Asymptotic behavior of solutions to nonclassical diffusion equations with degenerate memory and a time-dependent perturbed parameter
DOI:
https://doi.org/10.58997/ejde.2024.22Abstract
This article concerns the asymptotic behavior of solutions for a class of nonclassical diffusion equation with time-dependent perturbation coefficient and degenerate memory. We prove the existence and uniqueness of time-dependent global attractors in the family of time-dependent product spaces, by applying the operator decomposition technique and the contractive function method. Then we study the asymptotic structure of time-dependent global attractors as \(t\to \infty\). It is worth noting that the memory kernel function satisfies general assumption, and the nonlinearity \(f\) satisfies a polynomial growth of arbitrary order.
For more information see https://ejde.math.txstate.edu/Volumes/2024/22/abstr.html
References
F. Alabau-Boussouira, P. Cannarsa, G. Fragnelli; Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.
G. Barenblatt, Y. Zheltov, I. Kochina; Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mec., 24 (1960), 1286-1803.
S. Borini, V. Pata; Uniform attractors for a strongly damped wave equations with linear memory, Asymptotic Anal., 20 (1999), 263-277.
M. Campiti, G. Metafune, D. Pallara; Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.
P. Cannarsa, D. Rocchetti, J. Vancostenoble; Generation of analytic semi-groups in L2 for a class of second order degenerate elliptic operators, Control Cybern., 37 (2008), 831-878.
M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Silva, A. Y. Souza Franco; Exponential stability for the wave model with localized memory in a past history framework, J. Differential Equations, 264 (2018), 6535-6584.
M. M. Cavalcanti, L. H. Fatori, T. F. Ma; Attractors for wave equations with degenerate memory, J. Differential Equations, 260 (2016), 56-83.
P. J. Chen, M. E. Gurtin; On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614-627.
V. V. Chepyzhov, A. Miranville; On trajectory and global attractors for semilinear heat equations with fading memory, Indiana U. Math. J., 55 (2006), 119-167.
M. Conti, F. DellOro, V. Pata; Nonclassical diffusion with memory lacking instantaneous damping, Commun. Pur. Appl. Anal., 19 (2020), 2035-2050.
M. Conti, V. Pata; Asymptotic structure of the attractor for processes on time-dependent spaces, Nonlinear Anal. Real World Appl., 19 (2014), 1-10.
M. Conti, V. Pata, R. Temam; Attractors for process on time-dependent spaces: Applications to wave equations, J. Differential Equations, 255 (2013), 1254-1277.
M. Conti, E. M. Marchini, V. Pata; Nonclassical diffusion with memory, Math. Method. Appl. Sci., 38 (2015), 948-958.
C. M. Dafermos; Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.
J. C. O. Faria, C. M. Webler; Existence of a global attractor for the heat equation with degenerate memory, J. Dynam. Differential Equations, 35 (2023), 845-864.
H. Gao, L. Li, Z. Liu; Stability of degenerate heat equation in non-cylindrical/cylindrical domain, Z. Angew. Math. Phys., 70 (2019), 1-17.
S. Gatti, M. Grasselli, V. Pata; Lyapunov functionals for reaction-diffusion equations with memory, Math. Method. Appl. Sci., 28 (2005), 1725-1735.
C. Giorgi, M. G. Naso, V. Pata; Exponential stability in linear heat conduction with memory: a semigroup approach, Commun. Appl. Anal., 5 (2001), 121-133.
C. Giorgi, V. Pata, A. Marzocchi; Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA-Nonlinear Diff. Equ. Appl., 5 (1998), 333-354.
M. Grasselli, V. Pata; Uniform attractors for nonautonomous dynamical systems with memory, Progress Nonlinear Differ. Equ. Appl., 50 (2002), 155-178.
J. Jackle; Heat conduction and relaxation in liquids of high viscosity, Physica A, 162 (1990), 377-404.
P. Kloeden, T. Lorenz; Pullback incremental attraction, Nonauton. Dynam. Syst., 1 (2004), 53-60.
Y. Liu; Time-dependent global attractor for the nonclassical diffusion equation, Appl. Anal., 94 (2015), 1439-1449.
Q. Ma, X. Wang, L. Xu; Existence and regularity of time-dependent global attractors for the nonclassical reaction-diffusion equations with lower forcing term, Bound. Value Probl., 1 (2016), 1-11.
F. Meng, M. Yang, C. Zhong; Attractors for wave equation with nonlinear damping on timedependent space, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 205-225.
J. C. Robinson; Infinite-Dimensional Dynamical Dystems, Cambridge: Cambridge University Press, 2001.
J. L. Shomberg; Regular global attractors for wave equations with degenerate memory, Ural Math. J., 5 (2019), 59-82.
C. Sun, D. Cao, J. Duan; Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.
N. D. Toan; Uniform attractors of nonclassical diffusion equations lacking instantaneous damping on RN with Memory, Acta Appl. Math., 170 (2020), 789-822.
J. Wang, Q. Ma; Asymptotic dynamic of the nonclassical diffusion equation with timedependent coefficient, J. Appl. Anal. Comput., 11 (2020), 445-463.
X. Wang, L. Yang, C. Zhong; Attractors for the nonclassical diffusion equations with fading memory, J. Math. Anal. Appl., 362 (2010), 327-337.
Y. Xie, D. Liu, J. Zhang, X. Liu; Uniform attractors for nonclassical diffusion equations with perturbed parameter and memory, J. Math. Phys., 64 (2023), 022701.
Z. Xie, J. Zhang, Y. Xie; Asymptotic behavior of quasi-linear evolution equations on timedependent product spaces, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 2316-2334.
Y. Xie, Q. Li, K. Zhu; Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 23-37.
Y. Xie, Y. Li, Y. Zeng; Uniform attractors for nonclassical diffusion equations with memory, J. Funct. Space, 3-4 (2016), 1-11.
J. Yuan, S. Zhang, Y. Xie, J. Zhang; Attractors for a class of perturbed nonclassical diffusion equations with memory, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 4995-5007.
J. Zhang, Y. Xie, Q. Luo, Z. Tang; Asymptotic behavior for the semilinear reaction-diffusion equations with memory, Adv. Differ. Equations, 1 (2019), 510.
J. Zhang, Z. Liu, J. Huang; Upper semicontinuity of pullback D-attractors for nonlinear parabolic equation with nonstandard growth condition, Math. Nachr., 296 (2023), 5593-5616.
J. Zhang, Z. Xie, Y. Xie; Long-time behavior of nonclassical diffusion equations with memory on time-dependent spaces, Asymptotic Anal., (2023), doi: 10.3233/ASY-231887.
K. Zhu, Y. Xie, F. Zhou; Attractors for the nonclassical reaction-diffusion equations on timedependent spaces, Bound. Value Probl., 95 (2020), 1-14.
Downloads
Published
License
Copyright (c) 2024 Jiangwei Zhang, Zhe Xie, Yongqin Xie
This work is licensed under a Creative Commons Attribution 4.0 International License.
