Blowup for a Fourth-Order Parabolic Equation with Gradient Nonlinearity

Let u be a solution to the Cauchy problem for a fourth-order nonlinear parabolic equation ∂ₜ u+(-Δ)²u=-∇·(|∇ u|ᵖ⁻²∇ u) on RN, where $p>2$ and N≥ 1. In this paper we give a sufficient condition for the maximal existence time TM(u) of the solution u to be finite. Furthermore, we show that if TM(u)<∞, then \|∇ u(t)\|_L^∞( RN) blows up at t=TM(u), and we obtain lower estimates on the blow-up rate. We also give a sufficient condition on the existence of global-in-time solutions to the Cauchy problem.