Global Existence And Blowup Of Solutions For Fractional Porous Medium Equations

In recent years,existence and blow-up of solutions for fractional porous medium equations is a hot research topic,and it is also an important part of the theoretical study of fractional partial differential equations.This paper mainly studies the properties of the solutions for the fractional porous medium equation with spatial derivatives.One is the global existence and blow-up of the solutions for the fractional porous medium equation with singular potential term,and the other is the blow-up of the solutions for the Dirichlet initial boundary value problem of the fractional porous medium equation with power function reaction term.Due to the non-locality of the fractional Laplacian operator,the Caffarelli-Silvestre extension method is used to equivalently convert the original problem of non-locality into the definite problem of the local elliptic equation with dynamic boundary conditions.Thus,the global existence and blow-up of the solution for the original problem are equivalent to the global existence and blow-up of the solution for the definite problem of the local elliptic equation with dynamic boundary conditions.For the fractional porous medium equation with singular potential term,when the nonlinear exponent and the initial value satisfy a certain range,the global existence of the solution is proved.On this basis,the decay estimate and long time asymptotic behavior of the global solution of the equation and the blow-up behavior of the local solution are obtained by using the concave function method and the potential well.For the fractional porous medium equation with power function reaction term,based on the existence of global solution,the blow-up of the solution is proved by using the concave function method when the energy functional of the equation satisfies certain conditions.By the uniform boundedness of the global solution,the long time asymptotic behavior of the solution is obtained.The existence and blow-up theory of the solution for the fractional porous medium equation obtained in this paper provides mathematical theoretical support for the study of practical problems in practical life,and paves the way for the study of the related properties of other fractional partial differential equations.The paper has 1 pictures,and 69 references.