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arXiv 2004.04515math.AP
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具有捕食和食饵趋性的多维种群模型齐次平衡态附近的整体解

Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis

Mario Fuest

备注：
27 pages; changed title and incorporated referee comments
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摘要：

我们研究系统\begin{align*}\label{prob:star} \tag{$\star$} \begin{cases} u_t = D_1 Δu - χ_1 \nabla \cdot (u \nabla v) + u(λ_1 - μ_1 u + a_1 v) \\ v_t = D_2 Δv + χ_2 \nabla \cdot (v \nabla u) + v(λ_2 - μ_2 v - a_2 u) \end{cases} \end{align*}（除其他外）$D_1, D_2, χ_1, χ_2, λ_1, λ_2, μ_1, μ_2, a_1, a_2 > 0$在光滑，有界域$Ω\subset \mathbb R^n$，$n \in \{1, 2, 3\}$。没有任何进一步的限制，这些参数，我们证明，存在一个恒定的稳定的稳态$(u_\star, v_\star) \in [0, \infty)^2$，这意味着有$\varepsilon > 0$，这样，如果$u_0, v_0 \in W^{2, 2}(Ω)$是非负的$\partial_νu_0 = \partial_νv_0 = 0$的意义上的痕迹和\begin{align*} \|u_0 - u_\star\|_{W^{2,2}(Ω)} + \|v_0 - v_\star\|_{W^{2,2}(Ω)} < \varepsilon, \end{align*}，然后存在一个全球性的经典解决方案$(u, v)$的\eqref{prob:star}与初始数据$u_0, v_0$收敛到$(u_\star, v_\star)$在$W^{2, 2}(Ω)$。此外，收敛速度是指数的，除了$λ_2 μ_1 = λ_1 a_2$的情况下，它是唯一的代数。

英文摘要：

We study the system \begin{align*}\label{prob:star} \tag{$\star$} \begin{cases} u_t = D_1 Δu - χ_1 \nabla \cdot (u \nabla v) + u(λ_1 - μ_1 u + a_1 v) \\ v_t = D_2 Δv + χ_2 \nabla \cdot (v \nabla u) + v(λ_2 - μ_2 v - a_2 u) \end{cases} \end{align*} (inter alia) for $D_1, D_2, χ_1, χ_2, λ_1, λ_2, μ_1, μ_2, a_1, a_2 > 0$ in smooth, bounded domains $Ω\subset \mathbb R^n$, $n \in \{1, 2, 3\}$. Without any further restrictions on these parameters, we prove that there exists a constant stable steady state $(u_\star, v_\star) \in [0, \infty)^2$, meaning that there is $\varepsilon > 0$ such that, if $u_0, v_0 \in W^{2, 2}(Ω)$ are nonnegative with $\partial_νu_0 = \partial_νv_0 = 0$ in the sense of traces and \begin{align*} \|u_0 - u_\star\|_{W^{2,2}(Ω)} + \|v_0 - v_\star\|_{W^{2,2}(Ω)} < \varepsilon, \end{align*} then there exists a global classical solution $(u, v)$ of \eqref{prob:star} with initial data $u_0, v_0$ converging to $(u_\star, v_\star)$ in $W^{2, 2}(Ω)$. Moreover, the convergence rate is exponential, except for the case $λ_2 μ_1 = λ_1 a_2$, where it is is only algebraical.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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