Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension

Upper semicontinuity of strong attractors for the Kirchhoff wave model with structural nonlinear damping

“…5 Chueshov 3 found an exponent p ≡ N+2 N−2 depending on and showed that when the growth order p of the nonlinearity f(u) is up to the range 1 ≤ p ≤ p , problem (1.1)-(1.2) is well-posed and the related evolution semigroup has in natural energy space  = H 1 0 (Ω)×L 2 (Ω) a global and an exponential attractor, respectively. For the physical background and the related researches on the well-posedness and asymptotic behavior of the model of type (1.1), one can see 3,[6][7][8][9][10][11][12][13] and references therein.…”

Pullback Attractors for Nonautonomous Degenerate Kirchhoff Equations with Strong Damping

“…Under this condition, one can conclude that ϕð k∇uðtÞk 2 Þ ≥ c 0 > 0 if ∥∇uðtÞ∥ is bounded for t ∈ ℝ + . Recently, Ma et al [26] proved the existence of the global attractor in the case of degeneration for the autonomous Kirchhoff system.…”

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

“…. Since B 0 is a uniformly absorbing set, there exists a τ 0 ≥ τ such that U gn (τ 0 , τ )ξ n ∈ B 0 for all n ∈ N. Thus, by translation identity (15),…”

“…Ma and Zhong [13] showed that in non-supercritical case: 1 ≤ p ≤ p * , the related solution semigroup has a strong (H 1 0 × L 2 , H 1 0 × H 1 0 )-global attractor A, that is, the compactness and the attractiveness of A are in the topology of the stronger space H 1 0 × H 1 0 where A lies. For the related researches on this topic, one can see [14,15]. Uniform attractor (see Def.…”

“…Proof. Since the family of processes {U g (t, τ )}, g ∈ Σ satisfies translation identity (15), it is norm-to-weak continuous for each ∈ [0, 1] (see Lemma 4.1) and has a uniformly (w.r.t. g ∈ Σ and ∈ [0, 1]) absorbing set B (see Lemma 4.2), by Lemma 2.4, it is sufficient to prove Theorem 4.4 to show the precompactness of the sequence…”