I have problems proving that a set of temperature distributions is invariant. I've been looking a lot for material related to my problem, but I was unable to find the correct keywords or relate the relevant concepts to my problem.
In essence I'd like to know if the some set of temperature distributions is invariant. That is,
$$ T(x,0) \in \mathbb{T} \implies T(x,t) \in \mathbb{T},\quad t>0 $$
Here, $T(x,t)$ denotes the solution of the heat equation and $\mathbb{T}$ denotes the invariant set.
To this end, I wanted to show that the following holds for the heat equation,
$$ T(x,0) \geq 0,\quad \frac{\partial T(x,0)}{\partial t} \leq 0 \implies T(x,t) \leq T(x,0),\quad t>0 $$
Where $T$ is the solution of the heat equation with convective boundary condition
$$ \frac{\partial T(x,t)}{\partial t} = \nabla\cdot\left(k\nabla T(x,t)\right) - cT(x,t),\quad k>0,\quad c\geq 0,\\ n\cdot\nabla T(x,t) = -hT, \quad h>0 $$ Here, $n$ denotes the outward surface normal.
I have trouble showing the latter condition. Conceptually I think it should hold. However, as mentioned, I have trouble showing it rigorously / finding references that proof this. I would greatly appropriate I any of you could help me connect the missing dots!
