2004, 3(4): 849-881. doi: 10.3934/cpaa.2004.3.849

Asymptotic behavior of a parabolic-hyperbolic system

1. 

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

Received  December 2003 Revised  June 2004 Published  September 2004

We consider a parabolic equation nonlinearly coupled with a damped semilinear wave equation. This system describes the evolution of the relative temperature $\vartheta$ and of the order parameter $\chi$ in a material subject to phase transitions in the framework of phase-field theories. The hyperbolic dynamics is characterized by the presence of the inertial term $\mu\partial_{t t}\chi$ with $\mu>0$. When $\mu=0$, we reduce to the well-known phase-field model of Caginalp type. The goal of the present paper is an asymptotic analysis from the viewpoint of infinite-dimensional dynamical systems. We first prove that the model, endowed with appropriate boundary conditions, generates a strongly continuous semigroup on a suitable phase-space $\mathcal V_0$, which possesses a universal attractor $\mathcal A_\mu$. Our main result establishes that $\mathcal A_\mu$ is bounded by a constant independent of $\mu$ in a smaller phase-space $\mathcal V_1$. This bound allows us to show that the lifting $\mathcal A_0$ of the universal attractor of the parabolic system (corresponding to $\mu=0$) is upper semicontinuous at $0$ with respect to the family $\{\mathcal A_\mu,\mu>0\}$. We also construct an exponential attractor; that is, a set of finite fractal dimension attracting all the trajectories exponentially fast with respect to the distance in $\mathcal V_0$. The existence of an exponential attractor is obtained in the case $\mu=0$ as well. Finally, a noteworthy consequence is that the above results also hold for the damped semilinear wave equation, which is obtained as a particular case of our system when the coupling term vanishes. This provides a generalization of a number of theorems proved in the last two decades.
Citation: 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
[1]

Maurizio Grasselli, Hao Wu. Robust exponential attractors for the modified phase-field crystal equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2539-2564. doi: 10.3934/dcds.2015.35.2539

[2]

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

[3]

Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 211-238. doi: 10.3934/dcds.2004.10.211

[4]

Narcisse Batangouna, Morgan Pierre. Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system. Communications on Pure & Applied Analysis, 2018, 17 (1) : 1-19. doi: 10.3934/cpaa.2018001

[5]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31

[6]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[7]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559

[8]

Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719

[9]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189

[10]

Gianluca Mola. Global attractors for a three-dimensional conserved phase-field system with memory. Communications on Pure & Applied Analysis, 2008, 7 (2) : 317-353. doi: 10.3934/cpaa.2008.7.317

[11]

Tina Hartley, Thomas Wanner. A semi-implicit spectral method for stochastic nonlocal phase-field models. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 399-429. doi: 10.3934/dcds.2009.25.399

[12]

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

[13]

Pierluigi Colli, Danielle Hilhorst, Françoise Issard-Roch, Giulio Schimperna. Long time convergence for a class of variational phase-field models. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 63-81. doi: 10.3934/dcds.2009.25.63

[14]

Claudio Giorgi. Phase-field models for transition phenomena in materials with hysteresis. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 693-722. doi: 10.3934/dcdss.2015.8.693

[15]

Zhijian Yang, Yanan Li. Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2629-2653. doi: 10.3934/dcds.2018111

[16]

Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351

[17]

Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080

[18]

Filippo Dell'Oro. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1015-1027. doi: 10.3934/cpaa.2013.12.1015

[19]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051

[20]

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

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]