Statistical solutions for a nonautonomous modified Swift–Hohenberg equation

Abstract

We consider the nonautonomous modified Swift-Hohenberg equationon a bounded smooth domain Ω ⊂ R n with n ⩽ 3. We show that, if |b|<4 and the external force g satisfies some appropriate assumption, then the associated process has a unique pullback attractor in the Sobolev space H 2 0 (Ω). Based on this existence, we further prove the existence of a family of invariant Borel probability measures and a statistical solution for this equation.

Invariant sample measures and random Liouville type theorem for a nonautonomous stochastic <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian equation

“…Random Liouville type theorem. We now prove that the invariant sample measures {µ τ,ω } τ ∈R,ω∈Ω for φ satisfy a random Liouville type theorem (see more in [25,28,33,34]). Rewrite (1) as du = H(t, u, ω) = [∆ p u + f (x, t, u)]dt + αu • dW, where H : (τ, +∞) × W 1,p 0 × Ω → W −1,p ′ .…”

“…we find that φ(t − τ, τ, θ τ ω, ϕ) = u(t, τ, ω, ϕ), where u is a weak solution of the problem (1)-(3). As a consequence the problem reduces to the boundedness of τ → u(t, τ, ω, ϕ) over (−∞, t] and continuity of (τ, ϕ) → u(t, τ, ω, ϕ), which can be verified by similar methods to those in [26,25,28]. At last we extend the obtained invariant sample measures and develop the random Liouville type theorem to the nonautonomous stochastic p-Laplacian problem (1)- (3).…”

“…In order to construct the invariant sample measures, we need to introduce the generalized Banach limit, which is any linear functional, denoted by LIM t→−∞ (we only need to consider the case t → −∞, see more in [25,28]), defined for an arbitrary bounded real-valued function on (−∞, a] for some a ∈ R and satisfying…”

“…By applying this methods, invariant measures have been obtained for lots of concrete differential equations (see e.g. [27,25,28,33]).…”

Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

“…Based on these works, Zhao and Caraballo [36] used the trajectory attractor to construct trajectory statistical solutions for evolution equations. For more results on this issue of continuous systems, the interested reader is referred to [2,6,4,19,23,25,30,28,29,36,37,39,38,41], etc.…”