Instantaneous blow-up versus local solubility of the Cauchy problem for a two-dimensional equation of a semiconductor with heating

We consider the Cauchy problem for a model third-order partial differential equation with non-linearity of the form $|\nabla u|^q$. We prove that for $q\in(1,2]$ the Cauchy problem in $\mathbb{R}^2$ has no local-in-time weak solution for a large class of initial functions, while for $q>2$ a local weak solution exists.