A contribution to the analysis of time fractional stochastic functional partial differential equations and applications

Resumo

On the one hand, the classical heat equation∂tu= ∆udescribes heatpropagation in a homogeneous medium, while the time fractional diffusionequation∂αtu= ∆uwith 0< α <1 has been widely used to model anoma-lous diffusion exhibiting subdiffusive behavior. On the other side, when weconsider a physical system in the real world, we have to consider some in-fluences of internal, external, or environmental noises. Besides, the wholebackground of physical system may be difficult to describe deterministical-ly. Therefore, in this thesis, we will construct three models to show theapplications of the time fractional stochastic functional partial differentialequations.In Chapter 2, we study a stochastic lattice system with Caputo fractionalsubstantial time derivative, the asymptotic behavior of this kind of problemis investigated. In particular, the existence of a global forward attractingset in the weak mean-square topology is established. A general theorem onthe existence of solutions for a fractional SDE in a Hilbert space under theassumption that the nonlinear term is weakly continuous in a given sense isestablished and applied to the lattice system. The existence and uniquenessof solutions for a more general fractional SDEs is also obtained under aLipschitz condition.In Chapter 3, the local and global existence and uniqueness of mild solu-tions to a kind of stochastic time fractional impulsive differential equationsare studied by means of a fixed point theorem, and with the help of theproperty ofα-order fractional solution operatorTα(t) and the resolvent op-eratorSα(t). Moreover, the exponential decay to zero of the mild solutionsto this model is also proved. However, the lack of compactness of theα-order resolvent operatorSα(t) does not allow us to establish the existenceand structure of attracting sets, which is a key concept for understandingthe dynamical properties.Therefore, the second model of Chapter 3 is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolutionequations with time fractional differential operatorα∈(0,1). After estab-lishing the well-posedness of the problem, and a result ensuring the existenceand uniqueness of mild solutions globally defined in future, the existence ofa minimal global attracting set is investigated in the mean-square topology,under general assumptions not ensuing the uniqueness of solutions. Further-more, in the case of uniqueness, it is possible to provide more informationabout the geometrical structure of such global attracting set. In particular,it is proved that the minimal compact globally attracting set for the solution-1 s of the problem becomes a singleton. It is remarkable that the attractionproperty is proved in the usual forward sense, unlike the pullback conceptused in the context of random dynamical systems, but the main point is thatthe model under study has not been proved to generate a random dynamicalsystem.Chapter 4 is devoted to the well-posedness of stochastic time fractional2D-Stokes equations of orderα∈(0,1) containing finite or infinite delay withmultiplicative noise is established, respectively, in the spacesC([−h,0];L2(Ω);L2σ)) andC((−∞,0];L2(Ω;L2σ)). The existence and uniqueness of mild so-lution to such kind of equations are proved by using a fixed-point argument.Also the continuity with respect to initial data is shown. Finally, we con-clude with several comments on future research concerning the challengingmodel: time fractional stochastic delay 2D-Navier-Stokes equations withmultiplicative noise.