The study of pattern-forming instabilities in reaction–diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction–diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace–Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing–Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.

对生长或其他时间依赖域的反应扩散系统中模式形成不稳定性的研究出现在各种环境中，包括在发育生物学、空间生态学和实验化学中的应用。分析这种不稳定性是复杂的，因为任何空间均匀的基态都对时间有很强的依赖性，并且用于确定不稳定性开始的线性化扰动的结果结构本质上是非自治的。我们根据域的拉普拉斯-贝尔特拉米谱的时间演化和指定域演化的函数，获得了域上的反应-扩散系统中扩散驱动不稳定性的开始和结构的一般条件，这些不稳定性随时间演化。尽管研究问题具有普遍性，但我们的结果为用微分不等式表述的扩散不稳定性提供了充分条件，这些不等式既通用又易于实现。这些条件概括了文献中已知的大量结果，例如通常用作静态域上图灵不稳定性的充分标准的代数不等式，以及对特定类型的增长或特定域有效的近似渐近结果。我们在具有不同进化规律的各种域上展示了我们的一般图灵条件，特别是展示了如何在域随时间迅速变化或同质状态振荡时获得洞察力，例如在图灵的情况下—— Hopf 不稳定性。