Double grow-up and global convergence of solutions for a parabolic prescribed mean curvature problem

Abstract

The paper is concerned with a parabolic mean curvature type problem with a varying parameter λ. We study large time behavior of global solutions for different ranges of λ and initial data and present some new asymptotic results about global convergence and infinite time blow-up. In particular, it is shown that for suitable ranges of parameter λ and initial data, there exists a double grow-up phenomenon: the solution itself blows up at every interior point and its gradient blows up at the boundary of the domain as t→+∞. We also establish an interesting connection between global convergence and the non-classical solution of the associated stationary problem: if initial data are smaller than the non-classical solution, then the solutions must decay to zero in C1 norm as t→∞.