# American Institute of Mathematical Sciences

July  2002, 8(3): 781-794. doi: 10.3934/dcds.2002.8.781

## A degenerate evolution system modeling bean's critical-state type-II superconductors

 1 Department of Mathematics, Washington State University, Pullman, WA 99164, United States 2 Department of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, United States 3 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

Received  May 2001 Revised  December 2001 Published  April 2002

In this paper we study a degenerate evolution system $\mathbf H_t +\nabla \times [|\nabla \times \mathbf H|^{p-2}\nabla \times \mathbf H]=\mathbf F$ in a bounded domain as well as its limit as $p\to \infty$ subject to appropriate initial and boundary conditions. This system governs the evolution of the magnetic field $\mathbf H$ in a conductive medium under the influence of a system force $\mathbf F$. The system is an approximation of Bean's critical-state model for type-II superconductors. The existence, uniqueness and regularity of solutions to the system are established. Moreover, it is shown that the limit of $\mathbf H(x, t)$ as $p\to \infty$ is a solution to the Bean model.
Citation: H. M. Yin, Ben Q. Li, Jun Zou. A degenerate evolution system modeling bean's critical-state type-II superconductors. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 781-794. doi: 10.3934/dcds.2002.8.781
 [1] Goro Akagi. Doubly nonlinear evolution equations and Bean's critical-state model for type-II superconductivity. Conference Publications, 2005, 2005 (Special) : 30-39. doi: 10.3934/proc.2005.2005.30 [2] Frank Jochmann. Power-law approximation of Bean's critical-state model with displacement current. Conference Publications, 2011, 2011 (Special) : 747-753. doi: 10.3934/proc.2011.2011.747 [3] Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545 [4] Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025 [5] Lars Grüne, Hasnaa Zidani. Zubov's equation for state-constrained perturbed nonlinear systems. Mathematical Control and Related Fields, 2015, 5 (1) : 55-71. doi: 10.3934/mcrf.2015.5.55 [6] Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941 [7] Jesse Berwald, Marian Gidea. Critical transitions in a model of a genetic regulatory system. Mathematical Biosciences & Engineering, 2014, 11 (4) : 723-740. doi: 10.3934/mbe.2014.11.723 [8] Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure and Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015 [9] Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235 [10] Philippe Michel, Bhargav Kumar Kakumani. GRE methods for nonlinear model of evolution equation and limited ressource environment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6653-6673. doi: 10.3934/dcdsb.2019161 [11] V. Styles. A note on the convergence in the limit of a long wave vortex density superconductivity model to the Bean model. Communications on Pure and Applied Analysis, 2002, 1 (4) : 485-494. doi: 10.3934/cpaa.2002.1.485 [12] Takahisa Inui, Nobu Kishimoto, Kuranosuke Nishimura. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6299-6353. doi: 10.3934/dcds.2019275 [13] Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283 [14] Youcef Mammeri, Damien Sellier. A surface model of nonlinear, non-steady-state phloem transport. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1055-1069. doi: 10.3934/mbe.2017055 [15] Ikuo Arizono, Yasuhiko Takemoto. Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking. Journal of Industrial and Management Optimization, 2022, 18 (1) : 25-44. doi: 10.3934/jimo.2020141 [16] Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2441-2474. doi: 10.3934/cpaa.2021049 [17] Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753 [18] Mikhaël Balabane, Mustapha Jazar, Philippe Souplet. Oscillatory blow-up in nonlinear second order ODE's: The critical case. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 577-584. doi: 10.3934/dcds.2003.9.577 [19] Björn Birnir, Kenneth Nelson. The existence of smooth attractors of damped and driven nonlinear wave equations with critical exponent , s = 5. Conference Publications, 1998, 1998 (Special) : 100-117. doi: 10.3934/proc.1998.1998.100 [20] Xu Xu, Xin Zhao. Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4777-4800. doi: 10.3934/dcds.2020201

2020 Impact Factor: 1.392