Global Dynamics of Nonlocal Diffusion Systems on Time-Varying Domains
Bibliographic record
Abstract
We propose a class of nonlocal diffusion systems on time-varying domains, and fully characterize their asymptotic dynamics in the asymptotically fixed, time-periodic and unbounded cases. The kernel is not necessarily symmetric or compactly supported, provoking anisotropic diffusion or convective effects. Due to the nonlocal diffusion on time-varying domains in our systems, some significant challenges arise, such as the lack of regularizing effects of the semigroup generated by the nonlocal operator, as well as the time-dependent inherent coupling structure in kernel. By investigating a general nonautonomous nonlocal diffusion system in the space of bounded and measurable functions, we establish a comprehensive and unified framework to rigorously examine the threshold dynamics of the original system on asymptotically fixed and time-periodic domains. In the case of an asymptotically unbounded domain, we introduce a key auxiliary function to separate vanishing coefficients from nonlocal diffusions. This enables us to construct appropriate sub-solutions and derive the global threshold dynamics via the comparison principle. The findings may be of independent interest and the developed techniques, which do not rely on the existence of the principal eigenvalue, are expected to find further applications in the related nonlocal diffusion problems. We also conduct numerical simulations based on a practical model to illustrate our analytical results.
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How this classification was reachedexpand
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 | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".