非线性扩散方程解的复杂渐近行为

本文是一篇综述,简要介绍非线性扩散方程解渐近行为的研究结果与研究方法.以非线性扩散方程的Cauchy问题为主线,本文主要阐述研究其解复杂渐近行为的方法与结论.     

时变参数动力系统的两种跟踪性质

在时变参数动力系统中引入链传递,伪轨跟踪以及渐进伪轨跟踪的概念,并通过这些概念讨论时变参数动力系统的伪轨跟踪和渐进伪轨跟踪的性质.证明了扩张的时变参数动力系统满足伪轨跟踪性质蕴含其满足渐近伪轨跟踪性质;论证了时变参数动力系统的积系统满足伪轨跟踪和渐进伪轨跟踪性质的充要条件是其每一个分系统也满足相应的性质.最后构造出了一个时变参数动力系统的例子:(∑∞X,F),证明了(∑∞X,F)是拓扑传递的,并且满足渐近伪轨跟踪性质.     

Complexity of asymptotic behavior of solutions for the porous medium equation with absorption

In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption u t-u m + γup = 0, where γ≥ 0, m 1 and p m + 2/N . We will show that if γ = 0 and 0 μ 2N/(N(m-1)+2), or γ 0 and 1/(p-1)μ2N/(N(m-1)+2), then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S (R N ), there exists an initial-value u0 ∈C(RN) with lim x →∞ u 0 (x) = 0 such that φ is an ω-limit point of the rescaled solutions t μ/2 u(t β·, t), where β =[2-μ(m-1)]/4 .     

Experimental realization of two-dimensional single-layer ultracold gases of 87Rb in an accordion lattice

We experimentally realize two-dimensional (2D) single-layer ultracold gases of ⁸⁷Rb by dynamically tuning the periodicity of a standing wave, known as accordion lattice. In order to load ⁸⁷Rb Bose—Einstein condensate into single dark fringe node of the blue detuning optical lattice, we reduce the lattice periodicity from 26.7 μ to 3.5 μ with the help of an acousto-optic deflector (AOD) to compress the three-dimensional BEC adiabatically into a flat and uniform quasi-2D single-layer. We describe the experimental procedure of the atoms loading into the accordion lattice in detail and present the characteristics of the quasi-2D ultracold gases. This setup provides an important platform for studying in- and out-of equilibrium physics, phase transition and 2D topological matter.     

Banach空间中弱收敛序列系数的估计

利用.Jordan—von Neumann型常数C_t~′(X),C_(-∞)(X)和弱正交系数μ(X)对Banach空间中的弱收敛序列系数WCS(X)进行了估计,从而得到空间具有正规结构的充分条件,这些结论推广了最近一些文献中的结果.同时,还计算了Bynum空间l_(2.∞)。中上述常数的取值,来说明我们给定的条件是一个严格的推广.