摘要
在这篇博士学位论文中，我们主要考虑在更一般的非自治外力项作用下的无穷维动力系统的一致吸引子的存在性，这种更一般的非自治外力是指系统带有更一般的符号函数。
为了考虑[57]中作者提出的问题：“对于弱耗散系统，如果符号空间不紧，一致吸引子是否可能存在?”，并希望在符号空间更一般的情况下也能得到所讨论的几类非自治系统一致吸引子的存在性，我们引进了两类在通常拓扑下非紧的函数：满足条件(C~*)的函数类L_(c*)~2(R；X)和满足正规条件(C~*)的函数类L_(nc*)~2(R；X)，并讨论了所提出的两类新的函数的性质。同时给出了这两类函数与[22]中的平移紧函数和[58]中的正规函数之间的关系：证明了L_(c*)~2(R；X)是平移有界函数空间的闭子空间，且平移紧函数类是L_(c*)~2(R；X)的真子集(很容易构造例子f_0∈L_(c*)~2(R；X)但不是平移紧的)；证明了满足条件(C~*)的函数类L_(c*)~2(R；X)和平移紧函数类L_c~2(R；X)以及正规函数L_n~2(R；X)都满足正规条件(C~*)，即都是空间L_(nc*)~2(R；X)的子空间。进一步，我们构造例子说明存在函数f_0∈L_(nc*)~2(R；X)，但f_0(?)L_n~2(R；X)。
继而，我们用具体的例子说明了我们所研究的几类非自治系统在外力项所在的空间更一般的情况下，系统一致吸引子的存在性。首先，针对非自治弱耗散双曲方程，分别给出了带有次临界和临界非线性项的非自治双曲方程在外力项仅满足条件(C~*)的情况下其一致吸引子的存在性。从而部分地解决了[57]中作者所提出的问题，即，在符号空间不紧时，给出了弱耗散系统一致吸引子的存在性。其次，以2D Navier-Stokes方程为例证明了外力项在L_(nc*)~2(R；X)中时的一致吸引子的存在性，这也说明了正规函数类并非是强耗散系统最一般的符号空间。最后，对于非经典扩散方程，由于方程自身良好的性质，甚至在符号空间仅是平移有界函数L_b~2(R；X)(这是得到系统一致吸收集的必要条件)时，我们也证明了此类方程所对应的过程族的紧的一致吸引子的存在性。
In this doctoral dissertation, we mainly consider the existence of uniform attractors for non-autonomous infinite dimensional dynamical system when the external forces are translation non-compact.
In order to consider the problem in [57]:" Are there uniform attractors for the weakly dissipative systems when the symbol spaces are non-compact?", we introduce two classes of functions which are translation non-compact, denoted by L_(c*)~2((?); X) and L_(nc*)~2((?); X) respectively. We discuss their properties and give their relations with translation compact and normal functions. We show that L_(c*)~2((?); X) is a closed subspace of L_b~3((?); X), and L_c~2((?); X) is a proper subspace of L_(c*)~2((?);X); L_(c*)~2((?);X), L_c~2((?);X) and L_n~2((?);X) satisfy condition (C~*) and they are indeed proper subspaces of L_(nc*)~2((?); X). Moreover, we also show that there is a function in L_(nc*)~2((?);X) and not in L_n~2((?);X).
As applications, we show by some concrete examples that we can obtain the existence of uniform attractors for several classes of non-autonomous systems even when the symbol space is more general. More precisely, we firstly discuss that the existence of uniform attractors for non-autonomous hyperbolic equations, with subcritical and critical nonlinearity, when the external forces just satisfy the condition (C~*). This result solves partly the problem mentioned above, that is, we obtain the existence of uniform attractors for weakly dissipative system with non-compact external forces. Secondly, we obtain the existence of uniform attractors for 2-D Navier-Stokes equations with external forces only in L_(nc*)~2((?); X), which improves the previous results (e.g., [57]). Finally, using the fine properties of the equation, we prove the existence of compactly uniform attractors for the non-classical diffusion equation even for the translation bounded external force (note that the translation boundedness of external force is necessary to obtain a uniform absorbing set).
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