Instabilities and propagation properties in a fourth-order reaction-diffusion equation

In this paper we investigate the instability and the propagation properties of a class of reaction-diffusion equations of fourth order. Two examples are introduced, the extended Fisher Kolmogorov equation (EFK), and the Swift-Hohenberg equation (SH). Both have been studied before by related methods (see for example, Peletier and Rottschafer, 2004 [19]; Van Saarloos, 2003 [24]) but the analysis here will support the introduced linear mechanism in front selection. These two equations support a patterned front solutions, and the double eigenvalue mechanism is used to provide evidence for that and to determine a minimal front speed.