Strong Attractors for the Structurally Damped Kirchhoff Wave Models with Subcritical-Critical Nonlinearities

Da, Fang; Yang, Zuodong; Sun, Yue

doi:10.1007/s00245-022-09894-9

Cited by 3 publications

(2 citation statements)

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“…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).…”

Section: Definition 12 ([2]mentioning

confidence: 99%

“…From then on, there have been many researches on the well‐posedness and asymptotic behavior of the Kirchhoff type wave models with fractional dissipations:

{\left(-\Delta \right)}&amp;#x0005E;{\alpha }{u}_t

, with

0\le \alpha \le 1

or nonlinear dissipation

h\left({u}_t\right)

, with

h(s)s\ge 0,s\in \mathrm{\mathbb{R}}

(see, e.g. [17–28] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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

Ding

Yang

2023

Math Methods in App Sciences

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This paper investigates the complete regularity of the weak solutions, the existence of the strong (X, X 2𝛼 )-global and exponential attractors, and their stability on dissipative index 𝛼 for the structurally damped Kirchhoff wave equation:, together with the Dirichlet boundary condition, where the perturbed parameter 𝛼 ∈ (1∕2, 1) is called a dissipative index, X is energy space, and X 2𝛼 is strong solution space. We show that when the nonlinearity 𝑓 (u) is of supercritical growth:≤ p < p𝛼 , (i) the weak solutions of the model are just the strong ones; (ii) the global and exponential attractors of the related dynamical system (S 𝛼 (t), X) obtained in literature before are exactly the strong (X, X 2𝛼 )-ones, and the family of strong (X, X 2𝛼 )-global attractors { 𝛼 } 𝛼∈(1∕2,1) is upper semicontinuous on 𝛼 in X 2𝛼 -topology; (iii) for each 𝛼 0 ∈ (1∕2, 1), S 𝛼 (t) has a family of strong (X, X 2𝛼 )-exponential attractors {𝔄 𝛼 exp }, which is Hölder continuous at 𝛼 0 in X 2𝛼 0 -topology. The method developed here allows establishing the above-mentioned results, which breakthrough the restriction 1 ≤ p < p * on this topic in literature before.

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Section: Definition 12 ([2]mentioning

confidence: 99%

“…From then on, there have been many researches on the well‐posedness and asymptotic behavior of the Kirchhoff type wave models with fractional dissipations: