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 $10$; for $q_1