Time-Periodic Chemotaxis in Shifting Environments: Propagation and Planar Forced Waves in $\R^N$

Hoang-Hung Vo

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arXiv · 2610.36627

Time-Periodic Chemotaxis in Shifting Environments: Propagation and Planar Forced Waves in $\R^N$

Hoang-Hung Vo

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Abstract

We study time-periodic Keller--Segel dynamics in a translating KPP environment. In the parabolic--elliptic regime, we prove global boundedness, identify the sharp speed $c_*=2\sqrt{\bar a\,\bar r}$, establish persistence in the interior region $|s|<c_*$, $s\cdot e<c$, and derive the complementary sharp upper bounds; for a moving favorable slab, the sign of a principal Floquet exponent determines persistence or extinction under the stated weighted-resolvent condition. For every fixed $τ\ge0$, under constant population diffusivity and sufficiently weak taxis, we further construct locally unique planar forced time-periodic waves in $\mathbb R^N$. The main new analytic mechanisms are an exact Yukawa first-contact estimate, which controls the chemotactic drift at the KPP leading-edge scale without losing the sharp propagation threshold, and a two-sided weighted Fredholm--monodromy continuation framework for the temporally nonlocal signal equation, with explicit control of the far-field spectral gaps and the translational neutral mode. To the best of the author's knowledge, planar forced time-periodic waves in $\mathbb R^N$ for shifting Keller--Segel systems have not been investigated in any previous studies; earlier forced-wave results are confined to substantially more restrictive settings, in particular one-dimensional parabolic--elliptic models. Thus the present construction provides the first multidimensional planar forced-wave theory for a time-periodic shifting Keller--Segel system with temporal signal memory.

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BibTeX RIS

Hoang-Hung Vo. 2026-10-27. Time-Periodic Chemotaxis in Shifting Environments: Propagation and Planar Forced Waves in $\R^N$. https://arxiv.org/abs/2610.36627

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