Pith. sign in

REVIEW 1 major objections 4 minor 34 references

Uniform attractors of non-autonomous Kirchhoff wave models

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that supercritical non-autonomous Kirchhoff wave equations have compact uniform attractors for every perturbation parameter, and that these attractors are upper semicontinuous as the parameter varies.

desk verdict Solid, structurally standard extension of Chueshov to the non-autonomous supercritical case, but the deferred proof of Theorem 3.1 is the load-bearing question. read the letter →

arxiv 1908.06500 v1 pith:V2RRTDNE submitted 2019-08-18 math.AP

Zhijian Yang , Yanan Li , Na Feng This is my paper · ORCID
classification math.AP MSC 35B4135L70
keywords non-autonomousKirchhoffwaveequationstrongdampingsupercriticalnonlinearityuniformattractoruppersemicontinuitypartiallytopologypullbacktranslation-boundedexternalforce
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the non-autonomous, strongly damped Kirchhoff wave equation $u_{tt}-\Delta u_t-(1+\varepsilon\|\nabla u\|^2)\Delta u+f(u)=g(x,t)$ on a bounded domain, where the nonlinearity $f$ has supercritical growth with exponent $p$ strictly between $p^*=(N+2)/(N-2)$ and $p^{**}=(N+4)/(N-4)^+$. For every perturbation parameter $\varepsilon\in[0,1]$ it proves that the family of processes generated by the equation has a compact uniform attractor in the energy space $H=(H^1_0\cap L^{p+1})\times L^2$, and that these attractors vary upper semicontinuously with $\varepsilon$ in the partially strong topology. This matters because supercritical growth lies beyond the Sobolev critical exponent, where compact embeddings used for attractors normally fail; the paper shows that the long-time dynamics are nevertheless captured by a compact attracting set that does not jump as $\varepsilon$ changes. It thereby extends known attractor results from the autonomous and subcritical/critical settings to the supercritical non-autonomous setting.

What carries the argument

The machinery centres on the family of processes $U^\varepsilon_g(t,\tau)$ acting on $H$, with symbols in the compact hull $\Sigma=H(g_0)$. Three estimates from Theorem 3.1 carry the whole argument: the partial-regularity bound $\|\nabla u^\varepsilon_t(t)\|^2+\|u^\varepsilon_{tt}(t)\|^2_{H^{-1}}\le K_1(1+1/(t-\tau)^2)$ for $t>\tau$, the quasi-stability inequality (3.7) for differences of two solutions, and the energy identity (3.5). The quasi-stability inequality is used with a contractive-function criterion to prove uniform asymptotic compactness of the process family in $H^{-1}$; the partial-regularity bound makes the uniformly absorbing set $B_0$ bounded in $(H^1_0\cap L^{p+1})\times H^1_0$, and this stronger boundedness converts the $H^{-1}$ semicontinuity into partially strong semicontinuity via the metric $\rho$ on $B_0$. Finally, for asymptotic compactness in $H$ itself, the modified energy $\Gamma$ is shown to converge along the sequence of solutions, and uniform convexity of $L^{p+1}$ and $L^2$ upgrades the weak convergence at time zero to strong convergence in $H$.

What would settle it

Exhibit a translation-bounded force $g$ with $\partial_t g\in L^2_b(\mathbb{R};L^2)$ for which a solution with bounded data violates the asserted bound $\|\nabla u^\varepsilon_t(t)\|^2+\|u^\varepsilon_{tt}(t)\|^2_{H^{-1}}\le K_1(1+1/(t-\tau)^2)$ near $t=\tau$; such a counterexample would break Lemma 5.3 and Corollary 5.4. Alternatively, compute the uniform attractor in a one-dimensional or radially symmetric analogue and check whether the partially strong Hausdorff distance indeed tends to zero while the strong $L^{p+1}$ distance may not.

