具非线性边界源和内部吸收项的扩散方程解的熄灭性质

The aim of this project is to investigate the extinction properties of solutions to a class of diffusion equations with nonlinear boundary sources and inner absorptions. We will investigate how the nonlinear diffusion, the nonlinear boundary sources and the inner absorptions affect the extinction properties of solutions to the corresponding problem, study the asymptotic behaviors of solutions near the extinction time and derive the critical extinction exponents, by using iteration methods，theories in dynamical systems, techniques of integral estimates as well as modified comparison principle and super and subsolution methods. Various Sobolev type inequalities and trace inequalities will be used in order to deal with the boundary integral due to the appearance of the boundary flux. Based on this, we will also consider the extinction properties of solutions to problems with variable exponents, mainly concerned with the relationship between the critical extinction exponents and the variable exponents.

本项目旨在研究一类具非线性边界源和内部吸收项的扩散方程解的熄灭性质。我们通过综合利用迭代技巧、动力系统理论、积分估计技巧、修正的比较原理和上下解方法，考察非线性扩散、非线性边界源和非线性吸收项对相应问题解的熄灭性质的综合影响，研究解在熄灭时刻附近的渐近行为，揭示临界指标。为了克服边界条件带来的困难, 我们要综合运用各种形式的Sobolev型不等式和迹不等式来处理由非线性边界源产生的边界积分。在此基础上，我们还将考虑变指数扩散方程解的熄灭性质，重点考察临界熄灭指标与变指数的关系。

本项目旨在研究几类具有重要物理或生物背景的非线性扩散模型解的整体存在与有限时刻爆破性质。所研究的模型主要是具非局部源、非线性边界源和非局部边界条件的扩散方程。我们综合利用现代分析技巧、位势井方法、积分估计和凸方法研究不同性质的非线性项对解的整体存在与爆破性以及渐近性的综合影响。此外，我们还研究了一类带奇异零阶项的椭圆型方程。通过选取恰当的检验函数，我们得到了必要的先验估计；结合收敛技巧，证明了相应问题弱解的存在性及正则性。