Spreading speed and profile for nonlinear Stefan problems in high space dimensions
Spreading speed and profile for nonlinear Stefan problems in high space dimensions
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DOI：
10.1016/j.matpur.2014.07.008
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发表时间：
2015-03
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影响因子：
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通讯作者：
Yihong Du;H. Matsuzawa;Maolin Zhou
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作者：
Yihong Du;H. Matsuzawa;Maolin Zhou
考虑具有Stefan型自由边界条件的非线性扩散问题ut = Δ u+ f（u），其中非线性项f（u）是单稳态的、非线性的或燃烧型的.这些问题被用作（相应的柯西问题的）替代模型来描述生物或化学物种的扩散，其中自由边界代表扩展前沿。我们感兴趣的是它的长期传播行为，根据杜，Matano和王[10]的最近结果，在很大程度上是由径向对称解决定的。因此，我们将研究径向对称的情况，其中方程满足|X| < h（t），其中|X| = h（t）自由边界。我们假设扩散发生，即lim t→∞ <$h（t）=∞，lim t→∞ <$u（t，|X|）= 1.对于一维空间（N= 1）的情形，Du和Lou [8]证明了：对于某个c <$> 0，limt →∞ <$h（t）t= c <$.随后，本文的作者在[11]中得到了传播速度的更精确的估计，形式为lim t→∞ <$[h（t）− c <$t]= H <$∈ R 1。本文考虑了N≥ 2的情形，证明了存在一个对数移位，即存在与N无关的c <$> 0使得lim t→∞ <$[h（t）− c <$t+（N− 1）c <$log <$t]= h <$∈ R 1.同时，我们也得到了u（t，r）的扩展轮廓的一个相当清晰的描述。这些结果揭示了显着的差异，从相应的柯西问题所模拟的传播行为。
We consider nonlinear diffusion problems of the form u t= Δ u+ 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 lim t→∞ h (t)=∞, lim t→∞ u (t,| x|)= 1. For the case of one space dimension (N= 1), Du and Lou [8] proved that lim t→∞ 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 lim t→∞[h (t)− c⁎ t]= H ˆ∈ R 1. 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 lim t→∞[h (t)− c⁎ t+(N− 1) c⁎ log t]= h ˆ∈ R 1. 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.