Study On The Solutions Of Two Kinds Of Fractional Evolution Equations

In recent years,mathematical models with fractional differential operator and Kirchhoff functions have attracted the attention of many scholars in pure mathematical theory and practical applications.This type of equation has very important applications in physics,finance,population dynamics,fluid mechanics and other fields.Because the fractional differential operator is nonlocal,the study of fractional evolution equation is more challenging than that of integer evolution equation.This paper mainly applies functional analysis theory,combines Galerkin method,contraction mapping principle,concave function method,and some differential inequality techniques to study some dynamic properties of solutions for a class of fractional order pseudo-parabolic equation with singular potential and a class of p-Kirchhoff type equation with damping and source terms,respectively.In the second chapter of this paper,we investigate a class of fractional order pseudo parabolic equation with singular potential.This chapter uses the Galerkin method and the contraction mapping principle to obtain the local existence and uniqueness of the solution of the equation.Subsequently,the blowup of weak solutions under two different assumptions was discussed using the concave function method and some differential inequalities.The results of the blowup of weak solutions in finite time were obtained,and the upper and lower bounds of the blowup time were estimated.These results extend the results obtained in existing literature.In the third chapter of this paper,we investigate a class of p-Kirchhoff type equation with damping and source terms.Firstly,the Galerkin method combined with potential well theory was used to obtain the global existence of solutions for the equation at subcritical and critical initial energy levels.Secondly,the upper and lower bounds on the blow up time of weak solutions were estimated using the concave function method and some differential inequality techniques.Finally,the asymptotic behavior of the solution under certain assumptions was discussed,and results were obtained that decay exponentially.