非线性发展方程解的性质和图像处理中的应用

本课题首先针对非线性偏微分方程（组）初值和初边值问题解的交界面的扩张速率、完全与非完全爆破、熄灭、Fujita 型临界指数、第二临界指数、生命跨度、一致有界性、大时间渐近行为等问题以及拟线性抛物-抛物（抛物-椭圆）趋化模型和带有Logistic源的趋化模型的解的性质的研究；其次考虑几类非线性色散波方程解的局部适定性、整体解的存在唯一性、爆破准则、持久性质以及无限速度传播等问题；最后基于偏微分方程的方法，去研究图像去噪、图像修补、图像分割等问题。这一系列问题是非线性发展方程理论和图像处理研究中的前沿和热点问题之一。力争在将来几年内解决其中一些热点问题和尚未完全解决的公开问题。

In this project, firstly we deal with expansion rate of interface, complete and incomplete blow-up, extinction, Fujita critical exponent, secondary critical exponent, life span, uniform boundedness and asymptotic behavior in large time on solutions of initial value and initial boundary value problem to nonlinear parabolic equations,and properties of solutions to a quasilineaner parabolic parabolic chemotaxis system and a chemotaxis system with Logistic source; secondly, we study local well-posedness, existence and uniqueness of global solutions, blow-up criterion, property of persistence, and infinite speed of propagation for nonlinear dispersion wave equations; thirdly, based on partial differential equations, we study image denoising, image repair and image segmentation. These problems are one of front and hot problems in research on theory of nonlinear evolutionary equation and inmage processing. We will strive to solve some hot problems and open problems within a few years in the future.