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.