The Long-time Behavior For A Class Of Nonlinear Evolution Equations

In this paper, we have studied the long time behaviors of a class of nonlinear evolution equations. By proving the asymptotic regularity of the global weak solution for the problem, We have established the existence of global attractors for the following equations (?)utt-Δut-Δu-μΔutt+f(u)=g(x) R3×R+ (∏) (?)u|t=0=u0, ut|t=0=u1 R3×R+ Where f∈C1(R,R) is the external forces which satisfies the condition of critical Sobolev exponential growth, and g∈L2(R3) is gived.We have established a general method for verifying the the existence and regularity of global attractors, where the solution for the problem doesn't exist the higher regularity and the domain is unbounded.In this paper,we have gotten the existence and uniqueness of the global weak solution for the problem (∏), by using the Galerkin approximation method and the energy estimate.Then,by the use of the asymptotical smoothness methods like bounded domains, we decomposed u=v+w, proved that one's norm is arbitrarily small, then the other one possess the higher regularity in H1(R3)xH1(R3) when t is big enough. So outside of B(0, k), the H1(R3)×H1(R3) norm of the solution of (II) is arbitrarily small when k is big enough. Then D(As)→H1(B),(s>1/2) is compactness and we proved the existence of the global attractor.