双曲空间中一类非线性抛物方程组整体解的若干性质及其应用

Hyperbolic space is one of classical examples of a complete Riemannian manifold. It has an intrinsic structure which is different from Euclid space. On hyperbolic space, parabolic systems are degenerate which are difficult to discuss the properties of global solutions. In this project, we study several properties and applications of global solutions to nonlinear parabolic systems on hyperbolic space by using methods in partial differential equations, geometric analysis and heat kernel of hyperbolic space. Firstly, we will show the existence of nonnegative solutions of nonlinear parabolic systems. Secondly, global existence and blow-up of nonnegative solutions are obtained and using global solutions study practical problems such as numerical solutions and image processing and so on. Furthermore, differential Harnack estimates are obtained by constructing the objective functional on hyperbolic space. Research on the properties of solutions of nonlinear parabolic systems may reveal the relationship between the structure of hyperbolic space and the properties of solutions. It may expand and improve the theories of degenerate parabolic systems.

双曲空间是典型的完备黎曼流形，其几何结构与欧氏空间有较大区别。在双曲空间中，抛物方程组具有退化性，给研究解的性质带来实质性的困难。本项目拟深入研究双曲空间中一类非线性抛物方程组整体解的性质及其应用，用偏微分方程和几何分析中的方法及双曲空间中热核的性质，从证明非线性抛物方程组的非负解入手，获得非负解的整体存在性和有限时刻爆破性，并考虑整体解在实际问题中的应用如数值解和图像处理的研究等。最后利用构造目标泛函的方法，深入研究双曲空间中抛物方程组解的微分Harnack估计。开展本课题的研究，可进一步揭示双曲空间中几何结构对抛物方程组解的性质的影响，拓展和深化退化抛物方程组的理论研究，完善现有的理论结果。

本项目研究了双曲空间中一类非线性抛物方程组整体解的性质及其应用，利用偏微分方程、微分几何的方法及双曲空间中热核的性质，获得了非负解的整体存在性和有限时刻爆破性，并研究了双曲空间中抛物方程组解的微分Harnack估计；讨论了在D-同构形变下保持不变的切触度量流形上的若干向量场，验证了广义Ricci向量场的若干性质；讨论了Yang-Baxter方程的代数解。相信本课题的研究结论可进一步揭示双曲空间中几何结构对抛物方程组解的性质的影响，拓展和深化退化抛物方程组的理论研究，完善现有的理论结果。