Nonexistence results for parabolic equations involving the p-Laplacian and Hardy–Leray-type inequalities on Riemannian manifolds
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Date
2021-09Author
GOLDSTEIN, Gisèle Ruiz
GOLDSTEIN, Jerome Arthur
KÖMBE, İsmail
BAKIM, Sümeyye
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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.
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