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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 $\Delta |\Delta u (p,t_0) p,t_0)| = \Delta |\Delta \psi (p)$. 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$. Hope somebody could help me or just give me some ideas as to how to proceed further. Thanks 2 edited body Consider a closed Riemannian manifold$(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(.,t_0)| |u(x,t_0)| \le \psi(p)$psi(x)$ 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

1

# Solutions to Heat Equations with Obstacles!

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 $|u(.,t_0)| \le \psi(p)$ 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