Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$

Abstract

In this paper, we study the dynamics of a non-autonomous semilinear degenerate parabolic equation ut − div(σ(x)∇u) + f (u) = g(x, t) defined on a bounded domain Ω ⊂ R N with smooth boundary. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time; Then we prove that the solution process U (t, τ) is continuous from L 2 (Ω) to D 1 0 (Ω, σ) w.r.t. initial data; And finally show that the known (L 2 (Ω), L 2 (Ω)) pullback D λ-attractor indeed can attract in D 1 0 (Ω, σ)-norm. Any differentiability on the forcing term is not required.

Random dynamics of non-autonomous semi-linear degenerate parabolic equations on <inline-formula><tex-math id="M1">\begin{document} $\mathbb{R}^N$ \end{document}</tex-math></inline-formula> driven by an unbounded additive noise

“…When D N ⊂ R N is a general bounded or unbounded domain, the asymptotic behavior of semi-linear degenerate parabolic equation has been studied in [1,2,20,23,27] for the deterministic case. In particular, Li et al [23] investigated the higher-order integrability of the deterministic equations on bounded domains and obtained the (L 2 (D N ), D 1,2 0 (D N , σ))-attraction of the pullback attractor, by means of some critical embedding. In the random case, Yang et al [37] proved the existence of random attractor in L 2 (D N ).…”