This paper is concerned with the existence and stability of steady states of a reaction-diffusion-ODE system arising from the theory of biological pattern formation. We are interested in spontaneous emergence of patterns from spatially heterogeneous environments, hence assume that all coefficients in the equations can depend on the spatial variable. We give some sufficient conditions on the coefficients which guarantee the existence of far-from-the-equilibrium patterns with jump discontinuity and then verify their stability in a weak sense. Our conditions cover the case where the number of equilibria of the kinetic system (i.e., without diffusion) changes from one to three in the spatial interval, which is not obtained by a small perturbation of constant coefficients. Moreover, we consider the asymptotic behavior of steady states as the diffusion coefficient tends to infinity. Some examples and numerical simulations are given to illustrate the theoretical results.

中文翻译：

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非均匀介质中具有滞后的反应扩散ODE系统中模式的存在性和稳定性

本文关注的是由生物图案形成理论引起的反应扩散-ODE系统稳态的存在性和稳定性。我们对空间异质环境中模式的自发出现很感兴趣，因此假设方程中的所有系数都可以取决于空间变量。我们给这些系数提供了一些充分的条件，以保证存在具有跳跃间断的远离平衡的模式，然后在较弱的意义上验证其稳定性。我们的条件涵盖了动力学系统平衡点（即无扩散）的数量在空间间隔内从1变为3的情况，这不能通过对常数系数的小扰动来获得。而且，当扩散系数趋于无穷大时，我们考虑稳态的渐近行为。给出了一些例子和数值模拟来说明理论结果。