# Revista Matemática Iberoamericana

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**Volume 27, Issue 3, 2011, pp. 885–907**

**DOI: 10.4171/RMI/656**

Published online: 2011-12-06

Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall perturbations

Minoru Murata^{[1]}(1) Tokyo Metropolitan University, Japan

Let $M$ be a rotationally symmetric Riemannian manifold, and $\Delta$ be the Laplace-Beltrami operator on $M$. We establish a necessary and sufficient condition for the constant function 1 to be a semismall perturbation of $-\Delta +1$ on $M$, and give optimal sufficient conditions for uniqueness of nonnegative solutions of the Cauchy problem to the heat equation. As an application, we determine the structure of all nonnegative solutions to the heat equation on $M\times(0,T)$.

*Keywords: *Heat equation, rotationally symmetric Riemannian manifold, nonnegative solution, integral representation, uniqueness, semismall perturbation, Laplace operator, Martin boundary

Murata Minoru: Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall perturbations. *Rev. Mat. Iberoam.* 27 (2011), 885-907. doi: 10.4171/RMI/656