大的非均匀性能否产生靶标图样？
Can large inhomogeneities generate target patterns?
AI总结：
本文研究了含大缺陷振荡介质中靶标图样的存在性，利用粘性程函方程和加权Sobolev空间证明了慢代数衰减的大缺陷能产生靶标图样，并确定了其微小频率。
摘要：
我们研究了在具有弱局部耦合且存在杂质（或缺陷）的振荡介质中靶标图样的存在性。我们使用建立在平面上的粘性程函方程对这些系统进行建模，并将缺陷表示为扰动。与以往的结果不同，我们考虑大的缺陷，并使用具有慢代数衰减的函数来描述它们，即对于 $m \in (1,2]$，$g \sim {\mathcal O}(1/|x|^m)$。我们证明了这些缺陷能够产生靶标图样，并且正如强局域化杂质的情形一样，它们的频率在描述其强度的小参数的所有阶次之外都是微小的。我们的分析包括寻找靶标图样解的两个近似，一个在中间尺度上有效，另一个在远场中有效。这是使用加权 Sobolev 空间完成的，该空间使我们能够恢复相关线性算子的 Fredholm 性质；同时使用了隐函数定理，随后用它来证明存在性。通过匹配中间尺度和远场近似，我们随后确定了系统所选择的图样频率。
英文摘要：
We study the existence of target patterns in oscillatory media with weak local coupling and in the presence of an impurity, or defect. We model these systems using a viscous eikonal equation posed on the plane, and represent the defect as a perturbation. In contrast to previous results we consider large defects, which we describe using a function with slow algebraic decay, i.e., $g \sim {\mathcal O}(1/|x|^m)$ for $m \in (1,2]$. We prove that these defects are able to generate target patterns and that, just as in the case of strongly localized impurities, their frequency is small beyond all orders of the small parameter describing their strength. Our analysis consists of finding two approximations to target pattern solutions, one which is valid at intermediate scales and a second one which is valid in the far field. This is done using weighted Sobolev spaces, which allow us to recover Fredholm properties of the relevant linear operators, as well as the implicit function theorem, which is then used to prove existence. By matching the intermediate and far field approximations we then determine the frequency of the pattern that is selected by the system.
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