Exponential attractor for the Kirchhoff equations with strong nonlinear damping and supercritical nonlinearity

We study the existence of exponential attractor for the Kirchhoff equations with strong nonlinear damping and supercritical nonlinearity utt−σ(‖∇u‖2)Δut−ϕ(‖∇u‖2)Δu+f(u)=h(x)utt−σ(‖∇u‖2)Δut−ϕ(‖∇u‖2)Δu+f(u)=h(x). The main result is that the nonlinearity f(u)f(u) is of supercritical growth. In this case we establish an exponential attractor in natural energy space by using a new method based on the weak quasi-stability estimates (rather than the strong one as usual).