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Doubly nonlinear evolution equations and Bean's critical-state model for type-II superconductivity

Pages: 30 - 39, Issue Special, August 2005

 Abstract        Full Text (222.4K)              

Goro Akagi - Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan (email)

Abstract: This paper is intended as an investigation of the solvability of Cauchy problem for doubly nonlinear evolution equation of the form $dv(t)/dt + \partial \lambda^t(u(t)) \in 3 f(t)$, $v(t) \in \partial \psi(u(t))$, 0 < $t$ < $T$, where $\partial \lambda^t$ and $\partial \psi$ are subdifferential operators, and @'t depends on t explicitly. Our method of proof relies on chain rules for t-dependent subdifferentials and an appropriate boundedness condition on $\partial \lambda^t$ however, it does not require either a strong monotonicity condition or a boundedness condition on $\partial \psi$. Moreover, an initial-boundary value problem for a nonlinear parabolic equation arising from an approximation of Bean's critical-state model for type-II superconductivity is also treated as an application of our abstract theory.

Keywords:  Doubly nonlinear evolution equation, time-dependent subdifferential, reflexive Banach space, Bean model.
Mathematics Subject Classification:  Primary: 47J35, 34G25; Secondary: 35K55.

Received: September 2004;      Revised: April 2005;      Published: September 2005.