Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models

Abstract

The paper investigates the existence and the continuity of uniform attractors for the non-autonomous Kirchhoff wave equations with strong damping: utt − (1 + ∇u 2 )∆u − ∆ut + f (u) = g(x, t), where ∈ [0, 1] is an extensibility parameter. It shows that when the nonlinearity f (u) is of optimal supercritical growth p : N +2 N −2 = p * < p < p * * = N +4 (N −4) + : (i) the related evolution process has in natural energy space H = (H 1 0 ∩ L p+1 ) × L 2 a compact uniform attractor A Σ for each ∈ [0, 1]; (ii) the family of compact uniform attractor {A Σ } ∈[0,1] is continuous on in a residual set I * ⊂ [0, 1] in the sense of Hps(= (H 1 0 ∩ L p+1,w ) × L 2 )-topology; (iii) {A Σ } ∈[0,1] is upper semicontinuous on ∈ [0, 1] in Hps-topology.

Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

“…In addition, Ma, Wang and Xie [11] verified the existence of pullback attractors for problem u tt − ∆u t − φ ∇u 2 ∆u + f (u) = h(x, t) in H 1 0 (Ω) × L 2 (Ω). Furthermore, Li, Yang and Feng [9] established the existence and continuity of uniform attractors for problem…”