Solution Blow-up in a Nonlinear System of Equations with Positive Energy in Field Theory

Abstract

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 \(\Phi (t) = \int\limits_\Omega {|A{|^2}dx} \), where 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.