论文标题
对完全跨排水捕食者系统的存在理论和定性分析
Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system
论文作者
论文摘要
此手稿考虑了捕食者捕集系统$$的Neumann初始价值问题 \ left \ {\ begin {array} {l} u_t = d_1 u_ {xx} - χ_1（uv_x）_x + u（λ_1 -u + a_1 v），\\ [1mm] v_t = d_2 v_ {xx} +χ_2（vu_x）_x + v（λ_2-v-a_2 u）， \ end {array} \ right。 \ qquad \ qquad（\ star）$$作为空间域中的开放界间隔$ω$作为空间域，其中$ i \ in \ {1,2 \} $ the parameters $ d_i，a_i，λ_i$和$χ_i$是正面的。 由于两种相互相互作用的出租车型跨划出机制的同时出现，其中一种甚至很有吸引力，似乎还不清楚解决方案理论可以基于抛物线进化问题的经典结果来构建多远。为了创建一个能够提供全球存在结果以及有关定性行为的详细信息的分析设置，这项工作通过抛物线正则化来追求一种策略，在此过程中，（$ \ star $）通过某些涉及涉及薄膜类型的脱粒化扩散操作员的四级问题（$ \ star $）近似。 在设计期间，一个重大挑战与保持与（$ \ star $）正式相关的一些基本熵的结构保持一致性的野心；特别是，这将促使构建近似方案，包括两个免费参数，这些参数最终将以不同的方式固定，具体取决于$λ_2$相对于$a_2λ_1$的大小。
This manuscript considers a Neumann initial-boundary value problem for the predator-prey system $$ \left\{ \begin{array}{l} u_t = D_1 u_{xx} - χ_1 (uv_x)_x + u(λ_1-u+a_1 v), \\[1mm] v_t = D_2 v_{xx} + χ_2 (vu_x)_x + v(λ_2-v-a_2 u), \end{array} \right. \qquad \qquad (\star) $$ in an open bounded interval $Ω$ as the spatial domain, where for $i\in\{1,2\}$ the parameters $D_i, a_i, λ_i$ and $χ_i$ are positive. Due to the simultaneous appearance of two mutually interacting taxis-type cross-diffusive mechanisms, one of which even being attractive, it seems unclear how far a solution theory can be built upon classical results on parabolic evolution problems. In order to nevertheless create an analytical setup capable of providing global existence results as well as detailed information on qualitative behavior, this work pursues a strategy via parabolic regularization, in the course of which ($\star$) is approximated by means of certain fourth-order problems involving degenerate diffusion operators of thin film type. During the design thereof, a major challenge is related to the ambition to retain consistency with some fundamental entropy-like structures formally associated with ($\star$); in particular, this will motivate the construction of an approximation scheme including two free parameters which will finally be fixed in different ways, depending on the size of $λ_2$ relative to $a_2 λ_1$.
