非线性反应-对流-扩散方程解的渐近行为研究

This project is concerned with the nonlinear reaction -convection-diffusion equations, which are derived from the reaction-convection- diffusion progress in many fields exist widely in the physics, chemistry and ecology, population dynamics and so on.They have both profound practical background and important theoretical value. We will investigate the asymptotic behavior of solutions, such as the long time behavior of solutions and their supports, the extistence of global attractor, the removability of an isolated singularity of solutions, the bifurcations and the multiplicity of solutions to the steady state equations. We expect to establish the critical exponents theories, which can be used to discrib the asymptotic behavior of solutions, and give the effection of reaction, convection and diffusion on them. Existence of the multiple nonlinear items and singularities make the model more close to the reality, but also led to the failure of some known useful theory. So our research not only need the classical mathematical tool, but also need to make innovations in the research ideas and methods, and it will enrich and perfect the theory of partial differential equation in some extent.

本项目拟研究非线性反应-对流-扩散方程，这类方程来源于物理学、化学和生态学以及种群动力学等领域中广泛存在的反应-对流-扩散现象，既有深刻实际背景也有重要理论价值。本项目将在研究此类方程解的渐近行为，讨论解本身及其支集的长时间行为、方程全局吸引子存在性、孤立奇异点附近的渐近行为与奇点可去性，还将讨论方程的分歧现象及其稳态方程的多解性。本项目将建立利用方程中指数或系数刻画解性态变化的临界指标理论，明确给出反应项、对流项与扩散项对解渐近行为的影响。多重非线性与退化、奇异的存在，这使得模型更接近于实际，同时也导致部分已有研究理论失效，给研究带来本质的困难。因此本项目的研究既需要经典的数学工具，也需要在研究思路与方法上有所创新，将在一定程度上丰富已有的偏微分方程理论。

本项目研究了具有鲜明背景的非线性偏微分方程解的渐近行为。具体地，我们给出了具有耦合源项的非线性扩散方程组的解发生有限时刻爆破与整体存在的充分性条件，及爆破时间与速率的界估计；对于具有分数次Laplace算子情形、含Hardy和Henon型权函数情形的几类非线性椭圆问题， 建立了解的存在性、对称性和唯一性；证明了非周期二阶哈密顿系统的多个同宿解与Schrodinger-Poisson系统多个小解的存在性；证明了几种肿瘤生长模型偏微分方程自由边界问题稳态解存在性、分歧与稳定性，包括含死核结构的、链状的、含抑制因子的等情形；关于含非线性Kato类位势的发展加权p-Laplace方程，给出了奇点可去的条件以及正解在孤立奇点处渐近性态的完全分类，揭示了扩散项与非线性项间的相互影响和竞争。本项目研究在结果与方法上均有不同程序的创新，丰富了已有的偏微分方程理论。