Activation waves in a model of platelet aggregation: existence of solutions and stability of travelling fronts

Platelets cohere to one another to form platelet aggregates as part of the blood's clotting response. The ability of a platelet to participate in this process depends on its prior 'activation' by chemicals released into the blood plasma by other activated platelets. We study the piecewise-linear system of reaction-diffusion equations which, in one spatial dimension, describe the chemically-mediated spread of platelet activation. We establish the existence of classical solutions to this system of equations, and show that these solutions do not blow up in finite time. We also explicitly construct travelling front solutions and discuss their stability. Finally, we present numerical evidence which suggests that for a broad range of initial data with the correct limiting values at +/- infinity, the solution to the initial value problem rapidly evolves into the travelling front solution provided the front is linearly stable.