Global attractors for the 2D hyperbolic Cahn–Hilliard equations

Khanmamedov, Azer; Yayla, Sema

doi:10.1007/s00033-018-0908-y

Cited by 6 publications

(36 citation statements)

References 34 publications

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“…where P ∶ R + → R + is a nondecreasing function. After that, testing (2.1a) by (2u t + 𝛿u) in L 2 (Ω) and exploiting from Khanmamedov and Yayla, 20 for sufficiently small 𝛿, we infer that…”

Section: Theorem 3 Assume That Assumption (F) Holds Then For Every In...mentioning

confidence: 93%

“…Hence, in view of (3.13) and the arguments presented in Khanmamedov and Yayla, 20 we pass to limit in (3.15) to get lim inf…”

Section: Lemma 9 Assume That Condition (F) Holds and B Is A Bounded S...mentioning

confidence: 95%

“…For more detailed explanation related to Definition 1, we refer to Remark 2.2 of Khanmamedov and Yayla. 20 We start with the following lemmas whose proofs can be found in Khanmamedov. 21 Lemma 1 (Khanmamedov 21 ).…”

Section: The Classical Fourier Law (𝛾 = 0)mentioning

confidence: 99%

“…Remark If u ( t , x ) is a weak solution of (2.1), then we obtain that

normalΔ u = 0

H^{- 1} ((), 0, τ; H^{- 1 false/ 2} ((), \partial normalΩ))

for every τ ∈ (0, T ). For more detailed explanation related to Definition 1, we refer to Remark 2.2 of Khanmamedov and Yayla 20 …”

Section: Well‐posedness and Boundedness Of Weak Solutionsmentioning

confidence: 99%

“…Equation () and its viscous variant have been analyzed in various mathematical aspects including well‐posedness and long‐time dynamics of solutions. In this respect, we can cite the previous papers 14–20 . In particular, Khanmamedov and Yayla 20 established the existence of the global attractor for the weak solutions of () in the two‐dimensional case for the sub‐cubic nonlinear term.…”

Section: Introductionmentioning

confidence: 99%

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Long‐time dynamics of the 2D nonisothermal hyperbolic Cahn–Hilliard equations

Yayla

2022

Math Methods in App Sciences

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In this paper, we consider the hyperbolic relaxation of the nonisothermal Cahn-Hilliard equation based on either Fourier law or Maxwell-Cattaneo law for heat conduction. In the Maxwell-Cattaneo case, we reformulate the problem by using enthalpy instead of relative temperature. In both cases, we prove the existence of the global attractor for the weak solutions of the problem. Moreover, for both laws, we establish that every full trajectory in the global attractor converges to a single stationary point as t → ∞ and another single stationary point as t → −∞. Consequently, we infer that the global attractor is equal to a union of the unstable manifolds emanating from the stationary points.

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Section: Theorem 3 Assume That Assumption (F) Holds Then For Every In...mentioning

confidence: 93%

“…Hence, in view of (3.13) and the arguments presented in Khanmamedov and Yayla, 20 we pass to limit in (3.15) to get lim inf…”

Section: Lemma 9 Assume That Condition (F) Holds and B Is A Bounded S...mentioning

confidence: 95%

Section: The Classical Fourier Law (𝛾 = 0)mentioning

confidence: 99%

“…Remark If u ( t , x ) is a weak solution of (2.1), then we obtain that