Regularity of pullback attractors for nonautonomous nonclassical diffusion equations
type: Item
instance of: scholarly article
title: Regularity of pullback attractors for nonautonomous nonclassical diffusion equations
zbMATH Open document ID: 1391.35243
author: Yonghai Wang
author: Zilong Zhu
author: Pengrui Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-12-12T00:00:00Z
review text: The authors consider the nonautonomus nonclassical reaction-diffusion equation \[ \begin{aligned} u_{t}-\Delta u_t -\Delta u+f(u) & =g(t) \quad (r,\infty) \times \Omega, \\ u|_{\partial \Omega} & = 0\quad [r,\infty) \times\partial \Omega,\\ u|_{t=0} & =u_r \quad \{0\} \times \Omega,\end{aligned} \] in a bounded domain \(\Omega\subset\mathbb{R}^3\), where \(u_r\in H_0^1(\Omega)\). The nonlinearity \(f\) is continuous, \(f(0)=0\) and it satisfies the growth and dissipativity conditions \[ \begin{aligned} |f(u)-f(v)| & \leq c|u-v|(1+|u|^4+|v|^4),\\ \liminf_{|u|\to\infty}\frac{f(u)}{u} & >-\lambda_1, \end{aligned} \] where \(\lambda_1>0\) denotes the first eigenvalue of the Dirichlet Laplacian \(A:=-\Delta\) in \(\Omega\). Moreover, \(g\) satisfies \(g\in L^2_{\mathrm{loc}}(\mathbb{R};H^{-1}(\Omega))\) and \[ \sup_{r\leq t}\left(\int_{r-1}^r \|g(s)\|^2_{\mathcal{E}^{-\frac{2}{3}}}ds\right)<\infty, \] where \(\mathcal{E}^{-\frac{2}{3}}=D(A^{-\frac{1}{3}})\) denotes the fractional power space associated to \(A\). The existence of a pullback attractor in \(H_0^1(\Omega)\) is established and, under certain additional assumptions on \(g\), the boundedness of the pullback attractor in \(H^1_0(\Omega)\cap H^2(\Omega)\) is shown.
reviewed by: Stefanie Sonner
Mathematics Subject Classification ID: 35K70
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37B55
zbMATH DE Number: 6817529
zbMATH Keywords: nonclassical diffusion equation
zbMATH Keywords: pullback attractor
zbMATH Keywords: nonautonomous dynamical systems
MaRDI profile type: Publication
full work available at URL: j.jmaa.2017.10.075
OpenAlex ID: W2766837705
cites work: On the problem of diffusion in solids
cites work: Pullback attractors for a class of non-autonomous nonclassical diffusion equations
cites work: Dynamics of non-autonomous nonclassical diffusion equations on \(\mathbb R^n\)
cites work: Uniform attractors for non-autonomous nonclassical diffusion equations on \(\mathbb R^N\)
cites work: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]
cites work: Pullback attractors for asymptotically compact non-autonomous dynamical systems
cites work: Existence of pullback attractors for pullback asymptotically compact processes
cites work: On a theory of heat conduction involving two temperature
cites work: On the regularity of global attractors
cites work: Nonclassical diffusion with memory
cites work: Global attractors for nonlinear viscoelastic equations with memory
cites work: Uniform exponential attractors for a singularly perturbed damped wave equation.
cites work: Pullback attractors for three-dimensional non-autonomous Navier-Stokes-Voigt equations
cites work: Quasilinear Evolution Equations in Nonclassical Diffusion
cites work: Global attractors in \(H^1(\mathbb R^N)\) for nonclassical diffusion equations
cites work: Time dependent perturbation in a non-autonomous non-classical parabolic equation
cites work: Q3530628
cites work: Global attractors for a nonclassical diffusion equation
cites work: Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains
cites work: A cooling process according to two-temperature theory of heat conduction
cites work: On the dynamics of a class of nonclassical parabolic equations
cites work: Attractors for the nonclassical diffusion equations with fading memory
cites work: Asymptotic regularity for the nonclassical diffusion equation with lower regular forcing term
cites work: Attractors for a nonclassical diffusion equation
cites work: Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity
cites work: Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent
cites work: Attractors and asymptotic regularity for nonclassical diffusion equations in locally uniform spaces with critical exponent
DOI: 10.1016/J.JMAA.2017.10.075
description: scientific article; zbMATH DE number 6817529