Existence of multi-spikes in the Keller–Segel model with logistic growth
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Bibliographic record
Abstract
The Keller–Segel model is a paradigm to describe the chemotactic mechanism, which plays a vital role on the physiological and pathological activities of uni-cellular and multi-cellular organisms. One of the most interesting variants is the coupled system with the intrinsic growth, which admits many complex nontrivial patterns. This paper is devoted to the construction of multi-spiky solutions to the Keller–Segel models with the logistic source in 2D. Assuming that the chemo-attractive rate is large, we apply the inner-outer gluing scheme to nonlocal cross-diffusion system and prove the existence of multiple boundary and interior spikes. The numerical simulations are presented to highlight our theoretical results.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.008 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it