Global solvability for a singular nonlinear Maxwell's equations (Q2566921)

The authors study the initial-boundary value problem \[ \frac{\partial}{\partial t}[\mu(x, | H| )H] + \nabla\times[r(x,t)\nabla\times H] = F(x,t), \quad (x,t)\in Q_T, \] \[ N \times H(x,t) = N \times G(x,t), \quad (x,t)\in \partial \Omega \times (0,T], \] \[ H(x,0) = H_0(x), \quad (x,t)\in \Omega, \] where \(Q_T=\Omega\times(0,T]\), \(\Omega\) is a bounded domain in \(\mathbb R^3\) with Lipschitz continuous boundary \(\partial \Omega\), \(r(x,t)>0\), \(N\) is the outward unit normal on \(\partial \Omega\), \(G(x,t)\) and \(H_0(x)\) are given boundary and initial functions. The authors prove that the initial-boundary problem has a global weak solution and the solution is also unique.