Schwarz symmetrizations in parabolic equations on complete manifolds

We prove sharp estimates for solutions to parabolic equations on bounded domains in both complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature. Our results generalize Bandle’s comparison theorem [C. Bandle, On symmetrizations in parabolic equations, J. Analyse Math. 30 (1976) 98–112] to the Riemannian setting, and Talenti’s comparison for elliptic equations by Chen and Li [Talenti’s comparison theorem for Poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature, J. Geom. Anal. 33(4) (2023), Article ID: 123, 20 pp] and Colladay et al. [Comparison results, exit time moments, and eigenvalues on Riemannian manifolds with a lower Ricci curvature bound, J. Geom. Anal. 28(4) (2018) 3906–3927] to parabolic equations on manifolds.