Spreading Speed and Profile for Nonlinear Stefan Problems in High Space Dimensions

We consider nonlinear diffusion problems of the form ut = ∆u + f(u) with Stefan type free boundary conditions, where the nonlinear term f(u) is of monostable, bistable or combustion type. Such problems arise as an alternative model (to the corresponding Cauchy problem) to describe the spreading of a biological or chemical species, where the free boundary represents the expanding front. We are interested in its long-time spreading behavior in the radially symmetric case, where the equation is satisfied in |x| < h(t), with |x| = h(t) the free boundary, and limt→∞ h(t) = ∞, limt→∞ u(t, |x|) = 1. For the case of one space dimension (N = 1), Du and Lou [8] proved that limt→∞ h(t) t = c∗ for some c∗ > 0. Subsequently, sharper estimate of the spreading speed was obtained by the authors of the current paper in [11], in the form that limt→∞[h(t) − c∗t] = Ĥ ∈ R. In this paper, we consider the case N ≥ 2 and show that a logarithmic shifting occurs, namely there exists c∗ > 0 independent of N such that limt→∞[h(t) − c∗t + (N − 1)c∗ log t] = ĥ ∈ R. At the same time, we also obtain a rather clear description of the spreading profile of u(t, r). These results contrast sharply with those for the corresponding Cauchy problem, where the logarithmic shifting for the monostable case is significantly different from that for the bistable and combustion cases.