An attractor for the singularly perturbed Kirchhoff equation with supercritical nonlinearity

•We prove the existence of a global attractor on an appropriate phase space.•We do not make any growth restrictions on the nonlinearity f(u)f(u).•The smallness assumption for the initial data can be replaced by the smallness of ϵ.

We consider a quasilinear wave equation of Kirchhoff typeϵutt−(1+‖∇u‖2)Δu+ut+f(u)=g(x),ϵutt−(1+‖∇u‖2)Δu+ut+f(u)=g(x), where ϵ>0ϵ>0 is a small parameter. Without any growth restrictions on the nonlinearity f(u)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.