Watch

Extended reading notes

Core claim

Under Assumption 1.1, with external forces $g$ taken from the hull $\Sigma=H(g_0)$ of a translation-bounded force satisfying $g_0,\partial_t g_0\in L^2_b(\mathbb{R};L^2)$, the paper establishes in Theorem 4.3 that for each $\varepsilon\in[0,1]$ the family of processes $\{U^\varepsilon_g(t,\tau)\}_{g\in\Sigma}$ has a compact uniform attractor $\mathcal{A}^\varepsilon_\Sigma$ in $H$, with the structure $\mathcal{A}^\varepsilon_\Sigma=\bigcup_{g\in\Sigma}K^\varepsilon_g(s)$ for every $s\in\mathbb{R}$. Theorem 5.1 shows upper semicontinuity at every $\varepsilon_0\in[0,1]$ in the $H^{-1}$ topology, and Corollary 5.4 upgrades this to the partially strong topology, in which convergence is strong in $H^1_0\times L^2$ and weak in $L^{p+1}$. For each fixed symbol $g$, the kernel sections $K^\varepsilon_g(t)$ form a pullback attractor of the single process $\{U^\varepsilon_g(t,\tau)\}$, also upper semicontinuous in $\varepsilon$ in the partially strong topology. The key upgrade from weak to strong convergence on the attractor is carried by the convergence of the modified energy $\Gamma(u,u_t)$ combined with the uniform convexity of $L^{p+1}$.

Load-bearing premise

The load-bearing premise is that the non-autonomous version of Theorem 3.1, especially the partial-regularity bound with the factor $1+1/(t-\tau)^2$ and the quasi-stability inequality (3.7), holds by repeating the autonomous arguments, so that the time-dependent external force does not destroy the estimates; if this transfer fails, both the compactness of the uniform attractor and the partially strong semicontinuity collapse.

Editorial extensions

If this is right

  • For every fixed $g\in\Sigma$, the kernel sections $\{K^\varepsilon_g(t)\}_{t\in\mathbb{R}}$ form a pullback attractor of the process $\{U^\varepsilon_g(t,\tau)\}$ in $H$, so the dynamics of individual non-autonomous realizations are captured by a compact invariant family of sets.
  • The whole family $\{\mathcal{A}^\varepsilon_\Sigma\}_{\varepsilon\in[0,1]}$ is uniformly bounded and does not exhibit a jump at any $\varepsilon_0$: small changes in the nonlocal coefficient produce small changes in the Hausdorff distance between attractors in the partially strong topology.
  • In the autonomous case $g(x,t)\equiv g(x)$, the uniform attractor collapses to the global attractor $\mathcal{A}^\varepsilon$ of the solution semigroup, and the result yields upper semicontinuity of that global attractor in $\varepsilon$.
  • The external force needs only be translation bounded with $\partial_t g\in L^2_b(\mathbb{R};L^2)$, not translation compact, because higher partial regularity of the weak solutions substitutes for the missing compactness of the symbol space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step, not pursued here, is to quantify the rate of upper semicontinuity; the proof suggests it is controlled by $|\varepsilon_1-\varepsilon_2|$ in $H^{-1}$ and by its square root in the partially strong metric.
  • The same quasi-stability-plus-partial-regularity route could plausibly work for other nonlocal coefficients, such as $\varepsilon\|\nabla u\|_{L^q}^q$ or an additional nonlinear damping term, as long as the analogous regularity bound holds.
  • One could test the sharpness of the partially strong topology by constructing, in a one-dimensional analogue, sequences of attractor points whose $L^{p+1}$ components converge only weakly; the paper's abstract argument predicts exactly this behaviour.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper studies the non-autonomous Kirchhoff wave equation (1.1) with strong damping, a Kirchhoff coefficient depending on the gradient norm with perturbation parameter ε∈[0,1], and a nonlinearity f(u) of supercritical polynomial growth p*<p<p**. The external force g is assumed to satisfy g, ∂t g ∈ L²_b(R;L²), and the symbol space Σ is the weak-locally compact hull of a fixed force g0. The main results claim that for each ε∈[0,1] the family of processes {U^ε_g(t,τ)} has a compact uniform attractor A^ε_Σ in H=(H^1_0∩L^{p+1})×L², with the structural description A^ε_Σ=∪_{g∈Σ}K^ε_g(s) (Theorem 4.3); and that the family {A^ε_Σ} is upper semicontinuous in ε at every ε0∈[0,1] in H^{-1} topology and in the partially strong topology (Theorem 5.1 and Corollary 5.4). The proof uses a contractive-function criterion for uniform asymptotic compactness, a quasi-stability inequality for differences of solutions, and an energy-equation argument to upgrade H^{-1} convergence to H convergence. The paper also derives a pullback-attractor consequence for each fixed g.

