Exponential Attractors For The Structurally Damped Wave Equations

In this paper,we are concerned with the long-time dynamical behavior of the following wave equation with structural damping and supercritical nonlinearity:Where Q is a bounded domain in Rn with the smooth boundary(?)?,?>0,??[1/2,1].It is well known that the long-time behavior of dissipative dynamical systems generated by mathematical physics can be described in terms of the so called Global attractor.An exponential attractor,in contrast to a global attractor,enjoys a uniform exponential rate of convergence of its solutions once the solution is inside an invariant absorbing set.Because of this,exponential attractors possess a deeper and more practical property,and they remain more robust under perturbations and numerical approximations than global attractors.Obtaining certain asymptotic regularity is important and helpful for further understanding the properties of attractors.The structural damping(-?)?ut plays a dissipative role,which is stronger than the weak damping but weaker than the strong damping Recently,A.Savostianov,S.Zelik,has proved the existence of exponential attractor for the above equation in the case:the nonlinearity f=f(u).In this paper,we investigate the case:the nonlinearity f depends on ut.Firstly weprove that the existence of exponential attractor for the above equation in the natural energy space H01(?)× L2(?).Secondly we will show that:if we shift A with a(proper)fixed point ?(x),then A-?(x)will be bounded in a stronger space H1+?(?)×H?(?).Finally,when ?=1,we prove the optimal regularity A ? H01(?)× H01(?).