Comparison Principle And Dynamical Properties Of Reaction-Diffusion Systems

In this dissertation, the comparison principle and dynamical properties to reaction-diffusion systems in bounded domain of R~N or in unbounded domain of R~N are studied, and the existence and uniqueness to Cauchy problems for semilinear evolution equations with almost sectorial operators and with nonlocal initial conditions are also investigated. It consists of five chapters.We present research background and main results of this dissertation Chapter 1.In the first part of Chapter 2, we study the comparison principle for semi-linear reaction-diffusion systems with nonlocal boundary conditions and with time delays, and for fully nonlinear reaction-diffusion systems with nonlocal boundary conditions, respectively. The comparison principle of nonquasimono-tone reaction-diffusion systems in unbounded domains of R~N is established in the second part of Chapter 2. As a by-product we also prove the uniqueness of solutions of these systems.In Chapter 3, we first, using comparison principle established in Chapter 2 in combination with monotone iteration methods and Bootstrap technology, investigate the existence of periodic solutions and asymptotic behavior of time-varying solutions for semilinear periodic-reaction-diffusion systems with nonlocal boundary conditions and with time delays in bounded domain. Under some general hypotheses on Kernel functions and others, we show that periodic solutions of such system is locally asymptotically stable. Second, we consider a nonquasimonotone reaction-diffusion system in unbounded domains of R~N. The global existence and boundedness of solutions are established, and some sufficient conditions on the nonlinearities are presented, which ensure the positively Ljapunov stability of the zero solution with respect to H~2—perturbations by using theory of operator semigroups and methods of dynamical systems.In Chapter 4, we prove the existence and uniqueness of mild and classical solutions to Cauchy problem of abstract operator differential equation with...