The Existence And Uniqueness Of The Global Solutions And The Existence Of The Golbal Attractor With Periodic Boundary For The Fitzhugh-nagumo Equations

Hodgkin-Huxley (HH) models is proposed by Hodgkin and Huxley through the study of nerve impulse propagation in nerve fibers in1952, it is the differential equations which used to describe the relationship between the membrane potential and membrane currents of neuron axons. The equations are a classic model of excitable cell. The FHN model is a simplification of the HH model. In recent years, a lot of research has been done in this type of model, but the study of nerve impulse conduction equations with external stimulation are seldom, It not only has the great significance in theory, but also has the important practical background.The paper first discusses the initial value problem with the external stimulation periodic boundary for the Fitzhugh-Nagumo nerve impulse conduction equations, that is, the following equations under the periodic boundary and the initial conditions We give the proof of the existence and uniqueness of the global solutions.The paper second discusses the external stimulation coupled FHN equations, the following under the periodic boundary and the initial conditions We give the proof of the existence and uniqueness of the global solutions and the existence of the global attractor.Where is a known constant, f(u),f(u1),f(u2) are the known real functions, u,u1,u2are the electric potential of the nerve fibers, v,v1,v2relate to recovery mechanisms of the cell membrane,I1,(i=1,2,……6) are external stimulus.In particular, studies are as follows:1.We made simple sum-up and comment on the developing and actuality study on the Fitzhugh-Nagumo nerve impulse conduction equations and the coupled FHN equations relevant with this paper;2.We give some important definitions and lemmas,and the right part of the text symbol is explained;3.Using Galerkin method to prove the existence and uniqueness of the global solutions for the problems(1)-(4) and(5)-(10),and prove the existence of the global attractor for the problems (5)-(10);4.We make some prospects of the FHN nerve impulse conduction equations research.