Backward regularity of attractors for lattice FitzHugh-Nagumo system with double random coefficients

摘要

In this paper, we investigate the backward compact dynamical behavior for the stochastic lattice FitzHugh-Nagumo system with double random coefficients and multiplicative noise. It is proved, under some suitable conditions on the nonlinear term and the body force, that the system has a unique backward compact bi-spatial random (ℓσ2×ℓσ2,ℓσp×ℓσ2)-attractor for any p>2. By such a bi-spatial attractor we mean an invariant backward compact set in ℓσp×ℓσ2 that pullback attracts all nonempty subsets of ℓσ2×ℓσ2 under another topology of ℓσp×ℓσ2. The time-dependent uniform pullback asymptotical compactness of the solution operators in ℓσp×ℓσ2 is proved by virtue of a cut-off technique and considering two different attracting universes to overcome several difficulties caused by the lack of compact Sobolev embeddings in the case of a lattice system and the unknown measurability of random attractor.