PreprintCC BY-NC-SA 4.0arXiv
Blow-down and immediate onset of positivity in the Neumann problem for (super-)fast diffusion equations
Abstract
For the degenerate parabolic equation ut=∇· (um−1∇ u), the homogeneous Neumann problem in smoothly bounded domains Ω⊂ ℝn is considered in the (super-fast) diffusion range when m∈ (−∞,1). Upper and lower estimates for solutions are inferred by means of two Moser-type iterations, and optimality properties of these bounds are derived using comparison arguments. On the basis of the accordingly obtained information on regularization into spaces of bounded and uniformly positive functions, a result on global solvability in an associated initial-boundary value problem is achieved for all nonnegative and nontrivial initial data belonging to L1(Ω) ∩ ⋃s>n(1−m)/2 Ls(Ω).
Related
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BibTeX
@misc{tao2026blowdown,
title = {{Blow-down and immediate onset of positivity in the Neumann problem for (super-)fast diffusion equations}},
author = {Youshan Tao and Michael Winkler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15673},
eprint = {2610.15673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15673},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Blow-down and immediate onset of positivity in the Neumann problem for (super-)fast diffusion equations
AU - Youshan Tao
AU - Michael Winkler
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.15673
UR - https://arxiv.org/abs/2610.15673
AB - For the degenerate parabolic equation u_t=∇· (u^m−1∇ u), the homogeneous Neumann problem in smoothly bounded domains Ω⊂ ℝ^n is considered in the (super-fast) diffusion range when m∈ (−∞,1). Upper and lower estimates for solutions are inferred by means of two Moser-type iterations, and optimality properties of these bounds are derived using comparison arguments. On the basis of the accordingly obtained information on regularization into spaces of bounded and uniformly positive functions, a result on global solvability in an associated initial-boundary value problem is achieved for all nonnegative and nontrivial initial data belonging to L^1(Ω) ∩ ⋃_s>n(1−m)/2 L^s(Ω).
ER -