Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = \Delta u$, such that at some positive time $t_0>0$, I have $|u(x,t_0)| \le \psi(x)$ and $|u(p,t_0)| = \psi(p)$ and $ |\Delta u (p,t_0)| = |\Delta \psi (p)|$.
I have been considering two approaches:
1) One approach is to let $h(x,t) = \psi(x) - u(x,t)$. $h$ satisfies $\partial_t h = \Delta h - \Delta \psi$ then the question becomes: construct a solution $h$ such that at some positive time $t_0$, $h(.,t_0) \ge 0$ , $h(p,t_0) = 0$ and $\Delta h (p,t_0) = 0$. We know that the above Heat equation for $h$ has a heat kernel but I have not been able to construct such a solution ??? (We might need to use some assertions about the zero crossings of a heat type equation.)
2) Take a function $g$ that satisfies $|g(x)|\le \psi(x)$ , $|g(p)| = \psi(p)$ and $ \Delta g (p) = \Delta \psi (p)$ and solve the backward Heat equation for a short time but the problem is that the backward Heat equation is not well-posed. At least we must have $g$ analytic and furthermore satisfying some proper decay rates on its derivatives. The question is, can we always find such a function $g$ satisfying the properties we want for which the backward heat equation is solvable for a short time? (I do not really need uniqueness)
P.S.: My goal is to prove a similar thing when, $(M,g(t))$ satisfy the Ricci flow on a time interval $[0,T]$ $\psi: M \to R$ a positive obstacle and when the heat equation is the heat equation under Ricci flow namely$\partial_t u = \Delta_{g(t)} u$.
Hope somebody could help me or just give me some ideas as to how to proceed further.
Thanks
