Let be a rotationally symmetric Riemannian manifold, and be the Laplace-Beltrami operator on . We establish a necessary and sufficient condition for the constant function 1 to be a semismall perturbation of on , 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 .
Cite this article
Minoru Murata, Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall perturbations. Rev. Mat. Iberoam. 27 (2011), no. 3, pp. 885–907DOI 10.4171/RMI/656