Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = u(0, x)$ supported inside a compact ball $B_1(p)$, with a $\frac{1}{\text{dist}(x, p)^m}$ type singularity at $p$ and smooth and bounded away from $p$. We know that the heat equation has infinite speed of propagation, so no matter how small $t$ is, $e^{t\Delta}u_0(x) = u(t, x)$ will spread throughout $M$. I was wondering, is there some way to control $\Vert e^{t\Delta}u\Vert_{L^q (M \setminus B_r(p))}$ in terms of $\Vert u\Vert_{L^q (M \setminus B_r(p))}$? In other words, can we have some amount of quantitative control on the spread of heat? We could assume $t << r << 1$ if that helps. I am mainly interested in the case where $q = 1$.
My intuition for this question is as follows: if $r$ is very small, the heat flow cannot change the function $u$ too much (and also the manifold $M$ is compact), so we can expect something like $$\Vert e^{t\Delta}u\Vert_{L^1 (M \setminus B_r(p))} \leq C \Vert u\Vert_{L^1 (M \setminus B_r(p))}$$ for very small $r$, $C$ being independent of $r$.
