The Well-posedness Of The Stochastic Fractional Partial Differential Equations

This thesis is devoted to the well-posedness of the mild solution for a stochastic fractional delayed reaction-diffusion equation driven by Lévy space-time white noise, a stochastic space fractional partial differential equation driven by Lévy space-time white noise, a stochastic space fractional partial differential equation driven by fractional Brownian motion, a stochastic fractional reaction-diffusion equation driven by fractional Brownian motion on bounded domain and a stochastic space fractional partial differential equation driven by both fractional Brownian motion and Lévy pure jump noise. In addition, the well-posedness and random dynamics of the solution for a stochastic impulsive reactiondiffusion equation driven by additive white noise and a stochastic impulsive reactiondiffusion equation driven by fractional Brownian motion have been studied. The thesis contains four chapters.In Chapter 1, we first introduce the physical background, status and latest advanced in the fractional differential equations and differential equations with impulse. Then we present the definition and conclusions of the space fractional operator and fraction Green kernel, the Lévy space-time white noise, infinite dimensional fractional Brownian motion and random dynamical system. In the last, our main results are stated.In Chapter 2, the proper working function space is constructed for the stochastic fractional delayed reaction-diffusion equation driven by Lévy space-time white noise. The existence, uniqueness, time regularity and space regularity of the mild solution are obtained.The results show that the regularity of initial value and the order of fractional operator can effect the regularity of both time regularity and space regularity of the mild solution. Then a proper working function space is constructed for the stochastic space fractional partial differential equation driven by Lévy space-time white noise. The existence, uniqueness,time regularity and space regularity of the mild solution are proved. The results show that the regularity of initial value, the order of fractional operator and the order of partial derivatives can effect the regularity of both time regularity and space regularity of the mild solution.In Chapter 3, we prove the existence, uniqueness, time regularity and space regularity of the mild solution for the space fractional partial differential equation driven by fractional Brownian motion. The results show that the time regularity and space regularity of the mild solution depend on the regularity of initial value, the Hurst parameter,the order of fractional operator and the order of partial derivatives. Further, the well-posedness of the mild solution for the fractional reaction-diffusion equation driven by fractional Brownian motion is proved on the bounded domain. Then a stochastic space fractional partial differential equation driven by both fractional Brownian motion and Lévy space-time white noise is studied due to the high frequency finance datum. The existence, uniqueness, time and space regularity of the mild solution are shown. The results show that the regularity of initial value, the order of fractional operator and the order of partial derivatives can effect both time regularity and space regularity of the mild solution.In Chapter 4, we respectively study the the long time behavior of the stochastic impulsive reaction-diffusion equation driven by additive white noise and the stochastic impulsive reaction-diffusion equation driven by fractional Brownian motion. The existence and uniqueness of the solutions for the two equations are proved respectively. Under the condition of the finite fixed moment of impulse time, the impulsive effect is transformed into the initial data to construct a new stochastic asymptoic equation.The existence of random attractors for the random dynamical systems generated the new stochastic equation is proved respectively. In the last, we present some discussions on the results and consider a more general assumption for the impulse.