American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A degenerate evolution system modeling bean's critical-state type-II superconductors
  • DCDS Home
  • This Issue
  • Next Article
    One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms
July  2002, 8(3): 757-780. doi: 10.3934/dcds.2002.8.757

Existence and long time behaviour of solutions to obstacle thermistor equations

Walter Allegretto 1, , Yanping Lin 2, and  Shuqing Ma 1,

1. 

Department of Mathematical Sciences, University of Alberta, Edmonton A B, Canada T6G 2G1, Canada, Canada
2. 

Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received  March 2001 Revised  November 2001 Published  April 2002

In this paper we introduce an obstacle thermistor system. The existence of weak solutions to the steady-state systems and capacity solutions to the time dependent systems are obtained by a penalized method under reasonable assumptions for the initial and boundary data. At the same time, we prove that there exists a uniform absorbing set for nonnegative initial data in $L_2(\Omega)$. Finally for smooth initial data a global attractor to the system is obtained by a series of Campanato space arguments.
Keywords: capacity solutions, Obstacle thermistor equations, global attractors..
Mathematics Subject Classification: 35K55, 35R4.
Citation: Walter Allegretto, Yanping Lin, Shuqing Ma. Existence and long time behaviour of solutions to obstacle thermistor equations. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 757-780. doi: 10.3934/dcds.2002.8.757
[1]

Walter Allegretto, Yanping Lin, Shuqing Ma. Hölder continuous solutions of an obstacle thermistor problem. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 983-997. doi: 10.3934/dcdsb.2004.4.983
[2]

Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935
[3]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279
[4]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31
[5]

Mei-Qin Zhan. Global attractors for phase-lock equations in superconductivity. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 243-256. doi: 10.3934/dcdsb.2002.2.243
[6]

Monica Conti, Elsa M. Marchini, V. Pata. Global attractors for nonlinear viscoelastic equations with memory. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1893-1913. doi: 10.3934/cpaa.2016021
[7]

V. V. Chepyzhov, A. Miranville. Trajectory and global attractors of dissipative hyperbolic equations with memory. Communications on Pure and Applied Analysis, 2005, 4 (1) : 115-142. doi: 10.3934/cpaa.2005.4.115
[8]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
[9]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks and Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[10]

Gaocheng Yue, Chengkui Zhong. Global attractors for the Gray-Scott equations in locally uniform spaces. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 337-356. doi: 10.3934/dcdsb.2016.21.337
[11]

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

Igor Kukavica. On Fourier parametrization of global attractors for equations in one space dimension. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 553-560. doi: 10.3934/dcds.2005.13.553
[13]

Antônio Luiz Pereira, Severino Horácio da Silva. Continuity of global attractors for a class of non local evolution equations. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1073-1100. doi: 10.3934/dcds.2010.26.1073
[14]

Xiaojie Yang, Hui Liu, Chengfeng Sun. Global attractors of the 3D micropolar equations with damping term. Mathematical Foundations of Computing, 2021, 4 (2) : 117-130. doi: 10.3934/mfc.2021007
[15]

Sergey Dashkovskiy, Oleksiy Kapustyan. Robustness of global attractors: Abstract framework and application to dissipative wave equations. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021054
[16]

William Rundell. Recovering an obstacle using integral equations. Inverse Problems and Imaging, 2009, 3 (2) : 319-332. doi: 10.3934/ipi.2009.3.319
[17]

Xiaoli Li, Dehua Wang. Global solutions to the incompressible magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 763-783. doi: 10.3934/cpaa.2012.11.763
[18]

Monica Conti, Vittorino Pata. On the regularity of global attractors. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1209-1217. doi: 10.3934/dcds.2009.25.1209
[19]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1345-1377. doi: 10.3934/dcdsb.2021093
[20]

Miroslav Bulíček, Annegret Glitzky, Matthias Liero. Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 697-713. doi: 10.3934/dcdss.2017035

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (113)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy