Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

<abstract><p>In this paper, the dynamical behavior of the nonclassical diffusion equation is investigated. First, using the asymptotic regularity of the solution, we prove that the semigroup $ \{S(t)\}_{t\geq 0} $ corresponding to this equation satisfies the global exponentially $ \kappa- $dissipative. And then we estimate the upper bound of fractal dimension for the global attractors $ \mathscr{A} $ for this equation and $ \mathscr{A}\subset H^1_0(\Omega)\cap H^2(\Omega) $. Finally, we confirm the existence of exponential attractors $ \mathscr{M} $ by validated differentiability of the semigroup $ \{S(t)\}_{t\geq 0} $. It is worth mentioning that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.</p></abstract>

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Uniform Attractors for Nonclassical Diffusion Equations with Memory

Journal of Function Spaces ◽

10.1155/2016/5340489 ◽

2016 ◽

Vol 2016 ◽

pp. 1-11 ◽

Cited By ~ 6

Author(s):

Yongqin Xie ◽

Yanan Li ◽

Ye Zeng

Keyword(s):

Polynomial Growth ◽

Arbitrary Order ◽

Time Dependent ◽

Diffusion Equations ◽

Fading Memory ◽

Uniform Attractor ◽

Nonclassical Diffusion Equation ◽

Uniform Attractors ◽

With Memory ◽

Nonclassical Diffusion Equations

We introduce a new method (or technique), asymptotic contractive method, to verify uniform asymptotic compactness of a family of processes. After that, the existence and the structure of a compact uniform attractor for the nonautonomous nonclassical diffusion equation with fading memory are proved under the following conditions: the nonlinearityfsatisfies the polynomial growth of arbitrary order and the time-dependent forcing termgis only translation-bounded inLloc2(R;L2(Ω)).

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Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

Advances in Difference Equations ◽

10.1186/s13662-020-03146-2 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Yongqin Xie ◽

Jun Li ◽

Kaixuan Zhu

Keyword(s):

Polynomial Growth ◽

A Priori ◽

Upper Semicontinuity ◽

A Priori Estimate ◽

Global Attractors ◽

Diffusion Equations ◽

Nonlinear Anal ◽

Estimate Method ◽

Appl Math ◽

Nonclassical Diffusion Equations

AbstractIn this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order $p-1$ p − 1 ($p \geq 2$ p ≥ 2 ). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

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Global Attractors inH1(ℝN)for Nonclassical Diffusion Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2012/672762 ◽

2012 ◽

Vol 2012 ◽

pp. 1-16 ◽

Cited By ~ 13

Author(s):

Qiao-zhen Ma ◽

Yong-feng Liu ◽

Fang-hong Zhang

Keyword(s):

Polynomial Growth ◽

Arbitrary Order ◽

Global Attractors ◽

Growth Conditions ◽

Diffusion Equations ◽

Order Polynomial ◽

Nonclassical Diffusion Equations

We study the existence of global attractors for nonclassical diffusion equations inH1(ℝN). The nonlinearity satisfies the arbitrary order polynomial growth conditions.

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Exponential attractors for reaction–diffusion equations with arbitrary polynomial growth

Nonlinear Analysis ◽

10.1016/j.na.2008.10.128 ◽

2009 ◽

Vol 71 (3-4) ◽

pp. 751-765 ◽

Cited By ~ 9

Author(s):

Yansheng Zhong ◽

Chengkui Zhong

Keyword(s):

Polynomial Growth ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Exponential Attractors ◽

Arbitrary Polynomial

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Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.65 ◽

2021 ◽

pp. 1-18

Author(s):

Yuming Qin ◽

Bin Yang

Keyword(s):

Weak Solution ◽

Existence And Uniqueness ◽

Nonlinear Term ◽

Time Dependent ◽

Diffusion Equations ◽

Nonlocal Diffusion ◽

Force Term ◽

Pullback Attractors ◽

Critical Exponential Growth ◽

Nonclassical Diffusion Equations

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

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