具有非局部初始条件的抽象发展方程解的存在性及渐近性态

本项目的主要目的是运用非线性分析的思想方法结合算子半群理论和抽象空间微分方程理论系统研究几类具有非局部初始条件的抽象发展方程解的存在性及渐近性态，从而完善和发展抽象发展方程非局部问题的研究。首先，从解决具有积分初始条件的抽象发展方程 mild 解的存在性这一公开问题入手，在获得充分的经验和感性认知后再过渡到对具有一般非局部初始条件的发展方程 mild 解的存在性的研究。其次，利用算子半群的相关性质结合某些积分不等式来研究抽象发展方程非局部问题解的渐近性态。最后，我们将致力于研究脉冲 (包括瞬时脉冲和非瞬时脉冲) 对抽象发展方程非局部问题所带来的影响。此外，本项目将特别注重所得抽象结果对具体的发展型偏微分方程的应用。

The main objective of this project is to systematically study the existence and asymptotic behavior of solutions for some kinds of abstract evolution equations with nonlocal initial conditions by applying the thought and method of nonlinear analysis combined with the theory of operator semigroups and differential equations in abstract spaces, thereby improve and develop the study of abstract evolution equations with nonlocal initial conditions. Firstly, starting from solving the existence of mild solutions for a class of evolution equations with integral initial conditions, which is an open problem, and then transition to study the existence of mild solutions for the evolution equation with more general nonlocal initial conditions after get sufficient experience and perceptual cognition. Secondly, we investigate the asymptotic behavior of the solutions for nonlocal problems of abstract evolution equations by utilizing the related properties of operator semigroups and some integral inequalities. Lastly, we will be dedicated to study the effects of impulses (including instantaneous impulses and not instantaneous impulses) to nonlocal problems of abstract evolution equations. In addition, this project will focus particularly on the abstract results are applied to the concrete evolution partial differential equations.