On the smoothness of the solution of one nonlinear equation with gradient nonlinearity

We consider the Cauchy problem for a nonclassical third-order partial differential equation with gradient nonlinearity $$|\nabla u(x,t)|^q$$ . A solution of this problem is understood in a certain weak sense. Using Schauder estimates, we show that a local-in-time weak solution of the Cauchy problem has a certain smoothness with $$q>N/(N-1)$$ .