Solution blow-up in a nonlinear system of equations with positive energy in field theory

A problem for a nonlinear system of electromagnetic equations in the Coulomb calibration with allowance for sources of free-charge currents is considered. The local-in-time solvability in the weak sense of the corresponding initial-boundary value problem is proved by applying the method of a priori estimates in conjunction with the Galerkin method. A modified Levine method is used to prove that, for an arbitrary positive initial energy, under a certain initial condition on the functional $\Phi(t)=\int\limits_{\Omega}|\mathbf{A}|^2dx$, where $\mathbf{A}(x)$ is a vector potential, the solution of the initial-boundary value problem blows up in finite time. An upper bound for the blow-up time is obtained.