Local Existence for the Cauchy Problem of a Reaction-Diffusion System with Discontinuous Nonlinearity

摘要

The most famous model for nerve conduction is due to Hodgkin and Huxley. However, a mathematical analysis of their model has proven very difficult. The complexity of the Hodgkin and Huxley model has led a number of other authors to introduce simpler models. In this report we consider one such simplification. It has been demonstrated (experimentally) that impulses in the nerve axon travel with constant shape and velocity. Mathematically, this corresponds to traveling wave solutions. A number of authors have proven that the mathematical equations considered here do possess traveling wave solutions. Another property of impulses in the nerve axon is the existence of a threshold phenomenon. This corresponds to the biological fact that a minimum stimulus is needed to trigger an impulse. Here we prove some preliminary results which will be used in a later report when it is demonstrated that the equations under study do indeed exhibit a threshold phenomenon.