几类发展型偏微分方程解的性态研究

The study of nonlinear evolution equations shows that, existence of solutions, large time behavior and decay estimate are the main contents of the study. In the study of nonlinear evolution equations, a lot of achievements have been made in the study of evolution equations with Laplace operator, although fractional order Laplace operator and Laplace operator are different in nature, the formulation and research direction of problem for this kind of nonlocal operator can refer to that of Laplace operator. In this project, we first discuss the property of solution for nonlinear evolution equations, including the existence of global solution, blow up in finite time, asymptotic behavior of solutions for Henon equations with weighted function, as well as the existence of solutions and a priori estimates for a class of fractional Laplace equations. Then, we diacuss the existence and property of solution for two classes of fractional Laplace evolution equations with inhomogeneous term. The research in this project can further expand the scope of evolution equations theory, and it not only provides new research contents for nonlinear partial differential equations, but also lays theoretical basis for the application of reaction diffusion, fractal theory, image processing and anomalous diffusion.

非线性发展方程的研究表明：解的存在性、大时间性态及解的衰减性估计是发展方程研究的主要内容。非线性发展的研究中，带Laplace算子的发展方程研究已经取得了很多成果，而对于分数阶Laplace发展方程，尽管分数阶Laplace算子与Laplace算子有着本质的不同，但对这类非局部算子问题的提法及研究方向完全可以借鉴Laplace算子发展方程的内容。本项目中，我们首先讨论非线性发展方程组解的性态，包括带权函数的Henon型方程组整体解的存在性、有限时间爆破、解的渐近行为，以及一类分数阶Laplace发展方程组解的存在性、先验估计；然后，我们研究两类带非齐次项的分数阶Laplace发展方程解的存在性及性态。本项目的研究可以进一步拓展发展方程理论的范围，为非线性偏微分方程提供新的研究内容，也为反应扩散、分形学、图像处理和反常扩散等领域提供理论应用基础。

本项目研究了拟线性、分数阶发展方程及定常问题解的性质，主要获得了如下研究成果：.1.证明了分数阶发展方程定常问题解的存在性、唯一性及多重性等结果，发现了分数阶Laplace与Laplace之间的一些区别..2.运用抛物型方程的上下解方法、比较原理等证明了p-Laplace发展方程解的整体存在性，有限时刻爆破以及爆破速率的估计..3.主要运用变分法和一些非变分技巧证明发展方程定常问题解的存在性. 考虑了一类p-Kirchhoff型方程解的存在性. 考虑更为复杂的边界条件，研究了一类有界光滑区域上带临界指标的p-Laplace方程Robin边值问题解的存在性..4.证明了凸区域上一类调和映照方程正解的存在性、唯一性、正则性和凸性. 进一步，研究了凸区域上一类p-调和映照方程正解的Harnack不等式.