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 vertical bar Delta u(x, t)vertical bar(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.

Automated summary

In this paper, the Cauchy problem for a nonclassical, third-order partial differential equation with gradient nonlinearity is considered. The existence of local-in-time weak solutions is shown for certain values of q, while for other values, no solution exists or the solution experiences finite-time blow-up. Schauder-type estimates for potentials are used to investigate the smoothness of weak solutions.