On a 𝑝-Laplacian type of evolution system and applications to the Bean model in the type-II superconductivity theory

Abstract

In this paper we study the Cauchy problem for a p p -Laplacian type of evolution system H t + ∇ × [ | ∇ × H | p − 2 ∇ × H ] = F {H_t} + \nabla \times \left [ {{{\left | {\nabla \times H} \right |}^{p - 2}}\nabla \times H} \right ] = F . This system governs the evolution of a magnetic field H, where the displacement currently is neglected and the electrical resistivity is assumed to be some power of the current density. The existence, uniqueness, and regularity of solutions to the system are established. Furthermore, it is shown that the limit solution as the power p → ∞ p \to \infty solves the problem of Bean’s model in the type-II superconductivity theory. The result provides us information about how the superconductor material under the external force becomes the normal conductor and vice versa. It also provides an effective method for finding numerical solutions to Bean’s model.