An attractor for the singularly perturbed Kirchhoff equation with supercritical nonlinearity

We consider a quasilinear wave equation of Kirchhoff type ϵ u t t − ( 1 + ‖ ∇ u ‖ 2 ) Δ u + u t + f ( u ) = g ( x ) , where ϵ > 0 is a small parameter. Without any growth restrictions on the nonlinearity f ( u ) , we prove the existence of a finite-dimensional global attractor on an appropriate (bounded) phase space. The key step is the estimate of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation.