Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on RN

The paper studies the longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on RN:utt−Δut−M(‖∇u‖2)Δu+ut+g(x,u)=f(x). We establish the well-posedness, the existence of the global and exponential attractors in natural energy space H=H1(RN)×L2(RN) in critical nonlinearity case. These results improve not only the recent ones achieved by Yang (2007) [32], but also ones achieved by Conti, Pata and Squassina (2005) [9] to some extent.