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

Abstract

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.