Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems
Abstract
We consider the Neumann initial boundary value problem associated to the chemotaxis system labelprob:abstracttag$⋆$ ut = ((u+1)m−1 ux − u(u+1)m vx)x & in $(0, 1) × (0, ∞)$, vt = vxx − v + u, &in $(0, 1) × (0, ∞)$, where m ∈ ℝ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio u(u+1)m/(u+1)m−1 grows like u2/n for large u with n = dim((0, 1)) = 1. Nonetheless, we show that there is no critical mass phenomenon if m ≤ −1; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \eqref{prob:abstract}, we obtain the same conclusion for all m ∈ (−∞, −1] ∪ (0, ∞) and even for all m ∈ ℝ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding W1, n ↪ L∞, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation.
Versions
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BibTeX
@misc{cao2026absence,
title = {{Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems}},
author = {Xinru Cao and Mario Fuest},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.18006},
eprint = {2606.18006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.18006},
}
RIS
TY - UNPB
TI - Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems
AU - Xinru Cao
AU - Mario Fuest
PY - 2026
DA - 2026/06/16
PB - arXiv
DO - 10.48550/arxiv.2606.18006
UR - https://arxiv.org/abs/2606.18006
AB - We consider the Neumann initial boundary value problem associated to the chemotaxis system labelprob:abstracttag$⋆$ u_t = ((u+1)^m−1 u_x − u(u+1)^m v_x)_x & in $(0, 1) × (0, ∞)$, v_t = v_xx − v + u, &in $(0, 1) × (0, ∞)$, where m ∈ ℝ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio u(u+1)^m/(u+1)^m−1 grows like u^2/n for large u with n = dim((0, 1)) = 1. Nonetheless, we show that there is no critical mass phenomenon if m ≤ −1; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \eqref{prob:abstract}, we obtain the same conclusion for all m ∈ (−∞, −1] ∪ (0, ∞) and even for all m ∈ ℝ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding W^1, n ↪ L^∞, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation.
ER -