Time-dependent attractor for the Oscillon equation

Abstract

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation ∂ _tt $u(x,t) +H $ ∂ _t$ u(x,t) -\e^{-2Ht}$ ∂ _xx $ u(x,t) + V'(u(x,t)) =0, \quad (x,t)\in (0,1) \times \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.