Significance. If the main results are correct, they are a meaningful advance: they extend Chueshov's autonomous supercritical Kirchhoff attractor theory to non-autonomous external forces that are only translation bounded, and they extend the pullback-attractor results of Wang and Zhong to the supercritical growth range. The structural formula for the uniform attractor and the upper-semicontinuity statement are valuable and should be of interest to researchers in infinite-dimensional dynamical systems. However, the significance is conditional because the paper's foundational well-posedness and a priori estimates, Theorem 3.1, are not proved in the manuscript; they are imported from the autonomous case. The paper is purely analytical and does not provide numerical or machine-checked verification, but for this field that is not a deficiency.

major comments (1)
  1. [Section 3, Theorem 3.1] The a priori estimates (3.3), (3.4), (3.6), and (3.7) are the load-bearing foundation of the paper, but they are imported from the autonomous analysis of Chueshov [6] with the single sentence 'Repeating the same arguments as in [6]' and no derivation. The transfer is not formally automatic. In particular, estimate (3.4) requires differentiating (1.1) with respect to t and controlling the term ∂_t g under the assumption g, ∂t g ∈ L²_b(R;L²), while also proving the asserted 1/(t-τ)^2 singular bound for p up to p**. Likewise, the quasi-stability inequality (3.7) must hold uniformly in the symbol space Σ for two solutions driven by different external forces, with the forcing difference appearing exactly as ‖g1-g2‖²_{L²(τ,t;H^{-1})}. These estimates are used immediately: (3.7) drives the contractive-function asymptotic compactness in Theorem 4.3 through (4.13), and (3.4) is what makes the set B0 in Lemma 5.3 bounded in (H^1_0∩L^{p+1})×H^1_0. Until a complete proof, or a precise reference to a theorem that covers the non-autonomous setting, is supplied, the central claims of the paper are not established.
minor comments (4)
  1. [Title] The title contains a typo: 'model s' should be 'models'.
  2. [Section 5, Corollary 5.4(ii)] The statement 'for any fixed g ∈ Σ and ǫ ∈ Σ' should read 'for any fixed g ∈ Σ and ǫ ∈ [0,1]'.
  3. [Section 5, Lemma 5.2] The proof of (5.7) is too compressed regarding the supercritical nonlinearity. The step bounding |(f(u1)-f(u2), (-Δ)^{-1}zt)| by δ‖zt‖² + CK(‖z‖²+‖zt‖²_{H^{-1}}) should spell out how the norm ‖(-Δ)^{-1}zt‖_{p+1} is controlled for p>p* (using zt(t)∈L² and the embedding H²→L^{p+1} for p<p**), and how the δ‖zt‖² term is absorbed by the dissipation on the left-hand side of (5.6).
  4. [Section 4, Lemma 4.2] In the verification that Ψ(ξu) ≥ κΓ(ξu) - C(1+‖g‖²), the absorption of -δ‖ut‖² by (1-2δ/λ1)‖∇ut‖² and the independence of κ, δ from ε∈[0,1] are asserted rather than shown. This is likely correct, but a short explicit computation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found; the existence and upper-semicontinuity results rest on external criteria and a priori estimates, not on predictions equivalent to inputs.

