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

Abstract

<p style='text-indent:20px;'>This paper considers the asymptotic behavior of nonclassical diffusion equation with memory and lacking instantaneous damping on time-dependent space. The existence and regularity of time-dependent pullback global attractors are proved by using the contractive process method and a new analytical technique. It is remarkable that the nonlinearity <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> satisfies the polynomial growth of arbitrary order.</p>

Asymptotic behavior of solutions to nonclassical diffusion equations with degenerate memory and a time-dependent perturbed parameter

“…Moreover, in this article, we also incidentally consider asymptotic structure of time-dependent global attractors based on existing studies in [11,12,30]. Thus, it is a comprehensive and innovative problem for us to think about the existence, uniqueness and the asymptotic structure of time-dependent global attractors for the equation (1.1), and this article improves the existing work in [15,32,33,37].…”

“…We know from [20] that the method of the standard Kuratowski measure of non-compactness may be useful for verifying the asymptotic compactness of solution process generated by equation (1.15). But the contractive function method seems to be more concise for our problem, which is mainly based on our previous research [32,33,36]. Thus, we only need to prove that U (t, τ ) is a M t -contractive process by Theorem 2.14.…”

“…When ε(t) is zero (or a positive constant) and memory term is non-degenerate, it is easy to show that the equation (1.1) becomes the usual reaction-diffusion equation (or nonclassical diffusion equation) with memory, under these circumstances, the asymptotic behavior of solutions has been researched by many scholars in recent years (see [9,17,18,19,31,35,36,37]). Especially to deserve to be mentioned, more recently, the authors considered the existence, regularity and upper semicontinuity of global and uniform attractors for autonomous and non-autonomous nonclassical diffusion equation lacking instantaneous damping −∆u in bounded and unbounded domain when the nonlinearity satisfies critical exponential growth and polynomial growth of arbitrary order respectively, see [10,29,32,33,36,38,39].…”

Dynamical behavior of nonclassical diffusion equations with memory and lacking instantaneous damping

“…In addition, in earlier work [31, 32], the existence, regularity, and asymptotic structure of the time‐dependent global attractors are obtained for this equation when $$$ f $$$ meets polynomial growth of arbitrary order. In particular, in Xie et al [33], the existence and regularity of time‐dependent pullback global attractors are confirmed for () on bounded domain and the nonlinearity satisfies the polynomial growth of arbitrary order.…”