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

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!

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.
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.