Estimates on a heat process with fixed boundary data and zero initial conditions

Question

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!

Answer 1

You cannot get a nontrivial lower bound only in terms of $t$ and $dist(x,\partial\Omega)$. To see this, consider a domain with a tiny hole. In the limit when the hole shrinks to a point, the solution converges to that with no hole, i.e. the effect of the boundary condition on the boundary of the hole disappears.

Answer 2

I'm nor aware of such an estimate, so just a guess. May be there is an estimate of the form $$ u(x,t)\ge C_1\text{erfc}(-C_2d(x)/\sqrt t), $$ where $d(x)=\text{dist}(x, \partial\Omega)$. When $\Omega$ is a half space it can be checked directly, using the representation of solution as a double layer potential. And in the general case local behavior of the solution near $t=0$ should be the same.