Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion
Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion
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DOI：
10.1137/0520037
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发表时间：
1989-05
影响因子：
2
通讯作者：
Margaret C. Memory
中科院分区：
文献类型：
--
作者：
Margaret C. Memory
考虑一维空间中具有扩散项的标量时滞微分方程，其中扩散系数D为分岔参数。利用中心流形理论和Lyapunov-Schmidt方法描述了空间常数解随着D的减小而产生的两个分支。通过修改方程，可以反转这些分岔的顺序。然后证明了该类方程的紧吸引子的存在性，并研究了修正方程的部分吸引子的结构。已知解是全局$L^2 $有界的；构造了从一个中间空间到另一个中间空间的解算子的界，以得到$W^{2,2} $意义上的吸引子。
A scalar delay-differential equation with diffusion term in one space dimension, where the diffusivity D is a bifurcation parameter, is considered. The center manifold theory and the method of Lyapunov–Schmidt are used to describe two bifurcations from spatially constant solutions as D decreases. By modifying the equation the order of these bifurcations can be reversed. Then the existence of a compact attractor for a class of such equations is shown and the structure of part of the attractor for the modified equation is investigated. It is known that the solutions are globally $L^2 $-bounded; bounds on the solution operator from one intermediate space to another are constructed to obtain an attractor in the $W^{2,2} $ sense.