两类具非正则值的非线性抛物方程的研究

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.

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。

本项目以p(x)-Laplace方程和双非线性抛物方程为基本模型， 研究了带可积函数或 Radon 测度等低正则值的非线性椭圆、抛物方程解的存在性与不存在性机制，以及抛物方程整体解的渐近行为。研究与变指数Sobolev空间对应的椭圆、抛物容量理论，并用其来刻画外力项的奇异性，进而结合非线性项的增长次数给出了外力项为一般Radon测度的椭圆、抛物型p(x)-Laplace方程以及若干变分不等式解的存在性与不存在性结果；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，进而研究了上述两类非线性抛物方程解半群在较正则空间中的全局吸引子（拉回吸引子）存在性；利用指标理论研究了上述两类退化方程以及一类四阶退化方程全局吸引子的分形维数估计问题，在适当条件下证明了全局吸引子的无穷维性质。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。