Korteweg-de Vries(KdV)方程是描述浅水波现象的数学模型，在偏微分方程领域占有重要地位。本项目研究全空间中或带周期边界条件下部分耗散KdV方程的动力学行为，具体内容包括：系统的整体吸引子存在性、正则性、分形维数估计，以及耗散区域的大小、分布与结构带来的影响。本项目的最大特色是仅假设部分区域上具有耗散机制，允许某些区域没有耗散。因而在考虑系统的吸收集、紧性以及分形维数时都会遇到新的困难。我们首先建立带变系数低价项的线性KdV方程的各种定量唯一延拓性，由此导出部分耗散KdV方程的能观能控型不等式，得到不同区域上的系统能量之间的联系，进而证明有界吸收集的存在性；其次利用Goubet高低频分解证明系统的渐近正则性，结合尾端估计说明解半群的渐近紧性，从而得到整体吸引子的存在性；最后，通过建立Chueshov-Lasiecka拟稳定估计得到吸引子的有限分形维数。

Korteweg-de Vries (KdV) equation is a mathematical model of waves on shallow water surfaces. It plays an important role in the theory of partial differential equations. In this study, we are devoted to the long-term dynamics of KdV equations with localized damping on the whole space or bounded domain with periodic boundary condition. We shall discuss several issues, including the existence, regularity, finite fractal dimension of global attractor and the impact of size, distribution and structure of damping area. The main novelty of this study lies in that the dissipation region is a strict subset of the domain. Compared with previous works, there are new challenges in proving the existence of an absorbing set, asymptotic compactness and finite dimensionality. Firstly, we shall investigate several kinds of quantitative unique continuation of linear KdV equation with lower order terms, and then deduce the observability type estimates to find the connection of the energy on different regions. As a consequence, we obtain the existence of a bounded absorbing set. Secondly, we prove the asymptotic compactness of solution semigroup and then the existence of a global attractor by tailed estimates and an asymptotic regularity, which follows from Goubet’s high-low frequency decomposition. Finally, we establish the finite dimensionality by showing that the solution semigroup satisfies Chueshov-Lasiecka’s quasi-stable estimates.