Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary
Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary
复制标题
DOI：
10.1016/j.jde.2018.03.026
复制
发表时间：
2018-08
影响因子：
2.4
通讯作者：
Y. Kaneko;H. Matsuzawa
中科院分区：
文献类型：
--
作者：
Y. Kaneko;H. Matsuzawa
本文研究了形如ut = uxx + f（u），t> 0，ct < x< h（t）的非线性扩散方程的自由边值问题，其中f是满足f（0）= 0的C1函数，c> 0是给定的常数，h（t）是由Stefan条件确定的自由边值.该模型可用于描述种群密度为u（t，x）的新物种或入侵物种在一维生境中的扩散。自由边界x= h（t）表示扩展前沿。在这个模型中，我们在左移动边界x= ct处施加零Dirichlet边界条件.这意味着栖息地的左边界对物种来说是非常不利的环境，并且栖息地被以恒定速度c移动的左边界侵蚀。本文将文[23]中的一个可分性结果推广到一般的单稳态、双稳态和燃烧型非线性。我们证明了解的长时间动力学行为可以用统一的方式表示，即对于任何初始数据，唯一解恰好表现出扩散、消失和跃迁中的一种行为.当扩散发生时，我们也给出了解在整个区域上的渐近分布。这里的方法与[23]中使用的方法完全不同。
We will study a free boundary problem of the nonlinear diffusion equations of the form u t= u x x+ f (u), t> 0, c t< x< h (t), where f is C 1 function satisfying f (0)= 0, c> 0 is a given constant and h (t) is a free boundary which is determined by a Stefan-like condition. This model may be used to describe the spreading of a new or invasive species with population density u (t, x) over a one dimensional habitat. The free boundary x= h (t) represents the spreading front. In this model, we impose zero Dirichlet boundary condition at left moving boundary x= c t. This means that the left boundary of the habitat is a very hostile environment for the species and that the habitat is eroded away by the left moving boundary at constant speed c. In this paper we will extend the results of a trichotomy result obtained in [23] to general monostable, bistable and combustion types of nonlinearities. We show that the long-time dynamical behavior of solutions can be expressed by unified fashion, that is, for any initial data, the unique solution exhibits exactly one of the behaviors, spreading, vanishing and transition. We also give the asymptotic profile of the solution over the whole domain when spreading happens. The approach here is quite different from that used in [23].