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 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 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.

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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