Sharp Estimate of the Spreading Speed Determined by Nonlinear Free Boundary Problems
Sharp Estimate of the Spreading Speed Determined by Nonlinear Free Boundary Problems
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DOI：
10.1137/130908063
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发表时间：
2014-01
期刊：
影响因子：
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通讯作者：
Yihong Du;H. Matsuzawa;Maolin Zhou
中科院分区：
文献类型：
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作者：
Yihong Du;H. Matsuzawa;Maolin Zhou
本文研究具有自由边界的非线性扩散问题u_t=u_{xx}+f（u）。这些问题可以用来描述生物或化学物种的扩散，自由边界代表扩展的前沿。对于单稳态、非线性和燃烧类型的非线性，Du和Lou [``Spreading and vanishing in nonlinear diffusion problems with free boundaries，”J. Eur. Math. Soc.（JEMS），出现]获得了一个相当完整的描述的长期动力学行为的问题，并揭示了急剧过渡现象之间的扩展（$\lim_{t\to\infty}u（t，x）=1$）和消失（$\lim_{t\to\infty}u（t，x）=0$）。他们还利用传播发生时的半波确定了波前的渐近传播速度。本文对波前的传播速度给出了一个比杜和娄的上述工作更精确的估计，并描述了当传播发生时解如何接近半波。
We study nonlinear diffusion problems of the form $u_t=u_{xx}+f(u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. For monostable, bistable, and combustion types of nonlinearities, Du and Lou [``Spreading and vanishing in nonlinear diffusion problems with free boundaries,” J. Eur. Math. Soc. (JEMS), to appear] obtained a rather complete description of the long-time dynamical behavior of the problem and revealed sharp transition phenomena between spreading ($\lim_{t\to\infty}u(t,x)=1$) and vanishing ($\lim_{t\to\infty}u(t,x)=0$). They also determined the asymptotic spreading speed of the fronts by making use of semiwaves when spreading happens. In this paper, we give a much sharper estimate for the spreading speed of the fronts than that in the above-mentioned work of Du and Lou, and we describe how the solution approaches the semiwave when spreading happens.