Some Properties Of Solutions For Several Reaction Diffusion Equations

This thesis mainly studies some properties of the solutions for three classes of reaction diffusion systems, including global existence, blow-up and so on.There exists kinds of the reaction diffusion phenomena in both daily life and scientific research, which causes many scholars’ interest. After development, under the continuous efforts and work of many scientists, the research about the diffusion system has become an important branch of the partial equation theory. In recent years, scholars in the different fields like biology, medicine, phase transitions, traveling wave, image processing and so on have been attracted in the nonlocal diffusion systems. Nonlocal diffusion model are also applied widely in these areas. On the other hand, the biological chemotaxis model of reaction-diffusion systems also attracts the interest of many scholars, Keller and Segel proposed the basic system, and many scholars have improved the model of biological chemotaxis later and do many depth research, they also get many meaningful results.This thesis is consisted of five chapters:the first two chapters are the introduction and some basic knowledge. In the introduction, the background and current research for the three reaction diffusion equations are given. In the second chapter, some basic knowledge to be used in this thesis is given. The following three reaction diffusion systems are discussed in the next three chapters separately, they are nonlocal-diffusion system under the Cauchy condition, nonlocal p-Laplacian reaction diffusion system with the Neumann condition and chemotaxis system which is parabolic-parabolic.The third chapter considers the global existence and blow-up properties of the solutions for nonlocal diffusion system, also the critical Fujita exponent and the upper time of blow-up are given. By using the contradicted method, the sufficient conditions of blow up in finite time of solutions are obtained. And then the blow-up time is given by establishing the lower solutions, that is the solutions will blow up in the finite time when the initial values are under some conditions.The forth chapter takes nonlocal p-Laplacian diffusion system into consideration, which is with both reaction term and absorb source. The global existence and blow-up properties of solutions are considered. By using the comparison principle and the methods of establishing the upper and lower solutions, it is able to prove the conditions of global existence and blow-up.The fifth chapter is about the chemotaxis system. It is mainly considered the global existence and boundedness of solutions of this system, the model is used to describe the gradient movement of the bacteria into the oxygen. Several important properties of semi-group are used, which include the estimates of the sector operator and embedding theorem, as well as several important inequalities including Gagliardo-Nirenberg and Yong inequalities. It is proved that the under the corresponding initial-boundary value problem, the system possesses a unique global but also uniformly bounded solution.