Time-dependent attractor for the oscillon equation. (English) Zbl 1223.37100

Di Plinio, Francesco; Duane, Gregory S.; Temam, Roger

Discrete Contin. Dyn. Syst. 29, No. 1, 141-167 (2011).

The authors investigate the asymptotic behavior of the nonautonomous evolution problem generated by the oscillon equation
\[ \partial u(x, t) + H\partial_t u(x,t) - e^{-2Ht} \partial_{xx} u(x,t) + V'(u(x,t)) = 0, \quad (x,t) \in (0,1) \times\mathbb R, \]
with periodic boundary conditions, where \(H > 0\) is the Hubble constant and \(V\) is a nonlinear potential of arbitrary polynomial growth. They construct a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution. The existence of a regular global attractor \(A = A(t)\) is established. The kernel sections \(A(t)\) have finite fractal dimension.

Reviewer: I. V. Boikov (Penza)

MSC:

37L30	Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41	Attractors
83D05	Relativistic gravitational theories other than Einstein’s, including asymmetric field theories

Keywords:

oscillon equation; nonautonomous attractors; fractal dimension