权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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.

本课题主要讨论了具有拟线性扩散项的反应扩散方程组（E）u_l = Δ[（1 + αv + γu）u] + uf（u，v），v_l = Δ[（1 + βv + δv）v] + vg（u，v）的定常和非定常问题，这是一个著名的模拟两个物种之间生境隔离现象的方程组。在（E）中，u，v表示种群密度，f，g表示u和v之间的相互作用，如Lotka-Volterra竞争型或捕食者-被捕食者型。(1)非平稳问题。当系统具有交叉扩散效应时，整体解的存在性结果仅限于二维情形。我们证明了，当α，γ > 0，β = δ = 0时，（E）存在唯一的整体解，且对空间维数和初值的幅值没有任何限制.我们的策略是将系统解耦，分别研究反应扩散方程。我们将抛物型方程解的联合收割机基本估计与能量估计结合起来。该方法对δ > 0的情形也是有效的，从而当空间维数小于6时，证明了该方程的整体存在性. (2)固定问题。从数学生物学的观点来看，研究正稳定解及其个数是非常重要的。我们试图得到这类正解的多重性的一些条件。特别地，如果在具有线性扩散的竞争模型中相互作用非常大，或者如果在食饵-捕食者模型中交叉扩散之一非常大，则多重存在成立。