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


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.


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