两类具非正则值的非线性抛物方程的研究

项目名称： 两类具非正则值的非线性抛物方程的研究

项目编号： No.11301003

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 钮维生

作者单位： 安徽大学

项目金额： 22万元

中文摘要： 本项目旨在研究两类初始值和外力项为可积函数或 Radon测度的非线性抛物方程解的存在性与不存在性机制及渐近行为。拟利用椭圆、抛物容量刻画初始值、外力项的奇异性，进而结合非线性项的增长次数来研究抛物方程解的存在性与不存在性问题；灵活运用方程分解，迭代和渐近先验估计方法来研究抛物方程解的正则性，进而深入研究解半群（解过程）在较正则空间中的全局吸引子（一致吸引子，拉回吸引子）存在性。最后，我们利用指标理论来研究吸引子的分形维数估计。本课题的研究对于深入认识带可积函数或Radon测度等非正则值的非线性抛物方程解的存在性与不存在性机制及发展演化规律，具有重要的理论和实际意义。

中文关键词： 非线性抛物方程;；可积函数与测度；解的存在性与不存在性机制；吸引子；分形维数

英文摘要： This project is devoted to considering the existence and nonexistence mechanisms as well as the asymptotic behaviors of two types of nonlinear parabolic equations with integrable functions or Radon measures as the initial data and external force term. We use elliptic and parabolic capacities to describe the singularity of the data. Then combining the singularity of the data with the growth of the nonlinear term, we investigate the existence and nonexistence of solutions for the equations. Next, using decomposition, bootstrap arguments and the asymptotic a priori estimate method flexibly, we try to establish some delicate regularity results on the solutions， by which we investigate the existence of global attractors for the solution semigroups (uniform attractors, pullback attractors for the solution processes ) in some regular spaces. At last, using the index theory, we consider the fractal dimension of the attractors. This project has important theoretical and practical significance for us to understand deeply the existence and nonexistence mechanisms as well as the evolution of nonlinear parabolic equations with nonregular data.

英文关键词： Nonlinear parabolic equations；L^1 and measure data；Existence and nonexistence mechanisms；Attractor；Fractal dimension