This work is concerned with a two-component parabolic system accounting for a doubly cross-diffusive interaction mechanism which was was predicted in Physical Review Letters 91, 218102 (2003) as responsible for the occurrence of certain solitary propagating waves in so-called pursuit-evasion systems. This system formally possesses two basic entropy-like structures, but especially in the presence of large data the regularity features thereby implied seem insufficient to ensure global extensibility of local-in-time classical solutions provided by known results on classical solvability in general parabolic systems of not necessarily tridiagonal type. Attempting to nevertheless develop a basic theory of existence and qualitative behavior, the manuscript firstly constructs global solutions within a natural concept of weak solvability and for arbitrarily large data, and secondly derives a result on large-time stabilization toward homogeneous equilibria. A major challenge connected with this appears to consist in designing a suitable regularization which complies with the two requirements of asserting global solvability in the corresponding approximate systems on the one hand, and of retaining consistency with essential structural properties on the other. To adequately cope with this, a fourth-order regularization is pursued which, besides essentially respecting said entropy features, conforms to the fundamental sine qua non of positivity preservation by involving thin-film type degeneracies in the associated artificial diffusion operators. Here the use of embeddings enforces a restriction to spatially one-dimensional settings, in which an apparently novel refinement of Gagliardo-Nirenberg interpolation reveals a crucial $L^1$ compactness feature of corresponding cross-diffusive fluxes.

这项工作涉及双组分抛物占一个双向交叉扩散互动机制这是在预测物理评论快报91，218102（2003）在所谓的逃避系统中，它负责某些孤立的传播波的发生。该系统形式上具有两个基本的类似熵的结构，但是特别是在存在大数据的情况下，由此暗示的规律性特征似乎不足以确保由一般抛物线型系统中经典可解性的已知结果所提供的局部时间经典解的全局可扩展性。不一定是三对角型的。然而，试图发展存在和定性行为的基本理论，该手稿首先在弱可解性和任意大数据的自然概念内构造了整体解，其次得出了朝着均质平衡进行长时间稳定的结果。与此相关的主要挑战似乎在于设计合适的正则化，该正则化一方面符合在相应的近似系统中主张整体可溶性，另一方面又与基本结构性质保持一致的两个要求。为了适当地解决这一问题，追求了四阶正则化，除了基本尊重所述熵特征外，还符合基本必要条件通过在相关联的人工扩散运营商涉及薄膜型简并积极保存。在这里，嵌入的使用对空间一维设置施加了限制，在其中，Gagliardo-Nirenberg插值的一种明显新颖的细化揭示了一个关键${\mathrm{大号}}^{\mathrm{1个}}$ 相应的交叉扩散通量的紧密度特征。