Continuity-sets of pullback random attractors for discrete porous media equations with colored noise

Abstract

For a random non-autonomous porous media lattice system driven by nonlinear colored noise, we prove the unique existence and local compactness of a pullback random attractor. We then mainly study the continuity-set (the set of all points of continuity) of the pullback random attractor on the time-sample plane with respect to the Hausdorff metric. With some calculations, we find that the continuity-set has four geometrical numerical features:(1) The continuity-set is residual and dense in the time-sample plane.(2) The continuity-set is composed of infinitely many diagonal rays (with slope one).(3) The continuity-set is dense in any non-diagonal line (with an inclined angle not being 45∘).(4) Restrictions of the continuity-set on diamonds are composed of parallel line segments with increasing numbers.