Maximum Principle for Heat Equation on all of $\mathbb{R}$

Question

I have a rather simple question regarding the maximum principle pertaining to the heat equation. It is my understanding that the maximum principle informally states that the maximum value the heat equation can obtain is at either the spatial boundary $x_l$ and $x_r$, or (assuming a forward heat equation) at $t = 0$.

However, does this assertion still hold if our domain is $\mathbb{R}$? Can a maximum be obtained as $x \rightarrow \pm \infty$?

Answer 1

Under some assumptions (about the behavior at infinity) the maximum principle holds, see Theorem 2.3.6 in Evans's Partial Differential Equations.

Without these assumptions it fails, see e.g. the post Properties of heat equation suggested by Dan Doe.