Research On The Existence,Uniqueness And Stability Of Fractional Partial Differential Equations

The existence,uniqueness and stability of the solutions of the fractional partial dif-ferential equations are considered in this paper.Firstly,the following nonlinear fractional diffusion equations c0(?)t?u(x,t)=?u(x,t)+f(t,x,u),(x,t)? ? x(0,?),(1)u(x,0)= u0(x),x??(2)with boundary conditions(?)u/(?)v/=0,(x,t)?(?)?×(0,?),(3)and u(x,t)= 0,(x,t)?(?)2 ×(0,?)(4)are considered,respectively.The existence and uniqueness of the global classical solutions and the stability of the classical solutions of(1)-(2)-(3)or(1)-(2)-(4)are proved.Secondly,the following linear fractional partial differential equations u(x,t)= 0,(x,t)?(?)U ×(0,T],(6)u(x,0)= g(x),x?U(7)is considered.The existence and uniqueness of the weak solutions of(5)-(6)-(7)are proved.This paper is arranged as follows.In Chapter 1,the background of the fractional calculus and the recent research of fractional differential equations are given.Some preliminaries on the fractional calculus are studied as well.In Chapter 2,a class of nonlinear fractional diffusion equations with Dirichlet and Neumann boundary conditions are considered.Namely,(1)-(2)-(3)and(1)-(2)-(4)are studied.The existence and uniqueness of the global solutions of the equations are given.Lyapunov functions are used to prove the stability of the equations(1)-(2)-(3).The stability of the equations of(1)-(2)-(4)is proved by the property of the eigenvalues and the eigenfunctions of the elliptic equations.In Chapter 3,a class of linear fractional partial differential equations(5)-(6)-(7)is studied.The Galerkin approximation method and the energy estimate method,the fractional integration by parts,the fractional Gronwall inequality are used to prove the existence of the weak solutions of the equations.The fractional Gronwall inequality are also used to prove the uniqueness of the weak solutions of the equations.