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

The paper studies the existence of global strong attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity u t t − σ ( ‖ ∇ u ‖ 2 ) Δ u t − ϕ ( ‖ ∇ u ‖ 2 ) Δ u + f ( u ) = h ( x ) . It proves that in strictly positive stiffness factors and supercritical nonlinearity case, there exists a global finite-dimensional attractor in the natural energy space endowed with strong topology (rather than partially strong topology). The result extends the recent one achieved by Chueshov (2012).