非线性反应-对流-扩散方程解的若干定性问题研究

We aim at studying on qualitative theories with important physical significance of nonlinear reaction-convection-diffusion equations from physics, chemistry, differential geometry, dynamics of biological groups and ecology and so on. The qualitative problems of the nonlinear reaction-convection-diffusion equations we study in this project, include the singularly perturbed problems, traveling wave solutions, controllability and quenching phenomenon, many of which are hot issues that people are concerned about. Such as traveling wave solutions of the problem with delay(s), the stability of Sharp waves, generation and propagation of interfaces, metastable patterns and the controllability of nonlinear diffusion equations. We mainly discuss the effect and restricted relationship on the solutions among nonlinear source, convection and diffusion. The source, convection and diffusion terms containing multiple nonlinearity and the characteristics of the degeneracy and singularity make the models be more actual. But, at the same time, it leads to essential difficulties as well which make us to find new approaches, frameworks etc. To a certain extent, our results and methods will provide important reference for explaining some physical phenomena and enrich the theory of partial differential equations.

本项目针对来源于物理学、化学、微分几何、生物种群动力学以及生态学等许多学科领域中广泛存在着的非线性反应-对流-扩散方程, 旨在研究具有重要物理意义的定性理论. 本项目研究这类方程的若干定性问题, 包括奇摄动问题、行波解、能控性以及解的淬火现象, 其中很多都是人们所关注的热点问题. 如具时滞问题的行波解, Sharp波的稳定性, 分界面的产生、移动、亚稳态模式以及非线性扩散方程支配系统的能控性问题. 我们将着重探讨非线性源、对流和扩散项对解的性态的影响以及它们之间可能存在着的制约关系. 源、对流和扩散所包含的多重非线性以及退化性和奇异性的特征使得模型更接近于实际, 同时为研究带来本质性困难, 需要寻找新的研究思路. 本项目的研究结果和方法将对解释有关物理现象提供重要参考, 并在一定程度上丰富和完善偏微分方程的理论.

本项目针对来源于物理学、化学、微分几何、生物种群动力学以及生态学等许多学科领域中广泛存在着的非线性反应-对流-扩散方程, 旨在研究具有重要物理意义的定性理论. . 本项目从研究奇异抛物方程古典解、弱解的存在、唯一性出发，进一步讨论了具奇异边界流及奇异内部源的退化抛物方程解的淬火现象, 同时得到了加权p-Laplace椭圆方程径向解的多解性质. 其次, 在非线性扩散方程支配系统的能控性问题方面, 证明了具记忆项的抛物方程系统解的零能控性, 近似能控性及时间最优控制. 另外, 得到了施加移动控制时伪抛物方程解的零能控性. 此外, 针对近年来人们关注的一些特殊非线性扩散方程-液晶方程及磁流体方程, 讨论了解的全局适定性及长时间解的渐近行为. . 本项目的研究结果和方法将对解释有关物理现象提供重要参考, 并在一定程度上丰富和完善偏微分方程的理论.