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

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.