非散度型退化抛物方程（组）解的渐近行为

In recent years, the study for degenerate parabolic equations not in divergence form, which come from biology, physics, chemistry, etc, becomes a hot topic in the field of nonlinear PDEs , and is significant both for the PDE theory and applications. However, the related results still are far from complete. In this project, we will mainly study asymptotic behavior of degenerate parabolic equations not in divergence form, including (1) blow-up properties and large time behavior of solutions to those equations with nonlinear reactive gradient terms; (2) blow-up properties of strong coupled degenerate parabolic system with multi-nonlinear. In order to overcome the essential difficulties due to the particular structure of the involved equation, we have to improves the general Kaplan method, rescaling transform and concave method, etc, in our study.

非散度型退化抛物方程源于生物、物理、力学等领域，是近年偏微分方程领域倍受关注和十分活跃的研究课题，具有重要理论与实际意义。本项目拟研究非散度型退化抛物方程（组）解的渐近行为等问题，包括：（1）含非线性梯度反应项的非散度型方程解的爆破性质及大时间行为；（2）多重非线性非散度型抛物方程组解的爆破性质。拟通过改进传统Kaplan方法、rescaling变换、凹方法等研究手段来克服方程结构的特殊性给问题带来的本质困难。

退化抛物方程是近年来偏微分方程备受关注的课题。本项目较好地完成了退化抛物方程解的渐近行为的研究工作，这些包括带非线性零阶项和一阶项的非散度型方程解的长时间行为，以及非局部扩散下带权函数的抛物方程的临界指标及长时间行为等。对于多重非线性的非散度型退化抛物方程 ，我们给出了各类非线性项占优情形下解的长时间行为，得到了较为完整的结果；而对于带权函数的非局部扩散方程 ，我们发现在高维情形下，局部化因子a极大的改变了解的长时间行为。