Radial Symmetry and Strict Radial Decrease for Master Equations with Decreasing Radial Potentials
Abstract
We study the master equation $ (\partial_t -Δ)^{s} u(x,t) = R(|x|)f(u(x,t))\quad\mbox{in}\ \mathbb{R}^n\times\mathbb{R}, $ where s∈(0,1), R is positive and strictly decreasing in the radial variable, and f is positive, locally Lipschitz, and satisfies f'(0)<0. Existing symmetry results for this operator require f(0)=0 with f'(0)≥ 0 (or, more generally, that f be non-decreasing near the origin), because the sign of the moving-plane comparison inequality is then controlled by the nonlocal diffusion term. When f'(0)<0, the linear term in the comparison inequality appears at the same order as the nonlocal diffusion but with the opposite sign, destroying the comparison principle underlying the standard cut-off perturbation argument. We prove that, despite this obstruction, every positive bounded classical solution with uniform spatial decay is radially symmetric and strictly radially decreasing in x for each t. The proof reconciles the direct method of moving planes with this adverse sign by exploiting the interplay between the spatial decay of u and the strict radial monotonicity of R. This appears to be the first symmetry result for the master operator that accommodates a local damping mechanism at the origin.
Related
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BibTeX
@misc{niu2026radial,
title = {{Radial Symmetry and Strict Radial Decrease for Master Equations with Decreasing Radial Potentials}},
author = {Yahui Niu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20476},
eprint = {2609.20476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20476},
}
RIS
TY - UNPB
TI - Radial Symmetry and Strict Radial Decrease for Master Equations with Decreasing Radial Potentials
AU - Yahui Niu
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20476
UR - https://arxiv.org/abs/2609.20476
AB - We study the master equation $ (\partial_t -Δ)^{s} u(x,t) = R(|x|)f(u(x,t))\quad\mbox{in}\ \mathbb{R}^n\times\mathbb{R}, $ where s∈(0,1), R is positive and strictly decreasing in the radial variable, and f is positive, locally Lipschitz, and satisfies f'(0)<0. Existing symmetry results for this operator require f(0)=0 with f'(0)≥ 0 (or, more generally, that f be non-decreasing near the origin), because the sign of the moving-plane comparison inequality is then controlled by the nonlocal diffusion term. When f'(0)<0, the linear term in the comparison inequality appears at the same order as the nonlocal diffusion but with the opposite sign, destroying the comparison principle underlying the standard cut-off perturbation argument. We prove that, despite this obstruction, every positive bounded classical solution with uniform spatial decay is radially symmetric and strictly radially decreasing in x for each t. The proof reconciles the direct method of moving planes with this adverse sign by exploiting the interplay between the spatial decay of u and the strict radial monotonicity of R. This appears to be the first symmetry result for the master operator that accommodates a local damping mechanism at the origin.
ER -