# Fundamental solution to the heat equation with zero boundary values

let $\Omega\subset M$ be an open and unbounded set in a smooth manifold $M$ with boundary $\partial \Omega$. Now let $p_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\Omega$ which vanishes on the boundary $\partial \Omega$, i.e. for all $y\in \Omega$:

$(i) (\partial_t -\Delta_x )p_t(x,y)=0, \text{ }t>0, x\in \Omega,$

$(ii) p_t(\cdot, y)\rightarrow \delta_y, \text{ for }t\downarrow 0$

$(iii) p_t(x,y)=0,$ for $x\in\partial \Omega$.

I'm wondering if the function $p_t(x,y)$ has to be the heat kernel for $\Omega$, i.e. the minimal non-negative fundamental solution in y? If the domain $\Omega\subset M$ would be bounded, than the maximum principle would guarantee that $p_t(x,y)$ is minimal, i.e. is the heat kernel. But I do not know the answer in the unbounded case. Is $p_t(x,y)$ the minimal non-negative fundamental solution?

I would appreciate any help.

Best regards

## 1 Answer

Without imposing some growth condition at infinity, this is not to be expected. There are nonzero solution of the heat equation on the real line which satisfy a zero initial condition.