Global unsolvability of a nonlinear conductor model in the quasistationary approximation

We study initial-boundary value problems for a model differential equation in a bounded region with a quadratic nonlinearity of a special type typical for the theory of conductors. Using the test function method, we show that such a nonlinearity can lead to global unsolvability with respect to time, which from the physical standpoint means an electrical breakdown of the conductor in a finite time. For the simplest test functions, we obtain sufficient conditions for the unsolvability of the model problems and estimates of the blowup rate and time. With concrete examples, we demonstrate the possibility of using the method for one-, two- and three-dimensional problems with classical and nonclassical boundary conditions. We separately consider the Neumann and Navier problems in bounded $\mathbb{R}^N$ regions $(N\ge2)$.