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

Abstract

In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient nonlinearity |∇u(x, t)| q . The solution to this problem is understood in a weak sense. We show that for 1 < q ⩽ 3∕2, there are no local-in-time weak solutions of this problem with initial data u 0 (x) from the class U, whereas for q > 3∕2, such a solution exists. For 3∕2 < q ⩽ 2, the Cauchy problem proved not to have global-in-time weak solutions. For 3∕2 < q ⩽ 2, 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 U. 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 q = 2 and q ⩾ 4.