full rationale

Walking the derivation chain: Theorem 4.3 uses the external uniform-attractor criterion Lemma 2.5 and the contractive-function criterion Lemma 2.7 from Sun-Cao-Duan [27], together with the quasi-stability inequality (3.7) and a priori bound (3.3) imported from Chueshov [6]. Theorem 5.1 uses the Lipschitz stability Lemma 5.2, which is derived in the paper from the equation and estimate (3.3), and Lemma 5.3 uses the partial-regularity bound (3.4) to construct the absorbing set B0; Corollary 5.4 is then a metric interpolation from Theorem 5.1. None of these steps defines its conclusion in terms of its inputs: the attractor is not assumed, no parameter is fitted to the quantity being predicted, and the contractive function used in Theorem 4.3 is the difference expression from the quasi-stability inequality, which is a standard sufficient condition for precompactness rather than a restatement of the desired compactness. The self-citations [7] and [30]-[32] appear only as literature background on autonomous Kirchhoff models and do not carry any proof. The only load-bearing deferral is Theorem 3.1, whose partial-regularity and quasi-stability estimates are asserted by 'Repeating the same arguments as in [6]'; this is an omitted derivation and a potential correctness risk, not a circular reduction, because [6] is an external source and those estimates are inputs to the attractor argument, not equivalents of the attractor conclusion being proved.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: ε ∈ [0,1] is a model parameter, and R0, K, K1, CK are derived bounds, not chosen constants. No new entities are postulated: the symbol space Σ, kernels K^ε_g, and uniform attractors A^ε_Σ are standard constructions from the theory of non-autonomous dynamical systems (Chepyzhov-Vishik [4]), and the partially strong topology is imported from Chueshov [6]. Everything else is either standard functional analysis, quoted external criteria, or the structural Assumption 1.1. The one fragile input is Theorem 3.1, whose estimates are deferred to Chueshov [6]; if those fail, the absorbing set, the contractive-function argument, and the partially strong topology result all break.

assumptions (5)
  • domain assumption Assumption 1.1: f ∈ C^1 satisfies c0|s|^{p-1} - c1 ≤ f'(s) ≤ c2(1+|s|^{p-1}) with p* < p < p**, and g, ∂tg ∈ L2_b(R;L2).
    These growth and regularity assumptions are the input to the whole paper: they imply the energy bounds (4.5), the absorbing set (Lemma 4.2), and the regularity estimates used in the attractor proofs. The supercritical range p* < p < p** is exactly the regime the results cover.
  • domain assumption Theorem 3.1: unique weak solutions with energy bound (3.3), partial regularity (3.4), energy identity (3.5), and stability/quasi-stability estimates (3.6)-(3.7), obtained by 'repeating the same arguments as in [6]'.
    The paper does not derive these estimates; they are asserted with a reference to the autonomous analysis in Chueshov [6]. The quasi-stability inequality (3.7) feeds the contractive-function argument in Theorem 4.3, and the partial regularity (3.4) is needed for Lemma 5.3 and Corollary 5.4. This is the main unverified load-bearing input.
  • standard math External criteria: Lemmas 2.5 and 2.7 (uniform-attractor existence and contractive-function compactness) from Sun-Cao-Duan [27]; Lemma 2.8 from Lu-Wu-Zhong [12]; Lemma 2.9 (Simon compactness) from [25].
    These are quoted theorems from the literature. The paper's attractor existence and asymptotic compactness proofs rest on them. They are standard in the field and their statements are given in the text.
  • standard math Sobolev embeddings: H1_0 ↪ L^{2N/(N-2)} and, for the supercritical range p < p**, H^{2-θ} ↪ L^{p+1} for suitably small θ > 0, used in Lemma 5.2.
    Used to control the nonlinearity difference f(u1)-f(u2) in the Lipschitz stability estimate; the admissible range of p depends on N and on choosing θ small. Standard for N ≤ 4 with p < ∞ and for N ≥ 5 with p+1 < 2N/(N-4).
  • standard math Uniform convexity of L^{p+1} and L2 is used to upgrade weak convergence plus norm convergence to strong convergence in H at the end of Theorem 4.3.
    Standard functional analysis. The argument uses (4.24)-(4.26) to conclude (un, unt)(0) → (u, ut)(0) in H.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Uniform attractors of non-autonomous Kirchhoff wave models." pith.science (2026). https://pith.science/paper/V2RRTDNE

@misc{pith2026190806500,
  author       = {Pith},
  title        = {Pith review of: Uniform attractors of non-autonomous Kirchhoff wave models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V2RRTDNE}},
  note         = {Machine review of arXiv:1908.06500}
}
abstract

