Time-dependent attractor for the oscillon equation

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation ∂ttu(x, t)+H∂tu(x, t)-e-2Ht∂xxu(x, t)+V '(u(x, t)) = 0, (x, t) ε (0, 1) × ℝ, with periodic boundary conditions, where H > 0 is the Hubble constant and V is a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular global attractor A = A(t). The kernel sections A(t) have finite fractal dimension.