The Existence Of Uniform Attractors For Infinite Dimensional Non-autonomous Dynamical Systems

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_b³((?); X), and L_c²((?); X) is a proper subspace of L_c*²((?);X); L_c*²((?);X), L_c²((?);X) and L_n²((?);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_n²((?);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).