American Institute of Mathematical Sciences

Advanced
November  2006, 15(4): 1193-1214. doi: 10.3934/dcds.2006.15.1193

Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system

Elisabetta Rocca 1, and  Giulio Schimperna 2,

1. 

Dipartimento di Matematica, Università di Milano, Via Saldini, 50, I-20133 Milano, Italy
2. 

Dipartimento di Matematica "F.Casorati", Università di Pavia, Via Ferrata, 1, I-27100 Pavia, Italy

Received  December 2004 Revised  July 2005 Published  May 2006

A singular nonlinear parabolic-hyperbolic PDE's system describing the evolution of a material subject to a phase transition is considered. The goal of the present paper is to analyze the asymptotic behaviour of the associated dynamical system from the point of view of global attractors. The physical variables involved in the process are the absolute temperature $\vartheta$ (whose evolution is governed by a parabolic singular equation coming from the Penrose-Fife theory) and the order parameter $\chi$ (whose evolution is ruled by a nonlinear damped hyperbolic relation coming from a hyperbolic relaxation of the Allen-Cahn equation). Dissipativity of the system and the existence of a global attractor are proved. Due to questions of regularity, the one space dimensional case (1D) and the 2D - 3D cases require different sets of hypotheses and have to be settled in slightly different functional spaces.
Keywords: global attractor., Penrose-Fife model, parabolic-hyperbolic system, dissipativity.
Mathematics Subject Classification: Primary: 35B41, 35L70; Secondary: 80A2.
Citation: Elisabetta Rocca, Giulio Schimperna. Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1193-1214. doi: 10.3934/dcds.2006.15.1193
[1]

Alain Miranville, Elisabetta Rocca, Giulio Schimperna, Antonio Segatti. The Penrose-Fife phase-field model with coupled dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4259-4290. doi: 10.3934/dcds.2014.34.4259
[2]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[3]

Michiel Bertsch, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. Standing and travelling waves in a parabolic-hyperbolic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5603-5635. doi: 10.3934/dcds.2019246
[4]

Ken Shirakawa. Solvability for phase field systems of Penrose-Fife type associated with $p$-laplacian diffusions. Conference Publications, 2007, 2007 (Special) : 927-937. doi: 10.3934/proc.2007.2007.927
[5]

Gilbert Peralta, Karl Kunisch. Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3055-3083. doi: 10.3934/dcds.2018133
[6]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699
[7]

M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827
[8]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-26. doi: 10.3934/dcds.2019226
[9]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639
[10]

Young-Sam Kwon. Strong traces for degenerate parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1275-1286. doi: 10.3934/dcds.2009.25.1275
[11]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[12]

Valerii Maltsev, Michael Pokojovy. On a parabolic-hyperbolic filter for multicolor image noise reduction. Evolution Equations & Control Theory, 2016, 5 (2) : 251-272. doi: 10.3934/eect.2016004
[13]

G. Métivier, K. Zumbrun. Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 205-220. doi: 10.3934/dcds.2004.11.205
[14]

Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203
[15]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1
[16]

Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163
[17]

Xingwen Hao, Yachun Li, Qin Wang. A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations. Kinetic & Related Models, 2014, 7 (3) : 477-492. doi: 10.3934/krm.2014.7.477
[18]

Irena PawŁow. The Cahn--Hilliard--de Gennes and generalized Penrose--Fife models for polymer phase separation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2711-2739. doi: 10.3934/dcds.2015.35.2711
[19]

I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635
[20]

Rana D. Parshad, Juan B. Gutierrez. On the global attractor of the Trojan Y Chromosome model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 339-359. doi: 10.3934/cpaa.2011.10.339

2018 Impact Factor: 1.143

Tools

Metrics

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

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences