions as well as the additional, modifying terms, which create lower and upper solutions, can be calculated by solving linear inhomogeneous time-periodic parabolic problems on the half axis. operators on the half axis. The inverted linear parabolic partial differential operators have to be order preserving, and they should map exponentially decaying functions into exponentially decaying functions. For proving stability we use the Krein-Rutman Theorem.