Studies On Properties Of Solutions Of Delayed Reaction-diffusion Equations And Navier-Stokes Equations

Reaction-diffusion equations and Navier-Stokes equations are widely applied in many fields such as physics, chemistry, biology, economy and many engineering problems. However, we remark that the phenomena of time delays can not be controlled by linear reaction-diffusion equations in chemistry, biology and population dynamics and some other fields. In the thesis, we give a new kind of model—delayed reaction-diffusion equations—to study the above phenomena, and investigate the dynamical properties of the solution of the delayed reaction-diffusion equations when the diffusion coefficients are small enough and analyze the effect of diffusion and time delay, also, we prove that the traveling wave solution exists under some conditions, so we conditionly solve the open problem on the existence of traveling wave solution of the delayed reaction-diffusion equations. At the same time, we note that many engineering problems need be controlled by Navier-Stokes equations with small parameters, however, it is very difficult since there are many open problems in the basic theoretical aspect and the computational aspect, so we give a new stabilized FEM—stabilized Multiscale FEM—to study the numerical solutions of Stokes equations and Navier-Stokes equations, and prove that it satisfies inf-sup condition and get the optimal error. Also, the numerical results show that our numerical method can work for Navier-Stokes equations even if the viscosity coefficient is very small.The thesis consists of six chapters. In Chapter 1, we give the background and significance of the research, and we draw a coclusion in the last chapter. The main results achieved are detailed in Chapters 2-5 as follows:In Chapter 2, we firstly derive a new kind of delayed reaction-diffusion equations which describes a two-species predator-prey system with diffusion terms and stage structure. Secondly, by using the linearized method and the method of upper and lower solutions, we investigate the local and global stability of the constant equilibria, respectively, and consider boundedness and existence of the local solution by using the positive lemma. Also, we give the long time behavior of solution of the delayed reaction-diffusion equations with small parameters, and analyze the effect of diffusion and time delay. The results show that the free diffusion of the delayed reaction-diffusion equations has no effect on the populations when the diffusion is too slow; otherwise, the free diffusion has certain influence on the populations, however, the influence can be eliminated by improving the parameters to satisfy some suitable conditions.In Chapter 3, we derive a new kind of nonlocal reaction-diffusion equations, then, we study the stability of equilibria and the existence of traveling wave solutions of the singular and nonsingular nonlocal reaction-diffusion equations, we prove that the traveling wave solution of the nonsingular and singular nonlocal reaction-diffusion equations exists under some conditions, the key idea of our proof is to seek the system uniformly approximated to the original system under some conditions and prove that the new system has at least one positive solution by using the method of upper and lower solutions, so we partly solve the open problem on the existence of traveling wave solutions of the delayed reaction-diffusion equations.In Chapter 4, we investigate the numerical solution of 2D or 3D Stokes equations by improving stabilized FEM with bubble functions and give a new stabilized finite element method--stabilized Multiscale FEM, then, we consider the inf-sup condition of the new method for P1 ? P0element, and obtain the optimal error, i.e., achieve O ( h ) in H 1-norm and O ( h 2) in L2 -norm.In Chapter 5, we study the numerical solution of Navier-Stokes equations by using the stabilized Multiscale FEM, and give the inf-sup condition of the new method for P1 ? P0 element and nonconforming P1 ? P0 element, and obtain the optimal error. The numerical results show that the method can work for Navier-Stokes equations even if the viscosity coefficient is very small...