American Institute of Mathematical Sciences

Advanced
July  2003, 9(4): 949-968. doi: 10.3934/dcds.2003.9.949

Dissipativity of a conserved phase-field system with memory

Federico Mario Vegni 1,

1. 

Dipartimento di Matematica, Università degli Studi di Brescia, Via Valotti, 9 I-25133 Brescia, Italy

Received  July 2001 Revised  October 2002 Published  April 2003

We analyze a conserved phase-field system, characterized by heat memory terms: memory kernels $a$ and $b$ account for relaxation effects in the energy constitutive equation and in the Gurtin-Pipkin heat conduction law, respectively. This model consists of a hyperbolic integrodifferential equation for the temperature $\theta$ coupled with a nonlinear fourth order evolution equation for the phase variable $\chi$. With appropriate initial and boundary data, we prove that the system can be interpreted as a process. We show that it possesses an absorbing set, under three -thermodynamically consistent- conditions on the memory kernel $a$.
Keywords: continuous dependence, materials with memory, existence and uniqueness, nonautonomous dynamical systems, uniform energy estimates, Conserved phase-field models, uniform absorbing set..
Mathematics Subject Classification: 35G25, 45K05, 80A2.
Citation: Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949
[1]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55
[2]

Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807
[3]

P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 213-232. doi: 10.3934/dcds.2005.12.213
[4]

Pierluigi Colli, Gianni Gilardi, Philippe Laurençot, Amy Novick-Cohen. Uniqueness and long-time behavior for the conserved phase-field system with memory. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 375-390. doi: 10.3934/dcds.1999.5.375
[5]

Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134
[6]

P.E. Kloeden, Victor S. Kozyakin. Uniform nonautonomous attractors under discretization. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 423-433. doi: 10.3934/dcds.2004.10.423
[7]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[8]

Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative lattice dynamical systems with delays. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 643-663. doi: 10.3934/dcds.2008.21.643
[9]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120
[10]

Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025
[11]

Sergiu Aizicovici, Hana Petzeltová. Convergence to equilibria of solutions to a conserved Phase-Field system with memory. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 1-16. doi: 10.3934/dcdss.2009.2.1
[12]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499
[13]

Ming Mei, Yau Shu Wong, Liping Liu. Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (I) Existence and uniform boundedness. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 825-837. doi: 10.3934/dcdsb.2007.7.825
[14]

Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28 (1) : 221-261. doi: 10.3934/era.2020015
[15]

Michele Carriero, Antonio Leaci, Franco Tomarelli. Uniform density estimates for Blake & Zisserman functional. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1129-1150. doi: 10.3934/dcds.2011.31.1129
[16]

Raffaela Capitanelli, Maria Agostina Vivaldi. Uniform weighted estimates on pre-fractal domains. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1969-1985. doi: 10.3934/dcdsb.2014.19.1969
[17]

Christophe Pallard. Growth estimates and uniform decay for a collisionless plasma. Kinetic & Related Models, 2011, 4 (2) : 549-567. doi: 10.3934/krm.2011.4.549
[18]

Vladimir V. Chepyzhov, Monica Conti, Vittorino Pata. Totally dissipative dynamical processes and their uniform global attractors. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1989-2004. doi: 10.3934/cpaa.2014.13.1989
[19]

Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909
[20]

Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences