Consider the following evolution equation for $c$, given by a convection-diffusion equation over one spatial dimension with positive $D$ and constant $v$:
$c_t = Dc_{xx} - vc_x$
Physical reasoning—and arguably common sense—indicate that, by selecting $c(x,0) \geq 0$ everywhere, you should expect to always find $c(x,t) \geq 0$ for $t > 0$. The same intuition should hold for similar systems (i.e. the diffusion equation) and in higher dimensions.
Is there a way to generally identify evolution operators that possess this property, in the sense that they always output $c(x,t)\geq0$ for $t\geq0$ given $c(x,0) \geq 0$?
The same question can be framed from a mathematical modeling perspective:
How do I know whether or not my PDE model for transport of a positive quantity is plausible in the sense that it won't output negative values a priori?
A comment: This question is inspired by Pawula's theorem, which indicates that a Taylor series expansion of a master evolution equation for some probability density must either stop at the second-order term or before (giving a Fokker-Planck equation) or contain infinite terms. This is due precisely to the fact that expanding to a finite number of terms larger than second-order can generate equations that evolve the probability density into negative states: see Hannes Risken's book for a discussion of this.
Another motivation comes from the Ostrogradsky instability, which seems to rule out equations of motion depending on derivatives larger than second-order because it would allow negative energies. Perhaps both of these examples are related?