Strong pullback attractors for a nonclassical diffusion equation

Abstract

<p style='text-indent:20px;'>In this paper, we investigate the existence of pullback attractors for a nonclassical diffusion equation with Dirichlet boundary condition in <inline-formula><tex-math id="M1">\begin{document}$ H^2(\Omega)\cap H^1_0(\Omega) $\end{document}</tex-math></inline-formula>. First, we prove the existence and uniqueness of strong solutions for a nonclassical diffusion equation. Then we prove the existence of pullback attractors in <inline-formula><tex-math id="M2">\begin{document}$ H^2(\Omega)\cap H^1_0(\Omega) $\end{document}</tex-math></inline-formula> by applying asymptotic a priori estimate method.</p>

Pullback D$$ \mathcal{D} $$‐attractors for doubly nonlinear parabolic equations

“…Let X be a complete metric space, and {U(t, 𝜏)} = {U(t, 𝜏) ∶ t ≥ 𝜏} be a two-parameter family of mappings act on X ∶ U(t, 𝜏) ∶ X → X, t ≥ 𝜏. Definition 2.1 ( 8,11,12,(14)(15)(16) ). A two-parameter family of mappings {U(t, 𝜏)} is said to be a norm-to-weak continuous process in X if…”

“…The process {U(t, 𝜏)} is said to be pullback − asymptotically compact if for any t ∈ R, any  ∈ , and any sequence 𝜏 n → −∞, any sequence x n ∈ D(𝜏 n ), the sequence {U(t, 𝜏 n )x n } is relatively compact in X. Definition 2.4 ( 7,8,10,12,17,18 ). The family…”

“…It is said that  ∈  is pullback  − absorbing for the process {U(t, 𝜏)} if for any t ∈ R and any  ∈ , there exists a 𝜏 0 (t, ) ≤ t such that U(t, 𝜏)D(𝜏) ⊂ B(t) for all 𝜏 ≤ 𝜏 0 (t, ). Definition 2.3 ( 7,8,10,12,17,18 ). The process {U(t, 𝜏)} is said to be pullback − asymptotically compact if for any t ∈ R, any  ∈ , and any sequence 𝜏 n → −∞, any sequence x n ∈ D(𝜏 n ), the sequence {U(t, 𝜏 n )x n } is relatively compact in X.…”

“…By hypotheses (1.2)-(1.5), 𝛽 and 𝑓 are nonlinear functions with polynomial growth of arbitrary order. Here, 𝛽 is more general than in previous studies 1,2,4,12 (where 𝛽 is linear growth), which is an essential difficulty in proving the existence of attractor. To the problem (1.1), the key points are to obtain the norm-to-weak continuous and compactness of process generated by (1.1).…”

“…It has been extensively studied when $$$ \beta (u)=u,g\left(x,t\right)=g(x) $$$ and the existence of attractors have been proved in previous studies 6–11 . For more general equation () with $$$ g\left(x,t\right)=g(x) $$$, the existence of attractors are constructed in previous works, 1,2,4,12,13 for the non‐autonomous case, as far as I know, the existence of attractors has not been studied.…”