On the critical exponent "instantaneous blow-up / local solvability" in the Cauchy problem for a model Sobolev type equation.

A Sobolev type equation with non-isotropic spatial operator and power gradient nonlinearity is considered. It is shown, by means of the nonlinear capacity method, that for the powers 1<𝑞≤3/2 there is no weak solution to the Cauchy problem for a wide class of initial data. However, for q>3/2 a local weak solution exists, as the the constraint mapping method with Green's function shows.