Nonexistence results for parabolic equations involving the p-Laplacian and Hardy–Leray-type inequalities on Riemannian manifolds

Abstract

The main goal of this paper is twofold. The first one is to investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation on a noncompact Riemannian manifold M, ⎧ ⎪⎨ ⎪⎩ ∂u ∂t = Δp,gu + V(x)u p−1 + λuq in Ω × (0, T ), u(x, 0) = u0(x) ≥ 0 in Ω, u(x, t) = 0 on ∂Ω × (0, T ), where 1 < p < 2, V ∈ L1 loc(Ω), q > 0, λ ∈ R, Ω is bounded and has a smooth boundary in M and Δp,g is the p-Laplacian on M. The second one is to obtain Hardy- and Leray-type inequalities with remainder terms on a Riemannian manifold M that provide us concrete potentials to use in the partial differential equation we are interested in. In particular, we obtain explicit (mostly sharp) constants for these inequalities on the hyperbolic space Hn.