The Attractor And The Approximate Inertial Manifold For The (2+1)-dimensional Non-autonomous Long-short Wave Equations

In this paper, a number of basic theories of infinite-dimensional dynamical systemand the long time behavior of (2+1)-dimensional non-autonomous long-short waveequations were studied. We prove the existence of global smooth solution for thegeneralized (2+1)-dimensional dissipative long-short wave equations, the existence ofthe strongly compact uniform attractor for the (2+1)-dimensional non-autonomouslong-short wave equations, and an approximate inertial manifold for the(2+1)-dimensional non-autonomous long-short wave equations was constructed.The paper is organized in four chapters.Chapter1, we reviewed some basic concepts of infinite-dimensional dynamicsystems and distinguished between autonomous systems and non-autonomous ones.some basic theories about attractor and an approximate inertial manifold werereviewed. The backgrounds in which long-short wave equations arise were statedbriefly and the main purposes for this research were presented in this chapter as well.Chapter2, a kind of generalized (2+1)-dimensional non-autonomous long-shortwave equations were researched and we obtained the existence of global smoothsolution for the inertial boundary value problem by the uniform a prior estimates andthe Galekin method.Chapter3, the (2+1)-dimensional non-autonomous long-short wave equations withtranslation compact forces were studied and the existence of the strongly compactuniform attractor were obtained. In the first section, using the uniform a prior estimatesand the Galekin method we got the the unique existence of the solution; In the secondsection, first we proved The existence of uniform absorbing sets and the weaklycompact uniform attractor, then we got that the above weakly compact attractor wasactually the strongly one by the weak convergence method.Chapter4, on the basis of the uniform a prior estimates which used in the(2+1)-dimensional non-autonomous long-short wave equations of the third chapter, weconstructed the approximate inertial manifold by means of extending phase space andoperator projections methods.