The paper investigates the existence and upper semicontinuity of uniform attractors of the perturbed non-autonomous Kirchhoff wave equations with strong damping and supercritical nonlinearity: $u_{tt}-\Delta u_{t}-(1+\epsilon\|\nabla u\|^{2})\Delta u+f(u)=g(x,t)$, where $\epsilon\in [0,1]$ is a perturbed parameter. It shows that when the nonlinearity $f(u)$ is of supercritical growth $p: \frac{N+2}{N-2}=p^*<p<p^{**}=\frac{N+4}{(N-4)^+}$: (i) the related evolution process has a compact uniform attractor $\mathcal{A}_\ls^\e $ for each $\epsilon\in [0,1]$; (ii) the family of uniform attractor $\mathcal{A}_\ls^\e $ is upper semicontinuous on the perturbed parameter $\epsilon$ in the sense of partially strong topology.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [6]

    Chueshov, Long-time dynamics of Kirchhoff wave model s with strong nonlinear damping, J

    I. Chueshov, Long-time dynamics of Kirchhoff wave model s with strong nonlinear damping, J. Differential Equations , 252 (2012) 1229-1262

    work page 2012

  2. [29]

    Y . H. Wang, C. K. Zhong, Upper semicontinuity of pullbac k attractors for non-autonomous Kirchhoff wave models, Dis crete Contin. Dyn. Syst. 7 (2013) 3189-3209

    work page 2013

  3. [1]

    J. J. Bae, M. Nakao, Existence problem of global solution s of the Kirchhoff type wave equations with a localized weakl y nonlinear dissipation in exterior domains, Discrete Conti n. Dyn. Syst. 11 (2004) 731-743

    work page 2004

  4. [2]

    J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), 31-52

    work page 2004

  5. [3]

    M. M. Cavalcanti, V . N. D. Cavalcanti, J. S. P . Filho, J. A. Soriano, Existence and exponential decay for a Kirchhoff- C arrier model with viscosity, J. Math. Anal. Appl. 226 (1998) 40-60

    work page 1998

  6. [4]

    V . V . Chepyzhov, M. I. Vishik, Attractors for equations of mathematical physics, Amer. Math. Soc., Colloquium Publications, V ol. 49, Providence, RI, 2002

    work page 2002

  7. [5]

    Chueshov, I

    I. Chueshov, I. Lasiecka, Long-Time Behavior of Second O rder Evolution Equations with Nonlinear Damping, Memoirs o f AMS 912, Amer. Math. Soc., Providence, 2008

    work page 2008

  8. [7]

    P . Y . Ding, Z. J. Yang, Y . N. Li, Global attractor of the Kir chhoff wave models with strong nonlinear damping, Appl. Mat h. Lett. 76 (2018) 40-45

    work page 2018

Show all 34 references
  1. [8]

    X. Fan, S. Zhou, Kernel sections for non-autonomous stro ngly damped wave equations of non-degenerate Kirchhoff-ty pe, Appl. Math. Comput. 158 (2004) 253-266

    2004

  2. [9]

    M. M. Freitas, P . Kalita, J. A. Langa, Continuity of non-a utonomous attractors for hyperbolic perturbation of parab olic equations, J. Differential Equations, 264 (2018) 1886-194 5

    2018

  3. [10]

    Kalantarov, S

    V . Kalantarov, S. Zelik, Finite-dimensional attracto rs for the quasi-linear strongly-damped wave equation, J. D ifferential Equations, 247 (2009) 1120-1155

    2009

  4. [11]

    Kirchhoff, V orlesungen ¨ uber Mechanik, Lectures onMechanics, Teubner, Stuttgart, 1883

    G. Kirchhoff, V orlesungen ¨ uber Mechanik, Lectures onMechanics, Teubner, Stuttgart, 1883. 16

  5. [12]

    S. S. Lu, H. Q. Wu, C. K. Zhong, Attractors for non-autono mous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst. 13 (2005) 701-719

    2005

  6. [13]

    S. S. Lu, Attractors for non-autonomous 2D Navier-Stokes equations with less regular normal forces, J . Differential Equa- tions, 230 (2006) 196-212

    2006

  7. [14]

    S. S. Lu, Attractors for non-autonomous reaction-diff usion systems with symbols without strong translation comp actness, Asymptot. Anal. 54 (2007) 197-210

    2007

  8. [15]

    H. L. Ma, C. K. Zhong, Attractors for the Kirchhoff equat ions with strong nonlinear damping, Appl. Math. Lett. 74 (20 17) 127-133

