非局部拟抛物方程解的全局存在性与爆破性

Pseudo-parabolic equations, as a class of reaction-diffusion equations, model the stationary analysis of the crystalline semiconductors, which become nonlocal in the presence of a nonlocal relation between the flux density and intensity of the electric field for this semiconductor. In mathematics, it is important to study the global existence and blowup of solutions to reaction-diffusion equations. The aim of this project is to establish the criterion on the global existence and blowup of solutions to nonlocal pseudo-parabolic equations with a general nonlinear source. To achieve this, we will consider first the local existence and uniqueness of solutions to the abstract nonlocal pseudo-parabolic equations. Different from the previous literature, we will replace the Galerkin method with the elliptic regularity estimates and the theory of ordinary differential equations in abstract Banach spaces and represent the solution in an integral form. Second, we will study several classes of nonlocal pseudo-parabolic equations and establish a sufficient condition for blowup of solutions in finite time. Here we do not limit that the initial energy is lower than the ground state energy of the related stationary equation any more. Finally, we will extend the conclusions derived for the explicit nonlocal pseudo-parabolic equations to the abstract nonlocal case. The achievements which would be obtained in this project may supply a gap in the study of nonlocal pseudo-parabolic equations and then promote the development for the pseudo-parabolic equations.

作为一类反应扩散方程,拟抛物方程是分析晶体半导体非静态过程的一个数学模型,当考虑该晶体所处的电场强度与通量密度的非局部联系时,该模型就成为一个非局部拟抛物方程.数学上,解的全局存在性与爆破性是反应扩散方程研究的一个重要内容.本项目旨在对具有一般非线性源项的非局部拟抛物方程建立判别解的全局存在性与爆破性的方法.为达到这个目的,我们将首先研究抽象拟抛物方程解的局部存在唯一性,与现有方法不同的是,我们将用椭圆方程的正则性估计与抽象Banach空间常微分方程理论来代替Galerkin方法且给出解具体的积分表达式.然后,我们将研究几类具体的非局部拟抛物方程,获得其解在有限时刻爆破的充分条件,这里不再要求方程的初始能量低于相应平衡态的基态能量.最后,将相关结论推广到抽象的非局部拟抛物方程.本项目将获得的研究成果将填补非局部拟抛物方程研究的一些空白,从而促进拟抛物方程的发展.

非局部拟抛物方程模拟了晶体半导体的非静态过程中电场强度与通量密度之间的非局部联系.本项目主要研究了一些具体的非局部拟抛物方程和相关的非局部椭圆问题,获得了关于解的全局存在性与爆破性的一些结论.特别地,对一类Kirchhoff型拟抛物方程建立了其解在有限时刻爆破的充要条件.对相关的非局部椭圆问题,一个重要的结果是,我们用变分方法刻画了椭圆传输问题的一个特征值,且利用该特征值分析了有界区域上超临界的Kirchhoff型传输问题规范化解的存在性.相应的结果为非局部拟抛物方程的研究提供了参考.项目基本实现了预期的目标,为后续的研究打下了坚实的基础.