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

The paper is concerned with a parabolic mean curvature type problem with a varying parameter λλ. We study large time behavior of global solutions for different ranges of λλ 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 λλ 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→+∞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 C1C1 norm as t→∞t→∞.