Study On The Extinction And Blow-up Of Solutions For Several Kinds Of Fractional Diffusion Equations

In recent years,the fractional Laplace operator and the fractional diffusion equation has been widely used in different fields,such as sparse obstacles,financial mathematics,the layered materials,anomalous diffusion,population dynamics and the game theory and so on.Fractional Laplace operator is quasi differential operator,the local characteristics,and thus to describe with memory and genetic material provides valuable methods.Based on the fractional Laplace operator as well as broad application background of the fractional diffusion equation,this paper studies two kinds of the extinction and blow-up properties of solutions of the fractional diffusion equations,As well as the existence of the ground state solution of the corresponding static equations,the specific research contents are as follows:In Chapter 3,the extinction and non-extinction properties of the solutions of a class of the fractional p-Kirchhoff equations are studied.Firstly,the classical Gagliardo-Nirenberg inequality is extended to estimate the attenuation of the solution of the equation.Then,a sufficient condition of solutions which vanish in finite time is given by employing the fractional Gagliardo-Nirenberg inequality,the first eigenvalue of fractional p-Laplacian and differential inequalities techniques.Moreover,the decay estimates of solutions are obtained.Finally,under some suitable assumptions on the initial energy functional and the Kirchhoff function M,a sufficient condition for the solutions which can't vanish is obtained.In Chapter 4,the global existence and blow-up property of solutions for a class of degenerate Kirchhoff diffusion equations with logarithmic nonlinear term are discussed.On one hand,the existence of ground states solutions for the corresponding stationary Kirchhoff equations is obtained by restricting the discussion on Nehari manifold and analyzing the fibering map.On the other hand,under suitable assumptions,a sufficient condition of extinction at infinity and finite time blow-up of solutions is given by deploying potential well theory.