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# Research for Solutions of Several Fractional Differential Equations

Author: ZhangQuanGuo
Tutor: SunHongRui
School: Lanzhou University
Course: Basic mathematics
Keywords: Fractional differential equation Kirchhoff type problerrn Positive solution Critical point Vairational method iterative technique Mountain pass theorem Abstraet time fractional differential equation C0semigroup Analytie semigroup Mild solution Time fractional diffusion equation Cauehy problem Global existence Blow-up Fujita exponent
CLC: O175
Type: PhD thesis
Year: 2013
Quote: 0

### Abstract

 This thesis considers some problems of the fractional differential equation. First, we study the existence of solution of the fractional difierential equation in RN Second, we consider the abstract time fractional differential equation, some properties of the mild solution and the relation between classical solution and mild solution are obtained. Last, we investigate the global existence and blow-up of the solution of Cauchy problem for the time fractional nonlinear differential equation and system.In Chapter1, we firstly introduce the background and applications of frac-tional calculus, then state the main results of this thesis. Finally, we list the definition of fractional integral, derivative and their properties, fundamental prop-erties of some special functions are also given.In Chapter2is devoted to studying the existence of solution for two classes of nonlocal problems. In Section1, under the Borel probability measure M be-ing symmetric and nonsymmetric, we consider the existence of solution for the fractional differential equation in RN by using the mountain pass theorem and iterative technique. In Section2, for Kirehhoff type problem with0-Dirichlet boundary condition with one parameter and two parameters, the existence of the positive solution when the parameter is small are investigated, the main tools are mountain pass theorem combining with iterative technique.In Chapter3, we study the abstract time fractional Cauchy problem in Banach A’. First, for the linear problem CODαfu-Au=f(t), t>0,u(0)=u0, under the assumption that A is a infinitesimal generator of a C0semigroup or an analytic semigroup {T(t)}, we can define the operators {Pα(t)} and {Sα(t)} and analyze their fundamental properties. In addition, we get, the relationship between classical solution and mild solution and study the regularity of mild solution. Second, for the semilinear problem C0Dαtu-Au=f(t,u), t>0, u(0)=u0, under the assumption that, A is a infinitesimal generator of an analytic semigroup {T(t)}, we give the relationship between classical solution and mild solution in A’" and investigate the existence and unique of solution and the continuous dependence on initial data and parameters. Those properties are important to study the specific time fractional equation in next Chapter.In Chapter4, we investigate the blow-up and global existence of solutions of Caucliy problem for the time fractional differential equation and system. Based on the properties of t he abstract time fractional equation in C0(RN) which are proved in Chapter3, and using the test function methods and the contraction mapping principle, we consider the blow-up and global existence of solutions of Caucliy problem for the time fractional equation. Fujita exponent is determined. In ad-dition, the spatio-temporal equation and time fractional system Cauchy problem are discussed respectively. Similarly, Fujita exponent is determined.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations