Fractional Laplacian And Some Nonlinear Fractional Diffusion Equations

In this paper, we mainly study the problem of fractional Laplacian. Firstly, when s E (1,2) we study the fundamental solution and Possion core of (-△)s, We are based on the fundamental solution of the integer order Laplace operator, give the fundamental solutions of fractional Laplace operators, and prove one kind of situation. Secondly, study the problem of nonlinear fractional diffusion equations (-△)su=λg(u) (s ∈ (0,1)), there have been a lot of studies on the diffusion problem of these equations. In this paper, based on the fractional order Kato’s inequality, give the existence of solutions and the blow up phenomena in finite time.The structure of this paper is as follows:The first chapter, we introduce the research background of the fractional Laplacian and the main results of this paper; The second chapter, we introduce the definition and theorem,which we will use in the paper; The third chapter, proofs the fundamental solution of the fractional Laplacian,and give the existence of solutions and the blow up phenomena in finite time by Kato’s inequality.