This work provides analytical results towards applications in the field of invasive-invaded systems modeled with nonlinear diffusion and with advection. The results focus on showing regularity, existence and uniqueness of weak solutions using the condition of a nonlinear slightly positive parabolic operator and the reaction-absorption monotone properties. The coupling in the reaction-absorption terms, that characterizes the species interaction, impedes the formulation of a global comparison principle that is shown to exist locally. Additionally, this work provides analytical solutions obtained as selfsimilar minimal and maximal profiles. A propagating diffusive front is shown to exist until the invaded specie notes the existence of the invasive. When the desertion of the invaded starts, the diffusive front vanishes globally and the nonlinear diffusion concentrates only on the propagating tail which exhibits finite speed. Finally, the invaded specie is shown to exhibit an exponential decay along a characteristic curve. Such exponential decay is not trivial in the nonlinear diffusion case and confirms that the invasive continues to feed on the invaded during the desertion.

中文翻译：

---

具有平流的非Lipschitz多孔介质方程的侵入系统

这项工作为使用非线性扩散和平流建模的侵入性入侵系统领域的应用提供了分析结果。结果集中展示了利用非线性微正抛物算子条件和反应吸收单调性质的弱解的规律性、存在性和唯一性。表征物种相互作用的反应-吸收项中的耦合阻碍了全局比较原理的制定，该原理显示为局部存在。此外，这项工作提供了作为自相似最小和最大轮廓获得的分析解决方案。在被入侵物种注意到入侵物种的存在之前，显示出存在传播的扩散前沿。当被入侵者的遗弃开始时，扩散前沿整体消失，非线性扩散仅集中在传播速度有限的尾部。最后，入侵物种显示出沿特征曲线呈指数衰减。这种指数衰减在非线性扩散情况下并非微不足道，并证实入侵者在遗弃期间继续以入侵者为食。