Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $
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Bibliographic record
Abstract
<p style='text-indent:20px;'>In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula>,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \begin{cases} u_{t} = \Delta u - \chi\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in{{\mathbb R}}^N,\\ {v_t} = \Delta v -\lambda v+\mu u,\quad x\in{{\mathbb R}}^N,\,\,\, \end{cases} \;\;\;\;\;\;\;\;\left( 1 \right)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document}</tex-math></inline-formula> are positive constants and <inline-formula><tex-math id="M4">\begin{document}$ N $\end{document}</tex-math></inline-formula> is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption <inline-formula><tex-math id="M5">\begin{document}$ b&gt;\frac{N\chi\mu}{4} $\end{document}</tex-math></inline-formula>, the global existence of a unique classical solution <inline-formula><tex-math id="M6">\begin{document}$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $\end{document}</tex-math></inline-formula> of (1) with <inline-formula><tex-math id="M7">\begin{document}$ u(x,0;u_0, v_0) = u_0(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ v(x,0;u_0, v_0) = v_0(x) $\end{document}</tex-math></inline-formula> for every nonnegative, bounded, and uniformly continuous function <inline-formula><tex-math id="M9">\begin{document}$ u_0(x) $\end{document}</tex-math></inline-formula>, and every nonnegative, bounded, uniformly continuous, and differentiable function <inline-formula><tex-math id="M10">\begin{document}$ v_0(x) $\end{document}</tex-math></inline-formula>. Next, under the same assumption <inline-formula><tex-math id="M11">\begin{document}$ b&gt;\frac{N\chi\mu}{4} $\end{document}</tex-math></inline-formula>, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function <inline-formula><tex-math id="M12">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula> is bounded below by a positive constant independent of <inline-formula><tex-math id="M13">\begin{document}$ (u_0, v_0) $\end{document}</tex-math></inline-formula> when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function <inline-formula><tex-math id="M14">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>. We show that there is <inline-formula><tex-math id="M15">\begin{document}$ K = K(a,\lambda,N)&gt;\frac{N}{4} $\end{document}</tex-math></inline-formula> such that if <inline-formula><tex-math id="M16">\begin{document}$ b&gt;K \chi\mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ \lambda\geq \frac{a}{2} $\end{document}</tex-math></inline-formula>, then for every strictly positive initial function <inline-formula><tex-math id="M18">\begin{document}$ u_0(\cdot) $\end{document}</tex-math></inline-formula>, it holds that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \lim\limits_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big] = 0. $\end{document} </tex-math></disp-formula></p>
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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 | 0.001 | 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