非散度型方程源于生物、物理、力学等领域的，是近年偏微分方程领域倍受关注却又很不完善的研究课题，具有重要理论与实际意义。本项目拟研究几类非散度型方程解的渐近性质等问题，包括：（1）带非线性一阶项和零阶项的非散度型方程解的爆破性质及长时间行为；（2）与非散度型方程对应的超快扩散方程解的 Blow-down 行为（3）双退化非散度型方程解的梯度爆破、同宿轨道的存在性等问题。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。

In recent years, the study for non-divergence equations, 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 non-divergence equations, including (1) blow-up properties and large time behavior of solutions to those equations with nonlinear reactive.and absorption terms; (2) blow-down properties of super-fast diffusion equations which can be obtained by the non-divergence equations via a transformation; (3) gradient blow-up properties and the existence of homoclinic orbit for doubly degenerate non-divergence equations. In order to overcome the essential difficulties due to the particular structure of the involved equation, we have to improve the general Kaplan method, rescaling transform and concave method, etc, in our study.