Asymptotic behavior of the stochastic Landau-Lifshitz-Bloch equation

In this contribution we are concerned with the asymptotic behavior of the solution of the stochastic Landau-Lifshitz-Bloch equation with multiplicative and additive noise in the sense of Stratonovich, which describes the phase spins in ferromagnetic material around the Curie temperature in a random environment. We first establish the existence and uniqueness of the weak pullback mean random attractor in 𝐿 𝑝 ⁡ ( 𝛺 , ℱ ; 𝐻 1 ⁡ ( 𝒟 ) ) , 𝑝 ≥ 2 , in a bounded domain 𝒟 ⊂ ℝ 𝑑 or 𝒟 = ℝ 𝑑 with 𝑑 = 1 , 2 . Then, focusing on one-dimensional bounded domain 𝒟 ⊂ ℝ , we prove the existence and uniqueness of the pullback measure attractor in 𝒫 2 ⁡ ( 𝐻 1 ⁡ ( 𝒟 ) ) . Furthermore, we prove the Freidlin-Wentzell large deviation principle of the solution in 𝐶 ⁡ ( [ 0 , 𝑇 ] ; 𝐻 1 ⁡ ( ℝ ) ) ∩ 𝐿 2 ⁡ ( 0 , 𝑇 ; 𝐻 2 ⁡ ( ℝ ) ) of the equation evolving in the entire real line. The analysis overcomes challenges posed by the high-order regularity of phase space 𝐻 1 ⁡ ( 𝒟 ) and the non-compactness of the Sobolev embedding in unbounded domains via a cut-off argument and uniform tail-ends estimate of the solution.