  9. [16]

    S. Ma, C. K. Zhong, H. Li, Attractors for non-autonomous wave equations with a new class of external forces, J. Math. A nal. Appl. 337 (2008) 808-820

    2008

  10. [17]

    S. Ma, C. K. Zhong, The attractors for weakly damped non- autonomous hyperbolic equations with a new class of externa l forces, Discrete Contin. Dyn. Syst. 18 (2007) 53-70

    2007

  11. [18]

    Matsuyama, R

    T. Matsuyama, R. lkehata, On global solution and energy decay for the wave equation of Kirchhoff-type with nonlinea r damping term, J. Math. Anal. Appl. 204 (1996) 729-753

    1996

  12. [19]

    Moise, R

    I. Moise, R. Rosa, X. Wang, Attractors for noncompact no n-autonomous systems via energy equations, Discrete Conti n. Dyn. Syst. 10 (2004) 473-496

    2004

  13. [20]

    Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J

    M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J. Math. Anal. Appl., 353 (2009) 652- 659

    2009

  14. [21]

    Nakao, Z

    M. Nakao, Z. J. Yang, Global attractors for some quasi-l inear wave equations with a strong dissipation, Adv. Math. Sci. Appl. 17 (2007) 89-105

    2007

  15. [22]

    Nishihara, Decay properties of solutions of some qua silinear hyperbolic equations with strong damping, Nonlin ear Anal

    K. Nishihara, Decay properties of solutions of some qua silinear hyperbolic equations with strong damping, Nonlin ear Anal. 21 (1993) 17-21

    1993

  16. [23]

    Ono, Global existence, decay, and blowup of solution s for some mildly degenerate nonlinear Kirchhoff strings, J

    K. Ono, Global existence, decay, and blowup of solution s for some mildly degenerate nonlinear Kirchhoff strings, J . Differ- ential Equations, 137 (1997) 273-301

    1997

  17. [24]

    Ono, On global existence, asymptotic stability and b lowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math

    K. Ono, On global existence, asymptotic stability and b lowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods A ppl. Sci. 20 (1997) 151-177

    1997

  18. [25]

    Simon, Compact sets in the space Lp(0, T; B), Ann

    J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl. 146 (1986) 65-96

    1986

  19. [26]

    C. Y . Sun, D. M. Cao, J. Q. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006) 2645-2665

    2006

  20. [27]

    C. Y . Sun, D. M. Cao, J. Q. Duan, Uniform attractors for no n-autonomous wave equations with nonlinear damping, SIAM J . Appl. Dyn. Syst. 6 (2007) 293-318

    2007

  21. [28]

    B. X. Wang, Uniform attractors of non-autonomous discr ete reaction-diffusion systems in weighted spaces, Int. J. Bifurcation Chaos, 18 (2008) 659-716

    2008

  22. [30]

    Z. J. Yang, Long-time behavior of the Kirchhoff type equ ation with strong damping in RN , J. Differential Equations, 242 (2007) 269-286

    2007

  23. [31]

    Z. J. Yang, Y . Q. Wang, Global attractor for the Kirchhof f type equation with a strong dissipation, J. Differential E quations, 249 (2010) 3258-3278

    2010

  24. [32]

    Z. J. Yang, P . Y . Ding, Longtime dynamics of the Kirchhof f equation with strong damping and critical nonlinearity on RN , J. Math. Anal. Appl. 434 (2016) 1826-1851. 17

    2016

  25. [33]

    Zelik, Asymptotic regularity of solutions of a non-a utonomous damped wave equation with a critical growth expon ent, Comm

    S. Zelik, Asymptotic regularity of solutions of a non-a utonomous damped wave equation with a critical growth expon ent, Comm. Pure Appl. Anal. 4 (2004) 921-934

    2004

  26. [34]

    Zelik, Strong uniform attractors for non-autonomou s dissipative PDEs with non translation-compact external f orces, Dis- crete Contin

    S. Zelik, Strong uniform attractors for non-autonomou s dissipative PDEs with non translation-compact external f orces, Dis- crete Contin. Dyn. Syst.: B 20 (2015) 781-810. 18

    2015

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.