拟线性抛物型方程变号解的奇性分析

In recent years, the sigh-changing solutions of qusilinear partial differential equations which mainly come from biology, physics, chemistry, etc, are the topic under the spotlight but lack of theoretical study in the field of partial differential equations. Related research has an important practical value and theoretical significance. In this project, we study quasilinear parabolic equations with Neumann boundary value problems, mainly concern with the singular properties of sign-changing solutions, including (1) blow-up, extinction properties of sign-changing solutions of the p-Laplace equation with Neumann initial boundary value; (2) local existence, blow-up and extinction of sign-changing solutions to fourth order parabolic equations. Via using and improving Gamma-convergence, truncated function method and convex method, etc, we aim to overcome the essential difficulties that due to the lackness of comparison principle.

源于生物、物理、化学等领域的拟线性抛物型方程模型的非负解问题，是近年来偏微分方程领域备受关注却又缺少理论研究的课题，具有重要的应用价值和理论意义。本项目拟研究拟线性抛物型方程Neumann边值问题变号解的奇异性质等问题，包括：(1)p-Laplace方程Neumann边值问题变号解的爆破和熄灭；(2)四阶抛物型方程变号解的局部存在性以及解的爆破和熄灭。拟借鉴并改进Gamma-convergence方法、截断函数法、凸方法等研究手段来克服缺少比较原理给问题带来的本质性困难。

本项目主要研究拟线性二阶抛物型方程变号解的奇异性质，包括以下几个方面：(1)对具非局部源的快扩散p-Laplace方程的Neumann初边值问题，建立了判定变号解爆破或者整体存在的临界准则，同时给出了变号解在有限时刻熄灭的充分条件；(2)对具非局部源的慢扩散p-Laplace方程的Neumann初边值问题，得到了非正初始能量下变号解的爆破等有趣的结果；(3)对一类四阶抛物型方程，利用井位势方法及Galerkin方法，结合一系列精细的估计，得到反应项占优或梯度依赖位势占优等情形下，弱解的整体存在性结果。我们的研究结果和方法将在一定程度上丰富偏微分方程的理论并对解释某些物理现象提供重要参考。