The periodic-parabolic logistic equation on $\mathbb{R}^N$
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Bibliographic record
Abstract
In this article, we investigate the periodic-parabolic logisticequation on the entire space $\mathbb{R}^N\ (N\geq1)$: $$\begin{equation} \left\{\begin{array}{ll} \partial_t u-\Delta u=a(x,t)u-b(x,t)u^p\ \ \ \ & {\rm in}\ \mathbb{R}^N\times(0,T),\\ u(x,0)=u(x,T) \ & {\rm in}\ \mathbb{R}^N, \end{array} \right. \end{equation} $$where the constants $T>0$ and $p>1$, and the functions $a,\ b$ with$b>0$ are smooth in $\mathbb{R}^N\times\mathbb{R}$ and $T$-periodic in time. Underthe assumptions that $a(x,t)/{|x|^\gamma}$ and $b(x,t)/{|x|^\tau}$are bounded away from $0$ and infinity for all large $|x|$, wherethe constants $\gamma>-2$ and $\tau\in\mathbb{R}$, we study the existenceand uniqueness of positive $T$-periodic solutions. In particular, wedetermine the asymptotic behavior of the unique positive$T$-periodic solution as $|x|\to\infty$, which turns out to dependon the sign of $\gamma$. Our investigation considerably generalizesthe existing 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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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