Existence and Stability of a Boundary Layer with an Interior Spike in the Singularly Perturbed Shadow Gierer–Meinhardt System
Bibliographic record
Abstract
Abstract. The singularly perturbed Gierer–Meinhardt (GM) system in a bounded [Formula: see text]-dimensional domain ([Formula: see text]) is known to exhibit boundary layer (BL) solutions for a nonzero activator flux. It was previously shown that such BL solutions can be destabilized by decreasing the activator flux below a stability threshold. Moreover, numerical simulations previously indicated that solutions consisting of a boundary layer and an interior spike emerge after the destabilization of a BL solution. In this paper we use the method of matched asymptotic expansions to investigate the structure and stability of such “boundary layer spike” (BLS) solutions in the presence of an asymptotically small activator diffusivity [Formula: see text]. We find that two types of BLS solutions, one of which is unconditionally linearly stable and the other unstable, can be constructed provided that the activator flux is sufficiently small. In this way we determine that there is an asymptotically large range of activator flux values for which both the BL solution and one of the BLS solutions are linearly stable. Formal asymptotic calculations are further validated by numerically simulating the singularly perturbed GM system.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.004 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".