On a study on dynamics of localized pattern for a bistable reaction diffusion equations with a heterogeneous environment

在本研究中，我们研究了非线性扩散方程的自由边界问题。这样的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。如果非线性是单稳态、双稳态或燃烧型的，杜教授和娄刚教授对问题的长期动力学行为得到了相当完整的描述，并揭示了所谓的“扩散”和“消失”之间的尖锐转变现象。他们还利用传播时的“半波”确定了锋面的渐近传播速度。在这项研究中，我们给出了一个比Du和Lou的工作更精确的前锋传播速度的估计，并描述了当传播发生时，解如何逼近半波。对于(1)一维问题，(2)具有径向对称背景的高维问题和(3)一维对流扩散问题，我得到了这些结果。

In this research, we study free boundary problems for nonlinear diffusions equations. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. If the nonlinearity is monostable, bistable, or combustion type, Professor Du and Professor Lou obtained a rather complete description of the long-time dynamical behavior of the problem and revealed sharp transition phenomena between so called "spreading" and "vanishing". They also determined the asymptotic spreading speed of the fronts by using of "semi-waves" when spreading happens. In this research we we give a much sharper estimate for the spreading speed of the fronts than that in the work of Du and Lou, and we describe how the solution approaches the semi-wave when spreading happens. I obtained these results for (1)1 dimensional problem, (2)higher dimensional problem with radially symmetric setting and (3)1 dimensional advection-diffusion problem.