Global Attractors For Nonlinear Beam Equations Of Memory Alloys With Force

Two class of nonlinear partial differential equations arising from the study of shape memory alloys with force are studied in this paper, one of which contains the viscosity term, and the other contains the low order damping term. We can establish the priori estimates for the solution of both equations, then global exis-tence and uniqueness follows from the continuation argument and compact theory. Meanwhile, by using the classical energy estimate method and the maximal attrac-tor theory, the existence of global attractors of initial and boundary value problem for both equations are proved.The paper is composed of four chapters.In the first chapter, the concept of shape memory alloys and related results concerning to the equations of shape memory alloys are presented simply. Mean-while, the main works of this paper are simply introduced.The related concepts, notations and inequalities which will be used in this paper are presented in the second chapter.In the third chapter, with initial and boundary conditions, the existence and uniqueness of the global solutions to the equation with force and viscosity are proved by using continuation and compact theory in the first section. In the second section, by using the embedding theory, we can prove that the operator of the solution is uniformly compact, then by using the classical energy method, the existence of the absorbing set is proved, finally, the existence of global attractors of this problem follow from the maximal attractor theory.In the forth Chapter, with initial and boundary conditions, the existence and uniqueness of the global solutions to the equation with force and low order damping term are considered by using continuation and compact theory in the first section. In the second section, by using the classical energy method, the existence of the absorbing set is proved, then by using the embedding theory, we can prove that the operator of the solution is uniformly compact, finally, the existence of global attractors of this problem follow from the maximal attractor theory.