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 $\partial_t u+(-\Delta)^2u=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)$ on ${\bf R}^N$, where $p>2$ and $N\ge 1$. In this paper we give a sufficient condition for the maximal existence time $T_M(u)$ of the solution $u$ to be finite. Furthermore, we show that if $T_M(u)<\infty$, then $\|\nabla u(t)\|_{L^\infty({\bf R}^N)}$ blows up at $t=T_M(u)$, 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.