The modelling of linear and nonlinear reaction–subdiffusion processes is more subtle than normal diffusion and causes different phenomena. The resulting equations feature a spatial Laplacian with a temporal memory term through a time fractional derivative. It is known that the precise form depends on the interaction of dispersal and reaction, and leads to qualitative differences. We refine these results by defining generalized spectra through dispersion relations, which allows us to examine the onset of instability and in particular inspect Turing-type instabilities. These results are numerically illustrated. Moreover, we prove expansions that imply for one class of subdiffusion reaction equations algebraic decay for stable spectrum, whereas for another class this is exponential.

中文翻译：

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反应扩散系统和记忆：光谱，图灵不稳定性和衰变估计

线性和非线性反应-扩散过程的建模比正态扩散更微妙，并引起不同的现象。所得方程式具有通过时间分数导数的带有时间记忆项的空间拉普拉斯算子。众所周知，精确的形式取决于分散和反应的相互作用，并导致质的差异。我们通过色散关系定义广义谱来完善这些结果，这使我们可以检查不稳定性的发生，尤其是检查图灵型不稳定性。这些结果用数字表示。此外，我们证明了扩展，这对于一类子扩散反应方程而言，意味着稳定谱的代数衰减，而对于另一类，它是指数级的。