Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping

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

“…Using ( 14), (24), and |ðh, u t ðtÞÞ | ≤λ 1 /2ku t k 2 + 1/2λ 1 khð•, tÞk 2 , we find that 7 Advances in Mathematical Physics…”

“…In general, the exponent p * = n + 2/ðn − 2Þ + is called to be critical when someone studies the problem in H 1 0 ðΩÞÞ × L 2 ðΩÞ. Assuming the stiffness factor is nondegenerate (ϕðsÞ ≥ ϕ 0 > 0), References [18][19][20][21][22][23][24] also proved the existence of the attractor. In the case of possible degeneration of the stiffness coefficient and the case of supercritical source term (p * < p < ðn + 4Þ/ðn − 4Þ + ), the first result on the well-posedness we are aware of is given by Chueshov [25].…”

Strong attractors and their stability for the structurally damped Kirchhoff wave equation with supercritical nonlinearity

“…From then on, there have been many researches on the well-posedness and asymptotic behavior of the Kirchhoff type wave models with fractional dissipations: (−Δ) 𝛼 u t , with 0 ≤ 𝛼 ≤ 1 or nonlinear dissipation h(u t ), with h(s)s ≥ 0, s ∈ R (see, e.g. [17][18][19][20][21][22][23][24][25][26][27][28] and references therein).…”