Applied Mathematics and Optimization ( IF 1.8 ) Pub Date : 2023-11-28 , DOI: 10.1007/s00245-023-10077-3 Yutaro Chiyo , Fatma Gamze Düzgün , Silvia Frassu , Giuseppe Viglialoro
This work deals with a chemotaxis model where an external source involving a sub and superquadratic growth effect contrasted by nonlocal dampening reaction influences the motion of a cell density attracted by a chemical signal. We study the mechanism of the two densities once their initial configurations are fixed in bounded impenetrable regions; in the specific, we establish that no gathering effect for the cells can appear in time provided that the dampening effect is strong enough. Mathematically, we are concerned with this problem
$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+au^\alpha -bu^\alpha \int _\Omega u^\beta &{}\textrm{in}\ \Omega \times (0, T_{max}),\\ \tau v_t=\Delta v-v+u &{}\textrm{in}\ \Omega \times (0, T_{max}),\\ u_\nu =v_\nu =0 &{}\textrm{on}\ \partial \Omega \times (0, T_{max}),\\ u(x, 0)=u_0(x)\ge 0, v(x,0)=v_0(x)\ge 0, &{}x \in {\bar{\Omega }}, \end{array}\right. } \quad {\Diamond } \end{aligned}$$for \(\tau =1\), \(n\in {\mathbb {N}}\), \(\chi ,a,b>0\) and \(\alpha , \beta \ge 1\). Herein u stands for the population density, v for the chemical signal and \(T_{max}\) for the maximal time of existence of any nonnegative classical solution (u, v) to system (\(\Diamond \)). We prove that despite any large-mass initial data \(u_0\), whenever
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(The subquadratic case) \(1\le \alpha <2 \quad \text {and} \quad \beta >\frac{n+4}{2}-\alpha ,\)
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(The superquadratic case) \(\beta >\frac{n}{2} \quad \text {and} \quad 2\le \alpha < 1+ \frac{2\beta }{n},\)
actually \(T_{max}=\infty \) and u and v are uniformly bounded. This paper is in line with the result in Bian et al. (Nonlinear Anal 176:178–191, 2018), where the same conclusion is established for the simplified parabolic-elliptic version of model (\(\Diamond \)), corresponding to \(\tau =0\); more exactly, this work extends the study to the fully parabolic case Bian et al. (Nonlinear Anal 176:178–191, 2018).
中文翻译：
亚二次和超二次增长的全抛物线趋化模型中非局部阻尼效应的有界性
这项工作涉及趋化性模型，其中涉及亚二次和超二次生长效应的外部源与非局部抑制反应形成对比，影响化学信号吸引的细胞密度的运动。一旦两种密度的初始构型固定在有界的不可穿透区域中，我们就研究它们的机制；具体来说，我们确定，只要阻尼效应足够强，细胞就不会及时出现聚集效应。从数学上来说，我们关心这个问题
$$\begin{对齐} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+au^\alpha -bu^\alpha \int _\Omega u^\beta &{}\textrm{in}\ \Omega \times (0, T_{max}),\\ \tau v_t=\Delta v-v+u &{}\textrm{in} \ \Omega \times (0, T_{max}),\\ u_\nu =v_\nu =0 &{}\textrm{on}\ \partial \Omega \times (0, T_{max}),\ \ u(x, 0)=u_0(x)\ge 0, v(x,0)=v_0(x)\ge 0, &{}x \in {\bar{\Omega }}, \end{array }\正确的。} \quad {\Diamond } \end{对齐}$$对于\(\tau =1\)、\(n\in {\mathbb {N}}\)、\(\chi ,a,b>0\)和\(\alpha , \beta \ge 1\)。这里u代表人口密度，v代表化学信号，\(T_{max}\)代表任何非负经典解 ( u , v ) 对于系统 ( \(\Diamond \) )的最大存在时间。我们证明，尽管有任何大量初始数据\(u_0\)，每当
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（次二次情况）\(1\le \alpha <2 \quad \text {and} \quad \beta >\frac{n+4}{2}-\alpha ,\)
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（超二次情况）\(\beta >\frac{n}{2} \quad \text {and} \quad 2\le \alpha < 1+ \frac{2\beta }{n},\)
实际上\(T_{max}=\infty \)并且u和v是一致有界的。本文与Bian等人的研究结果一致。(Nonlinear Anal 176:178–191, 2018)，对于模型的简化抛物线-椭圆版本 ( \(\Diamond \) ) 也得出相同的结论，对应于\(\tau =0\)；更准确地说，这项工作将研究扩展到完全抛物线情况 Bian 等人。（非线性肛门176：178–191，2018）。
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