Blow-up of the solution to the Cauchy problem for one $(N+1)$-dimensional composite-type equation with gradient nonlinearity

We consider the Cauchy problem for a third-order nonlinear evolution equation with nonlinearity $|D_xu|^q$. Two exponents, $q_1=N/(N-1)$ and $q_2=(N+1)/(N-1)$, are found such that for $1<q\leqslant q_1$, there is no weak solution local in time for any $T>0$; for $q_1<q\leqslant q_2$, there is a unique weak solution local in time; however, there is no weak solution global in time, i.e., independently of the “value” of the initial function, the solution to the Cauchy problem blows up in a finite time.