含梯度项非线性抛物型方程与方程组解的大时间行为

This project mainly focused on the large-time behavior of solutions for some nonlinear parabolic equations and systems with gradient terms which describe the physical theory of growth and roughening of surfaces and population dynamics. It includes three types of nonlinear parabolic equation with dissipative gradient term, convective gradient term, Hamilton-Jacobi gradient term respectively, and the coupled nonlinear parabolic systems. They are hot topics for research in partial differential equation and exist different characteristic, but they have numbers of natural relations in mathematical theory. By mean of the common methods such as comparison principle, supersolution and subsolution, energy function, under the concept of blow-up of solution, we will study the conditions of blow-up and global existence of solutions, Fujita exponents, blow-up sets and so on for these equations in different domains. In particular, under the concept of gradient blow-up of solution, we will determine the conditions, the space profile, the time rate and the rate estimates of geadient blow-up, and analyse the influencing conditions for "gradient term" which play how substantial roles in determining the singular emergence and asymptotic behavior of solutions of equations. At the same time, we also study some large-time behavior of solutions by using a new perspective "gradient blow-up", such as critical exponents, simultaneous and non-simultaneous blow-up criteria and so on. Therefore, the launching of the project is not only conducive to make up systems for research works, enrich the theories of nonlinear partial differential equation, but also shows a kind of new research method for same problems, and provides the theoretical support for solving the related practical problems.

项目主要针对描述表面粗化生长与生态系统理论中含有梯度项的非线性抛物型方程与方程组解的大时间行为展开研究，包括含耗散梯度、对流梯度、Hamilton-Jacobi梯度的方程及通过耦合形成的方程组。它们都是偏微分方程研究的热点且有各自不同的特色，但在数学理论上又有着本质的联系。项目将借助于比较原理、上下解技术与能量函数等常规方法，不仅在解本身爆破概念下研究这些方程在不同的区域组成的问题解的爆破与整体存在条件、Fujita指标与爆破集等渐近行为，而且在解的梯度爆破概念下研究梯度爆破的条件、空间分布、时间比率与爆破率估计，分析"梯度"对解的奇性产生与渐近行为的影响条件。同时,还从"梯度爆破"的角度去考虑问题解的一些大时间行为，如临界指数、同时与非同时爆破准则等。因此，项目的开展不仅有利于研究内容形成系统,丰富偏微分方程理论，更为相同问题展现一个新的研究方法，也为解决实际问题提供理论支持。

本项目围绕计划书中的研究内容和目标展开研究，并基本完成研究计划。根据项目研究的要求，研究了几类非线性抛物方程和方程组解的大时间行为。利用偏微分方程的理论、后继函数的方法、数学分析的手段及数值计算，重点研究了含梯度函数的单个方程形成的抛物问题和由多个方程构成的抛物方程组问题，获得了这些问题解的爆破或不爆破条件、整体存在条件、临界曲线、爆破时间的上下界估计与有效性分析等。除此之外，本项目还研究了几类相关内容的抛物问题，讨论与探索这些抛物问题的结果在对含梯度函数的抛物方程或方程组研究中的作用。本研究创新了解决这些抛物问题的数学思想, 丰富了抛物型方程理论的研究方法, 为探索类似问题的研究提供了可靠的数学手段。.本项目获得了一些有意义的研究成果。研究成果由11篇论文组成, 其中SCI/EI期刊论文3篇, CPCI-S收录1篇，国内中文核心期刊论文6篇。另外完成2篇已经录用的中文核心期刊论文、1篇完成修改的中文核心期刊论文。基本完成了预期指标。