脉冲抽象随机偏微分方程的伪概周期型解研究

The project will first introduce the concept of piecewise p-th moment pseudo almost periodic functions, piecewise p-th moment weighted pseudo almost periodic functions, piecewise p-th moment Stepanov almost periodic functions, piecewise p-th moment Stepanov weighted pseudo almost periodic functions, piecewise p-th moment h-type -Stepanov weighted pseudo almost periodic functions, piecewise p-th moment ∞-type-Stepanov weighted pseudo almost periodic functions and piecewise distributional pseudo almost periodic type functions, and discuss the completeness of the function space, establish the corresponding composite theorem. Then by using the semigroup theory, evolution systems theory and the stochastic analysis theory , to study the piecewise p-th moment pseudo almost periodicity and piecewise distributional pseudo almost periodicity of first-order linear impulsive abstract stochastic partial equations. By focusing on the new composite properties and new constraints conditions of perturbation functions, and applying the fixed point theorem. To study the existence, uniqueness and stability of piecewise p-th moment pseudo almost periodic type mild solutions and piecewise distributional pseudo almost periodic type mild solutions for several first-order impulsive abstract stochastic partial differential equations in separable Hilbert spaces . This study will enrich and develop the almost periodic functions theory and the piecewise stochastic pseudo almost periodic function theory, to promote the further development of the theory of almost periodicity and piecewise pseudo almost periodicity for impulsive abstract stochastic differential equations and other disciplines.

本项目将首先引入逐段p-阶矩的伪概周期函数、加权伪概周期函数、Stepanov伪概周期函数、Stepanov加权伪概周期函数、h型-Stepanov加权伪概周期函数和∞型-Stepanov加权伪概周期函数，以及逐段依分布伪概周期函数类的概念, 并讨论其函数空间的完备性, 建立相应的复合定理。接着利用半群理论、发展系统理论和随机分析的理论, 展开一阶线性脉冲抽象随机偏微分方程的逐段p-阶矩伪概周期性及逐段依分布伪概周期性的研究。通过着重研究非线性扰动函数所满足的新的复合性质和新的限定条件，以及不动点定理的应用。来研究可分Hilbert空间中几类一阶脉冲抽象随机偏微分方程逐段p-阶矩伪概周期型mild解与逐段依分布伪概周期型mild解的存在性、唯一性与稳定性。本研究将丰富和发展随机概周期函数和逐段伪概周期函数理论，以促进脉冲抽象随机方程概周期性与和逐段伪概周期性理论及其它学科的进一步发展。

脉冲随机偏微分方程综合了脉冲扰动和随机扰动对系统的影响, 因此，对这类系统的伪概周期性质的研究具有重要的理论和现实意义。本项目通过引入逐段p-阶矩的伪概周期函数、加权伪概周期函数、Stepanov伪概周期函数、Stepanov加权伪概周期函数、h型-Stepanov加权伪概周期函数和∞型-Stepanov加权伪概周期函数，以及逐段依分布伪概周期函数类的概念, 研究了一阶线性脉冲抽象随机偏微分方程的逐段p-阶矩伪概周期性及逐段依分布伪概周期性。进而应用新的复合定理、算子理论、发展系统理论和随机分析理论以及不动点定理，获得了可分Hilbert空间中几类一阶非线性脉冲抽象随机偏微分方程逐段p-阶矩伪概周期型温和解与逐段依分布伪概周期型温和解的存在性与稳定性，包括有限区间上的一些特殊性结果。所以本研究丰富和发展了逐段伪概周期函数理论，为进一步探讨脉冲抽象随机偏微分方程伪概周期性理论奠定了坚实的工作基础。项目资助已发表论文16篇，其中SCI收录14篇。项目投入经费36万元，支出经费18.2283万元。剩余经费15.9717万元，将用于本项目研究的后续支出。