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.

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 Time-dependent attractors for non-autonomous non-local reaction–diffusion equations

2018 ◽  
Vol 148 (5) ◽  
pp. 957-981
Author(s):  
Tomás Caraballo ◽  
Marta Herrera-Cobos ◽  
Pedro Marín-Rubio
Keyword(s):  
Existence And Uniqueness ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Pullback Attractors ◽  
Non Local ◽  
Bounded Sets

In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.

scholarly journals Pullback Attractors for a Nonautonomous Retarded Degenerate Parabolic Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Fahe Miao ◽  
Hui Liu ◽  
Jie Xin
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Degenerate Parabolic Equation ◽  
Pullback Attractors ◽  
Asymptotic Compactness ◽  
Absorbing Sets ◽  
Degenerate Parabolic ◽  
Standard Galerkin Method

This paper is devoted to a nonautonomous retarded degenerate parabolic equation. We first show the existence and uniqueness of a weak solution for the equation by using the standard Galerkin method. Then we establish the existence of pullback attractors for the equation by proving the existence of compact pullback absorbing sets and the pullback asymptotic compactness.

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