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 $$1<q\leq q_1$$ , there is no weak solution local in time for any $$T>0$$ ; for $$q_1<q\leq q_2$$ , 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.