• Previous Article
    Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods
  • DCDS-B Home
  • This Issue
  • Next Article
    Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations
September  2006, 6(5): 1051-1076. doi: 10.3934/dcdsb.2006.6.1051

Robust exponential attractors for population dynamics models with infinite time delay

1. 

Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via C. Saldini, 50, I-20133 Milano, Italy

2. 

Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano

Received  November 2005 Revised  January 2006 Published  June 2006

We consider an integrodifferential reaction-diffusion system which finds application in population dynamics. The memory kernels accounting for delay effects can be of both weak and strong type. Rescaling the kernels with a time relaxation $\varepsilon>0$, we show that the original model gives raise to a one-parameter family of dynamical systems in a suitable phase-space, We prove that this family is characterized by a corresponding family of exponential attractors which is stable as the delay effects vanish, i.e., when $\varepsilon$ goes to $0$.
Citation: Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051
[1]

Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure and Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721

[2]

Valeria Danese, Pelin G. Geredeli, Vittorino Pata. Exponential attractors for abstract equations with memory and applications to viscoelasticity. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2881-2904. doi: 10.3934/dcds.2015.35.2881

[3]

Peter E. Kloeden, José Real, Chunyou Sun. Robust exponential attractors for non-autonomous equations with memory. Communications on Pure and Applied Analysis, 2011, 10 (3) : 885-915. doi: 10.3934/cpaa.2011.10.885

[4]

S. Gatti, M. Grasselli, V. Pata, M. Squassina. Robust exponential attractors for a family of nonconserved phase-field systems with memory. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 1019-1029. doi: 10.3934/dcds.2005.12.1019

[5]

Hunseok Kang. Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 939-959. doi: 10.3934/dcds.2008.20.939

[6]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations and Control Theory, 2022, 11 (1) : 125-167. doi: 10.3934/eect.2020105

[7]

A. Jiménez-Casas. Invariant regions and global existence for a phase field model. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 273-281. doi: 10.3934/dcdss.2008.1.273

[8]

Yida Ding, Abhishek Deshpande, Gheorghe Craciun. Minimal invariant regions and minimal globally attracting regions for variable-k reaction systems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022143

[9]

Corrado Mascia. Stability analysis for linear heat conduction with memory kernels described by Gamma functions. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3569-3584. doi: 10.3934/dcds.2015.35.3569

[10]

Xinyu Mei, Kaixuan Zhu. Asymptotic behavior of solutions for hyperbolic equations with time-dependent memory kernels. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022150

[11]

José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329

[12]

Jon Jacobsen, Taylor McAdam. A boundary value problem for integrodifference population models with cyclic kernels. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3191-3207. doi: 10.3934/dcdsb.2014.19.3191

[13]

Jianhong Wu, Weiguang Yao, Huaiping Zhu. Immune system memory realization in a population model. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 241-259. doi: 10.3934/dcdsb.2007.8.241

[14]

Andrea Caravaggio, Luca Gori, Mauro Sodini. Population dynamics and economic development. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5827-5848. doi: 10.3934/dcdsb.2021178

[15]

Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713

[16]

Hermano Frid. Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 585-593. doi: 10.3934/dcds.1995.1.585

[17]

Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235

[18]

Tomás Caraballo, José Real, I. D. Chueshov. Pullback attractors for stochastic heat equations in materials with memory. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 525-539. doi: 10.3934/dcdsb.2008.9.525

[19]

Monica Conti, Elsa M. Marchini, V. Pata. Global attractors for nonlinear viscoelastic equations with memory. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1893-1913. doi: 10.3934/cpaa.2016021

[20]

V. V. Chepyzhov, A. Miranville. Trajectory and global attractors of dissipative hyperbolic equations with memory. Communications on Pure and Applied Analysis, 2005, 4 (1) : 115-142. doi: 10.3934/cpaa.2005.4.115

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]