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Rigidity for the heat equation with density on Riemannian manifolds through a conformal change
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 12 January 2026, pp. 1-41
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We investigate uniqueness of solution to the heat equation with a density
$\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution 
$u$ vanishes identically, assuming that 
$u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, 
$L^p_{\phi}$. We distinguish between the cases 
$p \gt 1$ and 
$p = 1$ which required stronger assumptions on the manifold and the density function 
$\rho$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density 
$\rho$.
EQUATIONS WITH DIRICHLET BOUNDARY NOISE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 87 / Issue 1 / February 2013
- Published online by Cambridge University Press:
- 31 October 2012, pp. 174-176
- Print publication:
- February 2013
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