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This thesis focuses on the weak and general classic solutions for the linear, quasilinear and nonlinear fractional reaction-diffusion equations. It is composed of four chapters. The first chapter contains an brief introduction to background of the fractional calculus, devel-opment and research method of fractional reaction-diffusion equations and the main works. The second chapter presents some elementary knowledge about the fractional calculus and Sobolev spaces. In the third chapter, we show the existence and estimates of the weak solu-tions for quasilinear fractional reaction-diffusion equations. In the last chapter, the existence and uniqueness of the solutions for the nonlinear fractional reaction-diffusion equations are presented.In chapter 1, we firstly introduce the background of the fractional calculus. Further-more, we show development and research method of the fractional reaction-diffusion equa-tions. Finally, we introduce our main results briefly.In chapter 2, we cite and introduce some elementary knowledge about fractional calcu-lus and some abstract Sobolev spaces. Firstly, we introduce the fractional calculus, such as Riemann-Liouville fractional derivatives, Caputo fractional derivatives, their properties, and etc. Specially, some special function:Mittag-Leffler function which is very useful when deal with fractional differential equations are also given. Then we introduce the abstract space, such as Hl(Ω), H10(Ω), and etc.In chapter 3, we obtain the existence and estimates of weak solutions for the quasilinear fractional reaction-diffusion equations by the Rothe approach. We use simple case to illustrate how Rothe approach work on it. At last, we discuss a more general case.In chapter 4, we use the upper-lower solution method to study the nonlinear fractional reaction-diffusion equations With the help of monotone iterative technique, we obtain the existence and uniqueness results for the nonlinear case. At the last, we give two examples to illustrate the results.
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