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The work proposes and studies a one-dimensional model, which involvesnonlocal interactions and finite propagation speed. It shows that the generalreaction-diffusion equation, the Swift-Hohenberg equation and the generalKuramoto-Sivashinsky equation represent special cases of the proposed model forlimited spatial interaction ranges and for infinite propagation speeds. After adetailed validity study on the generalization conditions, the three equationsare extended to involve finite propagation speeds. Moreover linear stabilitystudies of the extended equations reveal critical propagation speeds and noveltypes of instabilities in all three equations. In addition, an extendeddiffusion equation is derived and studied in some detail with respect to finitepropagation speeds. The extended model allows for the explanation of recentexperimental results on non-Fourier heat conduction in non-homogeneousmaterial.Comment: 20 pages, 6 figure

Generalization of the reaction-diffusion, Swift-Hohenberg, and Kuramoto-Sivashinsky equations and effects of finite propagation speeds