Attractors for a class of perturbed nonclassical diffusion equations with memory

Abstract

<p style='text-indent:20px;'>In this paper, using a new operator decomposition method (or framework), we establish the existence, regularity and upper semi-continuity of global attractors for a perturbed nonclassical diffusion equation with fading memory. It is worth noting that we get the same conclusion in [<xref ref-type="bibr" rid="b7">7</xref>,<xref ref-type="bibr" rid="b14">14</xref>] as the perturbed parameters <inline-formula><tex-math id="M1">\begin{document}$ \nu = 0 $\end{document}</tex-math></inline-formula>, but the nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> satisfies arbitrary polynomial growth condition rather than critical exponential growth condition.</p>

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

“…The asymptotic behavior of solutions of Equation (1.1) has been studied by many scholars (see, e.g., earlier studies [5][6][7][8][9][10][11][12][13] and the references therein). In particular, if we ignore the impact of the instantaneous damping on the system, then Equation (1.1) becomes as the following form:…”

“…, respectively, where the nonlinear term only satisfies the critically exponential growth condition. Yuan et al [11] obtained the existence, regularity, and upper semicontinuity of global attractors for a perturbed nonclassical diffusion equation in H 1 0 (Ω) × L 2 𝜇 (R, H 1 0 (Ω)) and the nonlinearity 𝑓 satisfies arbitrary polynomial growth condition. However, when 𝜀 = 𝜀(t) is a time-dependent parameter, then Equation (1.1) becomes as the following form:…”

“…The asymptotic behavior of solutions of Equation (1.1) has been studied by many scholars (see, e.g., earlier studies [5–13] and the references therein). In particular, if we ignore the impact of the instantaneous damping on the system, then Equation (1.1) becomes as the following form: $$$$ {u}_t-\varepsilon \Delta {u}_t-{\int}_0^{\infty }k(s)\Delta u\left(t-s\right) ds+f(u)=g. $$$$ In previous studies [14, 15], the authors used an ingenious technique to obtain the existence of the global attractors in $$$ {H}_0^1\left(\Omega \right)\times {L}_{\mu}^2\left(\mathrm{\mathbb{R}},{H}_0^1\left(\Omega \right)\right) $$$ and uniform attractors $$$ {H}^1\left({\mathrm{\mathbb{R}}}^n\right)\times {L}_{\mu}^2\left(\mathrm{\mathbb{R}},{H}^1\left({\mathrm{\mathbb{R}}}^n\right)\right) $$$, respectively, where the nonlinear term only satisfies the critically exponential growth condition.…”

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

“…)) respectively, where the nonlinear term only satisfies the critically exponential growth condition. In [27], the authors obtained the existence, regularity and upper semicontinuity of global attractors for a perturbed nonclassical diffusion equation ( 14) in…”

“…There is two main difficulties to obtain the existence of pullback global attractors in the time-dependent product space. Firstly, because of the nonlinearity with arbitrary order exponential growth condition, the higher asymptotic regularity of the solutions of the equation (1) can not be obtained by using the method of [7,21]; Secondly, due to the influence of the time-dependent perturbed parameter ε(t) and the lack of instantaneous damping, it is impossible to directly construct the contractive function to prove the asymptotic compactness for the corresponding process {U (t, τ )} t≥τ of the equation (1)(see e.g., [16,22,27]). 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.…”