Consider the following heat process: For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that

$$ \partial_t p(t,x) - \Delta p(t,x) = 0, \quad x\in \Omega \\ p(t,x) = 1, \quad x\in \partial \Omega \\ p(0,x) = 0, \quad x\in \Omega. $$

In other words, we start heating up with a constant "1 - radiation" at the boundary. I'd like to ask what are the known estimates of the form:

$$ p(t,x) \geq Expression (t,x). $$ Is there, perhaps, a result giving the expression on the right hand side in terms of $t$ and $dist(x, \partial\Omega)$? Many thanks!