Triviality results for weighted porous medium heat equations on Riemannian manifolds

We study aweighted porous medium heat equation on completeRiemannian manifolds with nonnegative Ricci curvature. Our aim is to establish Liouville-type nonexistence for nonnegative solutions.We treat both the slow diffusion regime m > 1 and the fast diffusion regime 0 < m < 1. Classical solutions are considered in the slow diffusion regime, whereas the fast diffusion result is formulated for weak solutions. Under suitable integral growth assumptions on the weight function, formulated according to the natural space-time scales of the corresponding diffusion regimes, we establish Liouville-type nonexistence results. In the slow diffusion case, the assumptions are expressed in terms of the intrinsic time scale TR = R^2+α(m−1), whereas in the fast diffusion case the standard parabolic time scale TR = R^2 is used. The resulting criteria extend several known triviality results for semilinear heat equations to the porous medium setting on Riemannian manifolds.