On a $p$-Laplacian type of evolution system and applications to the Bean model in the type-II superconductivity theory
We study the Cauchy problem for an $p$-Laplacian type of evolution system ${\mathbf H}_{t}+\g [ | \g {\mathbf H}|^{p-2} \g {\mathbf H}|]={\mathbf F}$. This system governs the evolution of a magnetic field ${\bf H}$, where the current displacement 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 $p\rightarrow \infty$ 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 to become the normal conductor and vice visa. It also provides an effective method to find numerical solutions to Bean's model.