论文标题
在具有捕食者和猎物的多维种群模型中，均匀稳态附近的全球解决方案
Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis
论文作者
论文摘要
我们研究系统\ begin {align*} \ label {prob：star} \ tag {$ \ star $} \ begin {case} u_t =d_1Δu -χ_1\ nabla \ cdot（u \ nabla v） + u（λ_1 -μ_1 -μ_1u + a_1 v）\\ v_t =d_2Δv +χ_2\ nabla \ cdot（v \ nabla u） + v（λ_2 -μ_2 -V -a_2 v -a_2 u）\ end {cases} \ end {case} \ end eend {align*}（inign*}（interia）$ $ d_1，d_2，d_2，χ_1，χ_1，χ_2，χ_2，χ_2，λ_1，λ，λ，λ A_1，A_2> 0 $在平滑的，有界域中$ω\ subset \ Mathbb r^n $，$ n \ in \ {1，2，3 \} $。 在[0，\ infty）^2 $中没有对这些参数的进一步限制，我们证明存在稳定稳定的稳态$（u_ \ star，v_ \ star）\，这意味着存在$ \ varepsilon> 0 $，这样的话\partial_νv_0= 0 $在痕迹的意义上，\ begin {align*} \ | u_0 - u_ \ star \ star \ | _ {w^{2,2}（2,2}（ω）} + \ | v_0 -v_0 -v_ v_ v_ \ v_ \ star \ star \ | _________ {w^{2,2,2,2, varep*and*存在\ eqref {prob：star}的全局经典解决方案$（u，v），具有初始数据$ u_0，v_0 $ clive to $（u_ \ star，v_ \ star）$ in $ w^{2，2，2}（ω）$。 此外，收敛速率是指数级的，除了情况$λ_2μ_1=λ_1a_2 $，其中仅是代数。
We study the system \begin{align*}\label{prob:star} \tag{$\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.
