Solvability Of Nonlinear Time-Space Fractional Reaction-Diffusion Equations

In this thesis,we discuss existence,uniqueness,blowup alternative result and Mittag-Leffler-Ulam-Hyers stability of mild solutions of the initial value problem and existence,uniqueness and Mittag-Lefiler-Ulam-Hyers stability of mild solutions of nonlocal problem for nonlinear time-space fractional reaction-diffusion equations with fractional Laplacian,where Ω? RN is a bounded open domain with smooth boundary ?Ω,cDtα is the Caputo time-fractional derivative of order α ∈(0,1).(-△)β is fractional Laplacian with 0<β≤1.The constant T>0 and 0<t1<t2<…<tp<T.p ∈ N and ck≠0(k=1,2,…,p)are real numbers.Firstly,we obtain the local existence and uniqueness of mild solutions of the initial value problem for time-space fractional reaction-diffusion equations by using the principle of power contraction mapping in fractional Sobolev space.On this basis,the existence and alternative result of saturated mild solutions are discussed.Secondly,by using Schauder fixed point theorem in fractional Sobolev space,we obtain the existence and uniqueness of mild solutions of time-space fractional reaction-diffusion equations with nonlocal condition.Thirdly,we discuss the Mittag-Leffler-Ulam-Hyers stability of mild solutions of the initial value problem and the nonlocal problem for time-space fractional reactiondiffusion equations.Finally,by using the monotone iterative method of upper and lower solutions in ordered Banach space,we obtain the existence and uniqueness of mild solutions for the time-space fractional reaction-diffusion equations with nonlocal condition.