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Time-dependent attractor for the Oscillon equation

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  • 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.

    Mathematics Subject Classification: Primary: 37L30, 35B41; Secondary: 83D05.


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