A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions iii : general case

We consider the Stefan problem of nonlinear diffusion equation ut = Delta u + f(u) for t > 0 and x E S2(t)(C RN) with positive bistable nonlin-earity f. We first prove that for any initial data, the long-time behavior of the solution is classified into four cases: vanishing, small spreading, big spreading and transition. In particular, we show that when transition occurs for the so-lution u, there exists an x0 E RN such that u(t, center dot ) converges as t-+ oo to an equilibrium solution which is radially symmetric and radially decreasing with center x0. We next give some results about large-time estimates of the expanding speed of S2(t) for small and big spreading cases. As in our previous paper [12], it can be expected that, under a certain condition, every big spreading solution accompanies a propagating terrace. We have succeeded in understanding the large-time behavior of such a solution with terrace in terms of its level set.