scholarly journals Global Attractors inH1(ℝN)for Nonclassical Diffusion Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
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.

scholarly journals Exponential attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity

2021 ◽  
Vol 6 (11) ◽  
pp. 11778-11795
Author(s):  
Jianbo Yuan ◽  
◽  
Shixuan Zhang ◽  
Yongqin Xie ◽  
Jiangwei Zhang ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Polynomial Growth ◽  
Arbitrary Order ◽  
Dynamical Behavior ◽  
Global Attractors ◽  
Diffusion Equations ◽  
Asymptotic Regularity ◽  
Exponential Attractors ◽  
Nonclassical Diffusion Equation ◽  
Nonclassical Diffusion Equations

<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>

scholarly journals Uniform Attractors for Nonclassical Diffusion Equations with Memory

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
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)).

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

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).

Global Attractors for p-Laplacian Equations with Dynamic Flux Boundary Conditions

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Bo You ◽  
Chengkui Zhong
Keyword(s):  
Boundary Conditions ◽  
Polynomial Growth ◽  
Arbitrary Order ◽  
Global Attractors ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Flux Boundary Conditions ◽  
Long Time ◽  
Bounded Smooth ◽  
Laplacian Equations

AbstractThis paper studies the long-time behavior of solutions for p-Laplacian equations with a polynomial growth nonlinearity of arbitrary order and dynamic flux boundary conditions in d-dimensional bounded smooth domains. We first prove the existence of global attractors in L

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

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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