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We consider nonlinear diffusion problems of the form ut = Δu + f(u) with Stefan type free boundary conditions, where the nonlinear term f(u) is of monostable, bistable or combustion type. Such problems are used 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 which, by recent results of Du, Matano and Wang [10], is largely determined by radially symmetric solutions. Therefore we will examine the radially symmetric case, where the equation is satisfied in |x| < h(t), with |x| = h(t) the free boundary. We assume that spreading happens, namely 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] =ˆ Ĥ∈R¹. In this paper, we consider the case N ≥ 2 and show that a logarithmic shifting occurs, namely there exists c͙ > 0 independent of N such that limt→∞[h(t) − c∗t + (N − 1)c͙logt] =ˆ ĥ∈R¹. At the same time, we also obtain a rather clear description of the spreading profile of u(t,r). These results reveal striking differences from the spreading behavior modeled by the corresponding Cauchy problem. |
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