A diffusive logistic equation with U-shaped density dependent dispersal on the boundary
A diffusive logistic equation with U-shaped density dependent dispersal on the boundary
复制标题
边界上具有 U 形密度相关扩散的扩散 Logistic 方程
DOI：
10.12775/tmna.2018.047
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发表时间：
2019
影响因子：
0.7
通讯作者：
R. Shivaji
中科院分区：
文献类型：
--
作者：
Jerome Goddard;Q. Morris;Catherine Payne;R. Shivaji
我们研究稳态反应扩散方程的正解：\开始{方程 *} \开始{情况} - \Delta v = \lambda v（1-v），& x \in \Omega_0，\displaystyle \frac{\partial v}{\partial \eta} + \gamma \sqrt{\lambda}（v-A）^2 v =0，& x \in \partial \Omega_0，\end{cases} \end{equation*}其中$\Omega_0$是$\mathbb{R}^n$中的有界域;$n \ge 1$具有光滑边界$\partial\Omega_0 $，${\partial }/{\partial \eta}$是向外法向导数，$A \in（0，1）$是常数，$\lambda$，$\gamma$是正参数。当种群在栖息地边界上表现出U形密度依赖扩散时，这种模型出现在种群动力学的研究中。我们建立存在性，多重性和唯一性的结果，一定范围内的参数$\lambda$和$\gamma$。利用次上解的方法得到了该问题解的存在性和多重性结果。
We study positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} - \Delta v = \lambda v(1-v), & x \in \Omega_0, \\ \displaystyle \frac{\partial v}{\partial \eta} + \gamma \sqrt{\lambda} ( v-A)^2 v =0 , & x \in \partial \Omega_0, \end{cases} \end{equation*} where $\Omega_0$ is a bounded domain in $\mathbb{R}^n$; $n \ge 1$ with smooth boundary $\partial \Omega_0$, ${\partial }/{\partial \eta}$ is the outward normal derivative, $A \in (0,1)$ is a constant, and $\lambda$, $\gamma$ are positive parameters. Such models arise in the study of population dynamics when the population exhibits a U-shaped density dependent dispersal on the boundary of the habitat. We establish existence, multiplicity, and uniqueness results for certain ranges of the parameters $\lambda$ and $\gamma$. We obtain our existence and mulitplicity results via the method of sub-super solutions.