Conservation of mass for solutions of Leibenson's equation on Riemannian Manifolds

Abstract

We consider on a Riemannian manifold M the Leibenson equation labeleqabs∂ _tu=Δ_pu^q, where p>1 and q>0. When q(p−1)≥ 1, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound V(x₀, r)≤ exp(C r^p/(p−1)) for some x₀∈ M and all large enough r>0. When q(p−1)< 1, we prove this property assuming V(x₀, r)≤ Cr^N and p>N[1−q(p−1)], which matches the threshold in ℝⁿ with N=n. We also show that solutions on the hyperbolic space ℍⁿ have a finite extinction time in the case q(p−1)< 1, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case q(p−1)=1, we also prove a L^p−1- Liouville property, which partially answers a conjecture stated by I. Holopainen \cite{holopainen2000sharp}.

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BibTeX RIS

BibTeX

@misc{surig2026conservation,
  title = {{Conservation of mass for solutions of Leibenson's equation on Riemannian Manifolds}},
  author = {Philipp Sürig},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02732},
  eprint = {2609.02732},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02732},
}

RIS

TY  - UNPB
TI  - Conservation of mass for solutions of Leibenson's equation on Riemannian Manifolds
AU  - Philipp Sürig
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02732
UR  - https://arxiv.org/abs/2609.02732
AB  - We consider on a Riemannian manifold M the Leibenson equation labeleqabs∂ _tu=Δ_pu^q, where p>1 and q>0. When q(p−1)≥ 1, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound V(x_0, r)≤ exp(C r^p/(p−1)) for some x_0∈ M and all large enough r>0. When q(p−1)< 1, we prove this property assuming V(x_0, r)≤ Cr^N and p>N[1−q(p−1)], which matches the threshold in ℝ^n with N=n. We also show that solutions on the hyperbolic space ℍ^n have a finite extinction time in the case q(p−1)< 1, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case q(p−1)=1, we also prove a L^p−1- Liouville property, which partially answers a conjecture stated by I. Holopainen \cite{holopainen2000sharp}.
ER  -

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