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

Abstract

In 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).

Attractors for a class of perturbed nonclassical diffusion equations with memory

“…For verifying the relative compactness of a sequence for the semigroup generated by evolutionary equations, the following method is presented for using in our later discussion: Lemma 2.2. [12,22,23] Let X equipped the norm • X be a Banach space and B ⊂ X be a bounded subset. Furthermore, supposing that {S(t)} t≥0 is a continuous semigroups on X which fulfills the following conditions:…”

“…Lemma 2.5. [22] Let X be Hilbert space, I = [0, T ], ∀T 0, and the memory kernel µ(s) satisfies (7) and (8). Then for any η t ∈ C(I, L 2 µ (R + , X)), the following estimate…”

“…The Eq(1) which one time derivative appears in the highest order term is called pseudo-parabolic equations [10,11]. Asymptotic behavior of the Eq(1) has been studied in many existing documents over the last few decades (see, e.g., [2,3,19,17,20,16,22,23] and the references therein).…”

Asymptotic behavior of quasi-linear evolution equations on time-dependent product spaces

“…For solving these problems, a new analysis technique combined with operator decomposition method is used to obtain contractive function, and then the pullback asymptotic compactness for the process {U (t, τ )} t≥τ of the equation ( 12) is proved. Furthermore, by using this operator decomposition method (see [24]), asymptotic regularity of the solutions for the equation ( 12) is also proved. Then the regularity of time-dependent pullback global attractors for this equation on…”

Strong pullback attractors for a nonclassical diffusion equation

“…Besides, when the nonlinearity satisfies the arbitrary polynomial growth condition, Anh and Toan [3] investigated the existence of pullback attractors for a non-autonomous nonclassical diffusion equation in a non-cylindrical domain with the homogeneous Dirichlet boundary condition and got the upper semi-continuous of pullback attractors. Authors in [48] considered upper semi-continuity and regularity of attractors for a nonclassical diffusion equations by applying the operator decomposition method. In addition to the above results, the asymptotic behavior of solutions for a nonclassical diffusion equation with delay and memory has been extensively investigated by lots of authors in [6,34,4,53,5,17,16,2,7,8,9,10,11,12,43,39,40,49,51] and references therein.…”