Time-dependent attractor for the Oscillon equation

We investigate the asymptotic behavior of the nonautonomous evolution problem generated bythe Oscillon equation ∂ tt $u(x,t) +H $ ∂ tu(x,t) --2Ht ∂ xx u(x,t) +V'(u(x,t)) =0, (x,t)∈ (0,1) × ,with periodic boundary conditions, where $H>0$ is the Hubbleconstant and V is a nonlinear potential of arbitrary polynomialgrowth. After constructing a suitable dynamical framework to dealwith the explicit time dependence of the energy of the solution,we establish the existence of a regular global attractor=(t). The kernel sections (t) have finite fractal dimension.