Oscillation And Stability Of Impulsive Partial Differential Systems

Based on the fundamental theory of impulsive differential systems, employing Green formula, Gauss divergence theorem, Jensen inequality and Gronwall-Bellman inequality with impulse, we discuss the oscillation of impulsive partial differential systems by adopting reduction to absurdity and the stability in via of comparison theorem, respectively. Especially, we take an in-depth study on the oscillation of neutral impulsive parabolic systems and the stability of a class of nonlinear impulsive partial differential equations and obtain some useful conclusions. This essay is organized as follows:In the first chapter, we introduce the research significance of impulsive partial differential systems together with the current corresponding situation to this paper all over the word and present the problems and set an object for our study.Chapter 2 investigates the oscillation of impulsive partial differential systems without time delay. The research objects mainly fix on impulsive hyperbolic systems and impulsive parabolic systems. Section 2.1 studies the oscillation of impulsive hyperbolic systems under different boundary conditions and obtains some useful criteria via some kinds of second order impulsive differential inequality. Section 2.2 discusses the oscillation of impulsive parabolic systems and gets some conclusions by using first order impulsive differential inequalities.Chapter 3 emphasizes on the case that the considered systems contain not only impulse but also time delay. We mainly investigate impulsive parabolic systems with time delay and neutral impulsive parabolic systems. Section 3.1 discusses the oscillation of impulsive parabolic systems with time delay under Robin boundary condition and obtains some useful criteria via first order impulsive differential inequalities with time delay. Section 3.2 studies the oscillation of neutral impulsive parabolic systems under Neumann boundary condition and Robin boundary condition, respectively, and obtains some sufficient conditions for oscillation and strong oscillation via first order neutral impulsive differential inequalities.Chapter 4 discusses the stability of a class of nonlinear impulsive partial differential equations. The main idea is translating the stability of the considered nonlinear impulsive partial differential equation into that of the corresponding linear impulsive ordinary differential equation via Gronwall-Bellman inequality with impulse on the basis of comparison theorem. Some theoretical criteria for stability of nonlinear impulsive partial differential equations are given.Chapter 5 gives a premature explanation and vista by synthesizing the above analysis.