Blow-up of the solution to the Cauchy problem for one -dimensional composite-type equation with gradient nonlinearity

Abstract: We consider the Cauchy problem for a third-order nonlinear evolution equation with nonlinearity. Two exponents, and, are found such that for, there is no weak solution local in time for any ; for, 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. © Pleiades Publishing, Ltd. 2025.