论文标题
在变化环境中的抛物线式凯勒 - 塞格模型的强迫浪潮
Forced waves of parabolic-elliptic Keller-Segel models in shifting environments
论文作者
论文摘要
当前的论文与形式的转换环境中的凯勒 - 塞格化学脱位系统的强制波有关，\ begin {equination} \ begin {cases} u_t = u_t = u_ {xx}} -qu（uv_x）_x +_x +u（r（x-ct）-bu），\ quad x \ cr cr cr cr cr { 0 = v_ {xx} - νv+μu，\ quad x \ in \ mathbb {r} \ end {cases}（cases}（1）\ end {equation}，其中$χ$，$ b $，$ b $，$ν$，$ $ $ $ $ c \ in r $ c \ in R $ cy，in Resource $ contry ins the Resource $ ress $ R（x） $ r^*= \ sup_ {x \ in r} r（x）> 0 $，$ r（\ pm \ infty）：= \ lim_ {x \ to \ pm \ pm \ pm \ infty} r（x）$，并且$ r（ - r（ - \ f（ - \ f）假设$ b>2χμ$。如果$ r（ - \ infty）<0 <r（\ infty）$，则表明（1）具有连接$（\ frac {r^*} {b} {b} {b} {bfracμν\ frac {r^*} {r^*} {b} {b} {b}）$（0,0,0,0,0）$的强制波解决方案（\ frac {r^*} {b}）$ $ c> \ frac {χμR^*} {2 \sqrtν（b-χμ）} -2 \ sqrt {\ frac {r^*（b-χμ）} {b-χμ}}} $。 In the case that $r(\pm\infty)<0$, it is shown that (1) has a forced wave solution connecting $(0,0)$ and $(0,0)$ with speed $c$ provided that $χ$ is sufficiently small and $λ_\infty>0$, where $λ_\infty$ is the generalized principal eigenvalue of the operator $u(\cdot)\mapsto u_ {xx}（\ cdot）+cu_ {x}（\ cdot）+r（\ cdot）u（\ cdot）u（\ cdot）$在某些意义上。还进行了一些数值模拟。模拟表明在某些参数区域中存在强制波解的存在，而理论结果中未涵盖，这引起了几个要进一步研究的问题，并提供了理论结果的一些例证。
The current paper is concerned with the forced waves of Keller-Segel chemoattraction systems in shifting environments of the form, \begin{equation} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(r(x-ct)-bu),\quad x\in\mathbb{R}\cr 0=v_{xx}- νv+μu,\quad x\in \mathbb{R} \end{cases} (1) \end{equation} where $χ$, $b$, $ν$, and $μ$ are positive constants, $c\in R$, the resource function $r(x)$ is globally Hölder continuous, bounded, $r^*=\sup_{x\in R}r(x)>0$, $r(\pm \infty):=\lim_{x\to \pm\infty}r(x)$ exist, and either $r(-\infty)<0<r(\infty)$, or $r(\pm\infty)<0$. Assume that $b>2χμ$. In the case that $r(-\infty)<0<r(\infty)$, it is shown that (1) has a forced wave solution connecting $(\frac{r^*}{b},\fracμν\frac{r^*}{b})$ and $(0,0)$ with speed $c$ provided that $c>\frac{χμr^*}{2\sqrt ν(b-χμ)}- 2\sqrt{\frac{r^*(b-2χμ)}{b-χμ}}$. In the case that $r(\pm\infty)<0$, it is shown that (1) has a forced wave solution connecting $(0,0)$ and $(0,0)$ with speed $c$ provided that $χ$ is sufficiently small and $λ_\infty>0$, where $λ_\infty$ is the generalized principal eigenvalue of the operator $u(\cdot)\mapsto u_{xx}(\cdot)+cu_{x}(\cdot)+r(\cdot)u(\cdot)$ on $R$ in certain sense. Some numerical simulations are also carried out. The simulations indicate the existence of forced wave solutions in some parameter regions which are not covered in the theoretical results, induce several problems to be further studied, and also provide some illustration of the theoretical results.
