This work is concerned with the doubly degenerate cross-diffusion system { ut = (uvux)x − (u 2vvx)x + uv, vt = vxx − uv, (0.1) that has been proposed as a model for experimentally observable quite complex pattern formation phenomena in bacterial populations. It is shown that for any initial data satisfying adequate regularity and positivity assumptions, a no-flux initial-boundary value problem for (0.1) in a bounded real interval possesses a global weak solution which is continuous in its first and essentially smooth in its second component. This solution is seen to asymptotically stabilize in the sense that u(·, t) → u∞ and v(·, t) → 0 as t→ ∞ (0.2) with some nonnegative u∞ ∈ C (Ω) which can be obtained as the evaluation of a weak solution z ∈ C(Ω× [0, 1]) to a porous medium-type parabolic problem at the finite time 1. It is moreover revealed that for each suitably regular nonnegative function u⋆ on Ω, the pair (u⋆, 0), formally constituting an equilibrium of (0.1), is stable in an appropriate sense. This finally implies a sufficient criterion for the limit u∞ in (0.2) to be spatially heterogeneous. The latter properties are in sharp contrast to known asymptotic features of corresponding nutrient taxis systems involving linear non-degenerate diffusion, as for which the literature appears to exclusively provide results on solutions which approach spatially constant states in the large time limit.

中文翻译：

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空间同质性最终会在营养出租车系统中盛行吗？自主抛物线流中快速扩散衰减的结构支撑范式

这项工作涉及双退化交叉扩散系统 { ut = (uvux)x − (u 2vvx)x + uv, vt = vxx − uv, (0.1) 已被提议作为实验可观察的相当复杂模式的模型细菌种群中的形成现象。结果表明，对于满足充分正则性假设的任何初始数据，(0.1) 在有界实区间的无通量初边界值问题具有全局弱解，其第一次连续，第二次基本平滑成分。在 u(·, t) → u∞ 和 v(·, t) → 0 as t→ ∞ (0.2) 的意义上，该解被视为渐近稳定，其中可以得到一些非负的 u∞ ∈ C (Ω)作为多孔介质型抛物线问题在有限时间 1 的弱解 z ∈ C(Ω× [0, 1]) 的评估。此外还揭示，对于Ω上的每个适当的正则非负函数u⋆，形式上构成（0.1）的平衡的对（u⋆，0）在适当的意义上是稳定的。这最终意味着 (0.2) 中的极限 u∞ 在空间上是异质的。后者的特性与涉及线性非简并扩散的相应营养出租车系统的已知渐近特征形成鲜明对比，因为文献似乎专门提供了在大时间限制内接近空间恒定状态的解决方案的结果。