positivity preservation

A density cannot be negative. A temperature on the absolute scale cannot be negative. A concentration of dissolved sugar cannot be negative. If your PDE is supposed to model one of these, you want a guarantee that a solution starting nonnegative stays nonnegative forever — otherwise the model would be predicting nonsense like negative matter. Positivity preservation is exactly that guarantee, and for diffusion-type equations it is true and provable.

Precisely: for the heat equation u_t = k Laplacian u, if the initial data is nonnegative and the boundary data is nonnegative, then the solution is nonnegative everywhere for all later time. This is just the comparison principle applied with v = 0: zero is itself a solution, your data sits above zero, so by comparison your solution sits above zero too. The same reasoning gives more — strict positivity. By the strong maximum principle (or directly from the strictly positive heat kernel), nonnegative initial data that is not identically zero becomes strictly positive everywhere instantly, a vivid face of infinite propagation speed: a little heat anywhere warms everywhere, so nothing can stay exactly at zero. For reaction-diffusion systems one checks an invariant-region or quasi-positivity condition to keep each component nonnegative.

Why it matters: it is the sanity check that says your model is physically admissible, and it is essential when you discretize. A numerical scheme that fails to preserve positivity can produce negative densities or temperatures and crash a simulation; designing positivity-preserving schemes is a real concern in computation. Honesty note: not every PDE preserves positivity — higher-order equations like the biharmonic (Laplacian-squared) equation generally do NOT, which is one reason second-order diffusion is so much better behaved than fourth-order models.