We consider a sixth-order integro-differential equation with periodic boundary conditions. The equation describes the phase transition dynamics in ternary oil–water–surfactant systems. Using the energy methods and the basic dissipative estimate, we establish the required estimates. Furthermore, based on these estimates, we prove the existence of the attractors by a decomposition of the solution semigroup.

中文翻译：

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六阶积分微分方程的渐近动力学

我们考虑具有周期性边界条件的六阶积分微分方程。该方程描述了三元油-水-表面活性剂系统中的相变动力学。使用能量方法和基本耗散估计，我们建立了所需的估计。此外，基于这些估计，我们通过解半群的分解证明了吸引子的存在。