Analysis of nonlinear diffusion equations and related phase transition problems

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 + Δ v)v] + vg （u, v）。这是一个众所周知的系统，它模拟了两个物种之间的栖息地分离现象。(E) u中，v表示种群密度，f， g表示u与v之间的相互作用，如Lotka-Volterra竞争型或捕食-捕食型。(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仅限于二维情况。我们证明了，如果α， γ > 0和β = δ = 0，则(E)允许一个唯一的全局解，而不受初始数据的空间维数和幅值的限制。我们的策略是将系统解耦并分别研究反应扩散方程。将抛物方程的基本估计与自扩散抛物方程解的能量估计结合起来。该方法也适用于δ > 0；因此，当空间维度小于6时，显示全局存在性。(2)平稳问题。从数学生物学的角度来看，研究正平稳解并知道它们的数目是非常重要的。我们试图得到这种正解的多重性的一些条件。特别是在线性扩散的竞争模型中相互作用非常大，或在捕食者-猎物模型中交叉扩散的相互作用非常大时，多重存在性得以建立。

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.