On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient nonlinearity

In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient nonlinearity (Formula presented.). The solution to this problem is understood in a weak sense. We show that for (Formula presented.), there are no local-in-time weak solutions of this problem with initial data (Formula presented.) from the class (Formula presented.) whereas for (Formula presented.), such a solution exists. For (Formula presented.), the Cauchy problem proved not to have global-in-time weak solutions. For (Formula presented.), we show finite-time blow-up of unique local-in-time weak solution of the Cauchy problem without dependence from the initial functions from the class (Formula presented.) As a technique, we obtain Schauder-type estimates for potentials. We use them to investigate smoothness of the weak solution to the Cauchy problem for (Formula presented.) and (Formula presented.). © 2022 John Wiley & Sons, Ltd.