Exact Formula for Front Velocity

The optimal trajectory is that of a free particle, i.e., the straight line connecting the points X and y e S2q, where is the support of the initial condition  

Let /(x, denote the minimal distance between the point x and the set Qq. Then, in terms of the phenomenological parameters tq, D, and r, the action functional G(x, t) can be rewritten as 

The effect of diffusion with finite velocity, x r 0, is to decrease the propagation rate corresponding to the RD equation. The last restriction x r 1 arises from the fact that the speed of light c = yjDjx is the maximal velocity of propagation. In other words, front motion with a velocity greater than the speed of light c is impossible. 

In the limiting case, where the relaxation time tq is small compared with the chemical time r i.e., x r -C 1, we can neglect the effect that the velocity of the transport mechanism is finite. 

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