The power-concavity on the nonlinear parabolic flows

Ki-Ahm Lee, Seoul National University

Our object in this talk is to show that the concavity of the power of the solution is  preserved in parabolic p-Laplace equation, called power-concavity, and that the power is determined by the homogeneity of the parabolic operator. In the parabolic p-Laplace equation:
   u_t=div (|D u|^p-2D u)
for the density u, the concavity of u^(p-2)/p is considered, which indicates why the log-concavity has been considered in heat flow. In addition, the long time existence of the classical solution of the parabolic p-Laplacian equation can be obtained if the initial smooth data has the right power-concavity and a nondegenerate gradient along the initial boundary.