On a chemotaxis model with singular sensitivity: Convergence rate towards spiky steady state
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
This paper is concerned with the stability analysis for a model of chemotaxis:$ \begin{equation*} \begin{aligned} u_t & = u_{xx} - \chi[u(\ln w)_x]_x, &(x, t)&\in \mathbb{R}_+ \times \mathbb{R}_+, \\ w_t & = \varepsilon w_{xx} - u^\gamma w^m, &(x, t)&\in \mathbb{R}_+ \times \mathbb{R}_+. \end{aligned} \end{equation*} $By utilizing spatially weighted energy method, it is shown that the non-trivial steady state associated with the model under certain parametric constraints and the boundary conditions:$ \begin{equation*} \begin{aligned} &u_x = \chi u(\ln w)_x, \ \ w = b>0, &&(x, t) \in \{0\}\times \mathbb{R}_+, \\ &(u, w) \to (0, 0), && x\to\infty, \ t\in \mathbb{R}_+, \end{aligned} \end{equation*} $is locally asymptotically stable. By exploring the idea of 'gaining temporal integrability through faster spatial decay', the explicit (algebraic) convergence rate, in terms of the system parameters, of general solutions of the initial-boundary value problem towards the steady state is identified.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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