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反应差异系统的固定和非平稳问题，术语：（e）u_l =δ[（1 +αv +γu）uf（u，v），v_l =Δ[（1 +βV +ΔV）物种。在（e）u中，v表示种群密度，f，g表示U和V之间的相互作用，例如Lotka-volterra竞争类型或猎物per依者类型。（1）非平稳问题。当系统具有交叉扩散效应时，全球溶液的存在结果仅限于两个维情况。我们提供的是，如果α，γ> 0和β=δ= 0，则（e）允许独特的全局解决方案，而没有对空间维度和初始数据放大器的任何限制。我们的策略是分离系统并分别研究反应扩散方程。我们将抛物线基本估计值与抛物线方程解决方案解决方案的能量估计与自扩散结合在一起。此方法也适用于Δ> 0的情况；因此，当空间维度小于六个时，显示全局存在。（2）固定问题。从数学生物学的角度来看，研究积极的固定解决方案并知道它们的数量非常重要。我们试图为多种积极解决方案获得一些条件。特别是，如果相互作用非常大，则在竞争模型的线性扩散中，或者在猎物预言模型的情况下，交叉扩散非常大，则建立多重存在。