  The initial value problem P0, in all of the space, for the spatio - temporal
FitzHugh - Nagumo equations is analyzed. When the reaction kinetics of the
model can be outlined by means of piecewise linear approximations, then the
solution of P0 is explicitly obtained. For periodic initial data are possible
damped travelling waves and their speed of propagation is evaluated. The
results imply applications also to the non linear case.
