  The paper deals with the explicit calculus and the properties of the
fundamental solution K of a parabolic operator related to a semilinear equation
that models reaction diffusion systems with excitable kinetics. The initial
value problem in all of the space is analyzed together with continuous
dependence and a priori estimates of the solution. These estimates show that
the asymptotic behavior is determined by the reaction mechanism. Moreover it's
possible a rigorous singular perturbation analysis for discussing travelling
waves with their characteristic times.
