Researching On The Global Solutions Of Initial-boundary Value Problem For The Nerve Conduction Equations

Biological tissue is made up of cells, it gives certain reaction when the surrounding environment changes. In the biology, the characteristic of biological tissue is called physiology excitability, and the motivation of reaction is called stimulation. In the process of the study about the nervous system and muscle activity, the bioelectricity is discovered first, with the study of bioelectricity and then started on the research of discharge activity of the nervous system. In which, the Hodgkin-Huxley equations and Fitzhugh-Nagumo equations are well known in the nerve conduction equations. In recent years, a lot of research about the nerve conduction equations have been done, but the study about the Fitzhugh-Nagumo equations with external stimulation and the mixed equations of nerve conduct and nonlinear wave are seldom, so the main works of this paper are the two aspects, it has not only the important practical background, but also great significance in theory.In this paper, we first prove the existence and uniqueness of the global solutions and the existence of the global attractor for the Fitzhugh-Nagumo equations with the external stimulus ut＝uxx＋f(u)-v＋f1(x,t) vt=σu-γv+f2(x,t) with the initial conditions u(x,0)=u0(x),v(x,0)=v0(x). and under the inhomogeneous boundary conditions u(x,t)|(?)Ω＝g(t) where σ,γ,a are constants such thatσ>0,γ>1/2,a>1/2,f(u) is real function, x∈Ω, let Q be the bounded open interval of R, u is the characteristic of transmembrane voltage, v is a slow process of potassium activation and sodium inactivation, f1(x,t),f2(x,t) are the external stimulus.Secondly, we discuss the existence and uniqueness of the global solutions for the mixed equations of nerve conduct and nonlinear wave with periodic boundary utt-△ut,=-f(u)ut+g(x,t,u,(?)u,ut,(?)ut)+(?)σ((?)u) u(x,0)=u0(x),ut(x,0)=u1(x) u(x+2D,t)=u(x,t)(t∈[0,T], where T>0is fixed arbitrarily. Let Ω=[0,2D] be the bounded interval of R).The specific contents will go as follows:First, the paper introduces the present situation of the nerve conduction equations at home and abroad and the research contents of this paper;Second, we give the important definitions and lemmas about this paper;Third, we discuss the existence and uniqueness of the global solutions and the universal attractor existence for the Fitzhugh-Nagumo equations with the external stimulus;Fourth, we discuss the existence and uniqueness of the global solutions for the mixed equations of nerve conduct and nonlinear wave with periodic boundary;Fifth, we summary the contents of this paper and make some prospects of the nerve conduction equations in the future.