American Institute of Mathematical Sciences

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

Asymptotic behavior of a parabolic-hyperbolic system

M. Grasselli 1, and  V. Pata 1,

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.
Keywords: damped semilinear wave equation., absorbing sets, Phase-field models, exponential attractors, upper semi- continuity, universal attractors.
Mathematics Subject Classification: Primary 35B40, 35B41, 35L05, 35Q40, 37L25, 37L30, 80A22.
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, 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, 2005, 12 (5) : 1019-1029. doi: 10.3934/dcds.2005.12.1019
[3]

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
[4]

Ahmed Bonfoh, Ibrahim A. Suleman. Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3655-3682. doi: 10.3934/cpaa.2021125
[5]

Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 211-238. doi: 10.3934/dcds.2004.10.211
[6]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31
[7]

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, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120
[8]

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
[9]

Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6267-6284. doi: 10.3934/dcdsb.2021018
[10]

Pengyu Chen, Xuping Zhang. Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4325-4357. doi: 10.3934/dcdsb.2020290
[11]

Veronica Belleri, Vittorino Pata. Attractors for semilinear strongly damped wave equations on $\mathbb R^3$. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 719-735. doi: 10.3934/dcds.2001.7.719
[12]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189
[13]

Tina Hartley, Thomas Wanner. A semi-implicit spectral method for stochastic nonlocal phase-field models. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 399-429. doi: 10.3934/dcds.2009.25.399
[14]

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
[15]

Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2019, 18 (2) : 825-843. doi: 10.3934/cpaa.2019040
[16]

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
[17]

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
[18]

Pierluigi Colli, Danielle Hilhorst, Françoise Issard-Roch, Giulio Schimperna. Long time convergence for a class of variational phase-field models. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 63-81. doi: 10.3934/dcds.2009.25.63
[19]

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, 2018, 38 (5) : 2629-2653. doi: 10.3934/dcds.2018111
[20]

Xinyu Mei, Chunyou Sun. Attractors for A sup-cubic weakly damped wave equation in $ \mathbb{R}^{3} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4117-4143. doi: 10.3934/dcdsb.2019053

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (155)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy