重排方法在非线性（分数阶）抛物方程中的应用

In this research, we will employ rearrangement method to study nonlinear (fractional) parabolic equations. First of all, for a class of parabolic equations with nonlinear zero order terms and first order terms, research schemes are proposed based on the different growth in the gradient, which respectively allow us to use an exponential function and the exponential variable substitution as the breakthrough point to overcome the influence of the first order. Then we can prove the integral comparison result between parabolic equations and the symmetrized equations. By using the comparison results, we obtain the regularity and existence results for the solutions. Hence, the results of linear parabolic equations are generalized to the nonlinear case. And this will provide an effective research way for nonlinear parabolic problems with all the lower order terms. Second, for fractional parabolic equations with double nonlinearity (and with zero order terms), we propose a research scheme that transforms the double nonlinearity to one nonlinear term and then extends the equation to a class of local Dirichlet-Neumann problem. Based on Steiner symmetrization, we design a test function and use the maximum principle to overcome difficulties of zero order terms. Then we get the integral comparison result between the original fractional parabolic equations and the symmetrized equations. Moreover, we analysis the blowup or extinction time and rate of the solutions. The scheme breaks the limitations of the time discretization method for fractional parabolic equations and has more extensive applications.

本项目应用重排方法就非线性（分数阶）抛物方程展开研究。首先，对同时含有非线性零阶项和一阶项的抛物方程，根据关于梯度增长条件的不同，分别提出以寻求指数乘因子和指数变量替换为突破口来克服一阶项影响的研究方案，证得抛物方程与其对称方程解之间的积分比较关系，并以此研究解的正则性与存在性。不仅将以往线性抛物方程的结果推广到非线性方程上，而且给出包含所有低阶项的非线性抛物问题新的有效研究思路。其次，对（带有零阶项的）双重非线性分数阶抛物方程，提出转化双重非线性到同一非线性项上并将方程延拓为一类局部Dirichlet-Neumann问题来进行研究的技术方案，借助Steiner对称设计合适的试验函数并结合极值原理来克服零阶项所带来的研究困难，进而获得与对称方程解之间的积分比较结果。并以此来分析解的爆破或灭亡的时间和速率。该方案打破以往基于时间离散化方法对分数阶抛物方程进行研究的局限性，具有更广泛的应用。

本项目主要应用重排方法研究非线性抛物方程和分数阶抛物方程解的定性性质，包括解的存在性、有界性、唯一性和解的比较等问题。取得的成果主要包括：1. 借助Schwarz对称和Volterra型积分算子的性质, 我们证明了关于梯度具有一般增长的非线性椭圆方程Dirichlet问题和Neumann问题有界弱解的存在性。并在该结果的启发下，得到了与之相对应的非线性抛物方程与对称抛物方程解之间的积分比较结果。2. 对于非线性分数阶抛物方程，通过将双重非线性化为单一非线性并离散化使之成为带有零阶项的非线性分数阶椭圆方程，我们研究以下两方面内容：一是带有线性零阶项的分数阶椭圆方程解的存在性问题，特别利用重排方法证得了分数阶椭圆算子第一特征值的变分表示和分数阶Faber-Krahn不等式。二是得到了带有凹凸非线性零阶项的分数阶椭圆方程弱解的存在唯一性以及多重性。