PreprintCC BY 4.0HAL
Reaction-diffusion equations in periodic media: spreading speeds and spreading sets
Abstract
We consider solutions of space-periodic reaction-diffusion-advection equations in the whole space R N , with general unbounded initial support. We show the existence of spreading sets for the solutions at large time, and variational formulas for the spreading speeds. The location of the upper-level sets of a solution with unbounded initial support is estimated in terms of that initial support and the spreading set for the solutions with bounded initial supports. The results hold for the standard classes of Fisher-KPP, ignition, or bistable reactions. The general results apply to the cases of V -shaped and Λ-shaped initial supports. The proofs rely in particular on upper estimates of what we call retracting solutions.
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BibTeX
@misc{guo2026reactiondiffusion,
title = {{Reaction-diffusion equations in periodic media: spreading speeds and spreading sets}},
author = {Hongjun Guo and François Hamel and Luca Rossi},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18795},
eprint = {2609.18795},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05739515v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reaction-diffusion equations in periodic media: spreading speeds and spreading sets
AU - Hongjun Guo
AU - François Hamel
AU - Luca Rossi
PY - 2026
DA - 2026/09/10
PB - HAL
DO - 10.48550/arxiv.2609.18795
UR - https://hal.science/hal-05739515v1
AB - We consider solutions of space-periodic reaction-diffusion-advection equations in the whole space R N , with general unbounded initial support. We show the existence of spreading sets for the solutions at large time, and variational formulas for the spreading speeds. The location of the upper-level sets of a solution with unbounded initial support is estimated in terms of that initial support and the spreading set for the solutions with bounded initial supports. The results hold for the standard classes of Fisher-KPP, ignition, or bistable reactions. The general results apply to the cases of V -shaped and Λ-shaped initial supports. The proofs rely in particular on upper estimates of what we call retracting solutions.
ER -