Infinite domain (Cauchy problem)

If the domain is infinite, $x$ can take any value between $-\infty$ and $+\infty$. From a practical standpoint, this condition is that of a population that can disperse without finding virtually any spatial barrier. Note that in this case there is no real boundary. As the only phenomenon that drives changes in the density of organisms at each point is just movement, the following important condition must be verified

   Total number of organisms$$= \int_{-\infty}^{+\infty} {c(x,t)} \;dx = \int_{-\infty}^{+\infty}% {c_0(x)} \;dx = N$$   constant.$$$$

Therefore the solution $c(x,t)$ must be a bounded function at each time instant and such that

$${\lim _{x \to + \infty }}c(x,t) = {\lim _{x \to - \infty }}c(x,t) = 0.$$

It is possible to prove that the solution of the Cauchy problem of eq. 14 is unique if we consider bounded functions only.