On Singular Solutions of a Cauchy Problem for an $$(N+1)$$-Dimensional Equation of a Nonlinear Thermal-Electrical Model

We consider the problem of the existence of weak singular solutions of a Cauchy problem for an $$(N+1)$$ -dimensional nonlinear Sobolev-type equation of thermal-electrical model of semiconductor with gradient nonlinearity or the form $$|D_xu|^q$$ . In particular, we show that, for $$1 N/(N-1)$$ , we construct a weak solutio global in time of the Cauchy problem. Moreover, we construct solutions singular in spatial variables and weak local in time. These solutions are not $$W^{1,q}_{\operatorname{loc}}$$ -integrable in $$\mathbb{R}^N$$ for $$1 N/(N-1)$$ .