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Long Time Behavior of Random and Nonautonomous Fisher–KPP Equations: Part I—Stability of Equilibria and Spreading Speeds
Journal of Dynamics and Differential Equations ( IF 1.3 ) Pub Date : 2020-04-25 , DOI: 10.1007/s10884-020-09847-2
Rachidi B. Salako , Wenxian Shen

In the current series of two papers, we study the long time behavior of nonnegative solutions to the following random Fisher–KPP equation,

$$\begin{aligned} u_t =u_{xx}+a(\theta _t\omega )u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$(1)

where \(\omega \in \Omega \), \((\Omega , {\mathcal {F}},{\mathbb {P}})\) is a given probability space, \(\theta _t\) is an ergodic metric dynamical system on \(\Omega \), and \(a(\omega )>0\) for every \(\omega \in \Omega \). We also study the long time behavior of nonnegative solutions to the following nonautonomous Fisher–KPP equation,

$$\begin{aligned} u_t=u_{xx}+a_0(t)u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$(2)

where \(a_0(t)\) is a positive locally Hölder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on \(a(\omega )\), we show that the constant solution \(u=1\) of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval \([2\sqrt{{\underline{a}}}, 2\sqrt{{\bar{a}}}]\), where \({\underline{a}}\) and \({\bar{a}}\) are the least and the greatest means of \(a(\cdot )\), respectively, and hence the spreading speed interval is linearly determinate. It is shown that the solution of (1) with a nonnegative initial function which is bounded away from 0 for \(x\ll -1\) and is 0 for \(x\gg 1\) propagates at the speed \(2\sqrt{{\hat{a}}}\), where \({\hat{a}}\) is the mean of \(a(\cdot )\). Under some assumption on \(a_0(\cdot )\), we also show that the constant solution \(u=1\) of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that \(a(\omega )\) and \(a_0(t)\) are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher–KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).



中文翻译:

随机和非自治Fisher-KPP方程的长时间行为:第一部分-平衡和扩散速度的稳定性

在当前的两篇论文系列中,我们研究了以下随机Fisher-KPP方程的非负解的长时间行为,

$$ \ begin {aligned} u_t = u_ {xx} + a(\ theta _t \ omega)u(1-u),\ quad x \ in {{\ mathbb {R}}},\ end {aligned} $ $(1)

其中\(\ omega \ in \ Omega \)\((\ Omega,{\ mathcal {F}},{\ mathbb {P}})\)是给定的概率空间,\(\ theta _t \)\(\ Omega \)上的遍历度量动力学系统,并且每个\(\ omega \ in \ Omega \)具有\(a(\ omega)> 0 \)。我们还研究了以下非自治Fisher-KPP方程的非负解的长时间行为,

$$ \ begin {aligned} u_t = u_ {xx} + a_0(t)u(1-u),\ quad x \ in {{\ mathbb {R}}},\ end {aligned} $$(2)

其中\(a_0(t)\)是局部正Hölder连续函数。在本系列的第一部分中,我们研究了正平衡的稳定性和传播速度。在\(a(\ omega)\)的适当假设下,我们证明(1)的常数解((u = 1 \))关于严格的正摄动是渐近稳定的,并证明(1)具有确定性扩展速度间隔\([2 \ sqrt {{\ underline {a}}} \,2 \ sqrt {{\ bar {a}}} \\},其中\({\ underline {a}} \)\( {\ bar {a}} \)\(a(\ cdot)\)的最小和最大手段,因此,扩展速度间隔是线性确定的。结果表明,(1)的非负初始函数的解以\(x \ ll -1 \)的边界为0且对于\(x \ gg 1 \)的边界为0,的传播速度为\(2 \ sqrt {{\\ hat {a}}} \),其中\({\ hat {a}} \)\(a(\ cdot)\)的平均值。在关于\(a_0(\ cdot)\)的一些假设下,我们还表明(2)的常数解\(u = 1 \)是渐近稳定的,并且(2)允许有界的扩展速度区间。不假定\(a(\ omega)\)\(a_0(t)\)被一些正常数限制在上下。在这一部分中获得的结果是新的,并且扩展了有关Fisher-KPP方程扩展速度的文献中的现有结果。在系列的第二部分中,我们将研究(1)和(2)的过渡前沿的存在和稳定性。

更新日期:2020-04-25
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