The Well-posedness Of Solution For Several Kind Of Fractional Differential Equations And The Study Of Infinite-dimensional Dynamical Systems

In this paper, we study the impulsive fractional differential equations, thefractional differential long-short wave equations, a class of fractional Schr dingerequations and the fractional random Ginzburg-Landua equations. We prove theexistence of mild solution for impulsive hybrid fractional differential equations and theexistence and uniqueness of piecewise continuous asymptotically periodic solution forimpulsive fractional differential equations. We prove the existence of global smoothsolution for the fractional long-short wave equations and the existence of uniformattractor for the fractional non-autonomous long-short wave equations. We prove theexistence and stability of standing waves for2-coupled nonlinear fractionalSchr dinger system. At last, we prove the existence of random attractor for fractionalGinzburg-Landau equations with additive noises. The content studied in the paperclosely contacts of physics, biology, stochastic analysis and so on. It has veryimportant theoretical significance and practical application value.The paper has seven chapters.In chapter1, we give the important background, development and application ofstudied equations. We also recall some known results and describe the ground work inthe paper.In chapter2, we study the the existence of mild solution for impulsive hybridfractional differential equations. Firstly, we prove the theorem of existence of mildsolution for fractional hybrid differential equations using the Dhage fixed pointtheorem. We give the actual example for application and value of the existencetheorem.In chapter3, we study existence and uniqueness of asymptotically periodic mildsolution for impulsive fractional differential equations. We prove the existence anduniqueness by the fixed point theorem.In chapter4, we consider the fractional differential long-short wave equationswith initial and periodic boundary conditions. We prove the existence of the globalsmooth solution for it using uniform priori estimates and the Gal rkin method.In chapter5, we study the fractional non-autonomous long-short wave equationswith initial and periodic boundary conditions. Firstly, we get uniform priori estimates by using the Gronwall, the Sobolev, the Young and the fractional order calculusinequalities. Secondly, we prove the existence and uniqueness of the solution for it byGal rkin method. Lastly, we prove the existence of uniform attractor for it using thetheory of uniform attractor for non-autonomous dynamical systems.In chapter6, we study the2-coupled nonlinear fractional Schr dinger system withinitial conditions. We prove the existence and stability of standing waves for it by themethod of concentration-compactness and commutator estimates.In chapter7, we study the long time behavior of the solutions for the fractionaldifferential Ginzburg-Landau equations with initial and periodic boundary conditions.We prove the existence of it by the standard method.