** I simplified the question: **

On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.

I would like to know whether this result is also true if applied on an unbounded domain:

Let $a>0$ and $g$ a continuous function with $g(a,t)>0$ for all $t \in [0,T]$ and $u(x,0) = u_0(x) >0$ for all $x \in (a,\infty)$ with $u_0 \in C^{\infty} \cap L^{\infty}.$

Let $u(x,t)$ be the solution to the heat equation $\left(\partial_t - \partial_{x}^2 \right)u=0$ with the above boundary data.

Does there exist then a version of the maximum principle saying that $u(x,t)>0$ for $(x,t) \in [a,\infty) \times [0,T]$?