Analysis of nonlinear diffusion equations and related phase transition problems

非线性扩散方程及相关相变问题分析

基本信息

批准号：
12640224
负责人：
YAMADA Yoshio
金额：
$ 2.24万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

In our project we have mainly discussed the stationary and non-stationary problems for the following reaction diffusion systems with quasilinear diffusion terms:(E) u_l = Δ[(1 + αv + γu)u] + uf (u, v), v_l = Δ[(1 + βv + δv)v] + vg (u, v).This is a well-known system which models the habitat segregation phenomenon between two species. In (E) u, v denote the population densities and f, g represent the interaction between u and v such as Lotka-Volterra competition type or prey-predator type.(1) Non-stationary problem. When the system has a cross-diffusion effect, the existence result of global solutions was restricted to the two dimensional case. We have proved that, if α, γ > 0 and β = δ = 0, then (E) admits a unique global solution without any restrictions on the space dimension and the amplitude of initial data. Our strategy is to decouple the system and study reaction-diffusion equations separately. We combine parabolic fundamental estimates with energy estimates of solutions of parabolic equation with self-diffusion. This method is also valid for the case δ > 0; so that the global existence is shown when the space dimension is less than six.(2) Stationary problem. From the view-point of mathematical biology, it is very important to study positive stationary solutions and to know their number. We have tried to get some conditions for the multiplicity of such positive solutions. In particular, the multiple existence is established if interactions are very large in case of competition model with linear diffusion or if one of cross-diffusion is very large in case of prey-predator model.

在我们的项目中，我们主要讨论了以下具有拟线性扩散项的反应扩散系统的平稳和非平稳问题：(E) u_l = Δ[(1 + αv + γu)u] + uf (u, v), v_l = Δ[(1 + βv + δv)v] + vg (u, v)。这是一个众所周知的系统，用于模拟两个物种之间的栖息地隔离现象。在 (E) 中，u、v 表示种群。密度，f，g代表u和v之间的相互作用，如Lotka-Volterra竞争型或捕食者型。(1)非平稳问题，当系统存在交叉扩散效应时，全局解的存在性结果为。限于二维情况，我们已经证明，如果 α, γ > 0 且 β = δ = 0，则 (E) 承认一个唯一的全局解，对空间维度和初始数据的幅度没有任何限制。是为了我们将系统解耦并分别研究反应扩散方程，并且将抛物线基本估计与具有自扩散的抛物线方程解的能量估计结合起来，使得当空间维数小于六。 (2)平稳问题从数学生物学的角度来看，研究正平稳解并了解其数量是非常重要的。解决方案。特别是，如果在线性扩散的竞争模型中相互作用非常大，或者在捕食者-捕食者模型中交叉扩散之一非常大，则多重存在成立。