Analysis of nonlinear diffusion equations and related phase transition problems

在本课题中，我们主要讨论了具有拟线性扩散项的反应扩散系统：(E)u_L=Δ[(1+αv+γu)u]+uF(u，v)，v_L=Δ[(1+βv+δv)v]+vg(u，v)。在(E)中，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-捕食者型。(1)非平稳问题。当系统存在交叉扩散效应时，整体解的存在性结果仅限于二维情形。我们证明了，如果α，γ&gt；0且β=δ=0，则(E)存在唯一的整体解，而不受空间维度和初始数据幅度的任何限制。我们的策略是将系统解耦，分别研究反应扩散方程。我们将抛物型基本估计与自扩散抛物型方程的解的能量估计结合起来。这种方法同样适用于δ&gt；0；的情形，从而证明了当空间维度小于6时整体解的存在性。(2)平稳问题。从数学生物学的观点来看，研究正定态解并知道它们的个数是非常重要的。我们试图得到这种正解的多解性的一些条件。特别地，对于线性扩散的竞争模型，如果相互作用很大，或者在捕食-捕食模型的交叉扩散很大的情况下，建立了多重存在性。

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.