Qualitative Study Of Mathematical Model Arising In The Nuclear Reactors

This paper mainly study the mathematical model arising in the nuclear reactors. Nuclear reactions plays an important role in the contemporary science and technology. In this paper, by introducing diffusion term and temperature negative feedback term, we present the new model, which will be closer to realistic model of nuclear reactions.In Chapter one, we state some backgrounds, methods and the main re-sults about linear and nonlinear partial differential equations. We introduced the reaction-diffusion equations from the nuclear reactor,such as development and some of the research results, and we introduce the diffusion and temperature of negative feedback term.We obtain an improved mathematical model (reaction-diffusion equations). Reaction-diffusion equations in biology, chemistry, physics, medicine and other disciplines have a wide range of applications, in particular, the following appeared in the nuclear reactor of a reaction-diffusion equations whereΩis a bounded domain of RN,N≥1,(?)Ωis smooth boundary, a,b,c are positive number, u denote neutrons,v denote the temperature of the nuclear reaction, b denote temperature positive feedback term,nis unit outward normal on(?)Ω.This model shows a nuclear reactor with external heat exchange(α≠0),or the reaction of insulation in the container with the outside world occurred in (α= 0).On partial differential equations (elliptic, parabolic) or equations the exis-tence, uniqueness, regularity of solutions of such issues have been one of people's attention and there are rich research results. In 1996, Gu Y.G., Wang M.X.study the following initial boundary value problem and obtain the existence of solutions and uniqueness. Nuclear reactions in the contemporary science and technology plays an important role, this paper we introduced diffusion term and temperature feedback term, which render the model more realistic model of nuclear reactions.We modify the model as follows: whereΩis a bounded domain of RN, N≥1, (?)Ωis smooth boundary. In Chapter two,in order to prove the main results of this paper, we give some basic knowledge including some definitions and theorems.In Chapter three, firstly, we study the existence of positive solutions to elliptic problem,and adopted the fixed point theorem on cone and some a priori estimates in order to get the existence of non-negative solutions of the steady-state reaction-diffusion equations. In general,to prove the existence of upper and lower solutions for elliptic equations we mainly use technology tools, such as degree theory. But we want to apply these tools to prove the existence of classical solutions of the steady-state problem(2). There are some difficulties. The first difficulty is the function u and von the border to zero, so we use upper and lower solutions in technology, we can not rule out the trivial solution existence. Secondly, it is not easy to get the priori estimates, therefore, the author use the fine analytical skills combined with a number of embedding theorem,the fixed point theorem on cone and prove non-trivial non-negative stationary solution of the problem (2).In section two, we study the existence of positive solutions of the problem (2).we improve the mathematical model of (2)(reaction-diffusion equations),convert the original problem into the equivalent system by applying Green's function,select the appropriate solution space,and prove the existence of nonnegative solutions to the integral system with the contraction mapping principle in Banach Space,then the problem (2) non-negative classical solution exist.Finally we obtain an asymptotic property of the nonnegative solutions of (Laplace) equation with (Dirichlet) boundary value by eigenvalue function,and (Gronwall)inequality. The main results are as follows: We mainly concern the problem: whereΩ∈RN is a bounded domain,whit smooth boundary (?)Ω,a,b,c,d,e is pos-itive constant,b is temperature positive feedback term,d is temperature negative feedback term.Theorem 1. Ifλdenotes the principal eigenvalue of Laplace with Dirichlet boundary value. N=2or 3 and a>(bc)/d,then the problem of(3)has at least a non-negative classical solution. whereΩC RN is a bounded domain,whit smooth boundary (?)Ω.Theorem 2.Let u0∈C(Ω),v0∈C(Ω).Then problem(4)has a unique non-negative classical solution.Theorem 3.Let(u,v)is the non-negative solution of problem(4).If the follow-ing conditions are satisfied, Then the solution of problem(4)satisfies the following estimates:...