American Institute of Mathematical Sciences

Advanced
April  2010, 26(2): 625-647. doi: 10.3934/dcds.2010.26.625

Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory

Jan Prüss 1, , Vicente Vergara 2, and  Rico Zacher 3,

1. 

Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, Theodor-Lieser-Strasse 5, D-06120 Halle, Germany
2. 

Universidad de Tarapacá, Instituto de Alta Investigación, Antofagasta N. 1520, Arica, Chile
3. 

Fachbereich Mathematik und Informatik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle

Received  January 2009 Revised  April 2009 Published  October 2009

In this paper we study a temperature dependent phase field model with memory. The case where both the equation for the temperature and that for the order parameter is of fractional time order is covered. Under physically reasonable conditions on the nonlinearities we prove global well-posedness in the $L_p$ setting and show that each solution converges to a steady state as time goes to infinity.
Keywords: Cahn-Hilliard equation with memory, integro-differential equations, phase field system with memory, Łojasiewicz-Simon inequality, Maximal regularity, convergence to equilibrium..
Mathematics Subject Classification: Primary: 45K05; Secondary: 45M0.
Citation: Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625
[1]

Monica Conti, Stefania Gatti, Alain Miranville. A singular cahn-hilliard-oono phase-field system with hereditary memory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3033-3054. doi: 10.3934/dcds.2018132
[2]

Jean-Baptiste Burie, Ramsès Djidjou-Demasse, Arnaud Ducrot. Slow convergence to equilibrium for an evolutionary epidemiology integro-differential system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019225
[3]

Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423
[4]

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
[5]

Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581
[6]

Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002
[7]

Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019040
[8]

Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367
[9]

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
[10]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741
[11]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[12]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[13]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[14]

Xinlong Feng, Yinnian He. On uniform in time $H^2$-regularity of the solution for the 2D Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5387-5400. doi: 10.3934/dcds.2016037
[15]

Kelong Cheng, Cheng Wang, Steven M. Wise, Zixia Yuan. Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020186
[16]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113
[17]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145
[18]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630
[19]

Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220
[20]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences