Generalized Transition Fronts for One-Dimensional Almost Periodic Fisher-KPP Equations

This paper investigates the existence of generalized transition fronts for Fisher-KPP equations in one-dimensional, almost periodic media. Assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction, we show that such fronts exist if and only if their average speed is above an explicit threshold. This hypothesis is satisfied in particular when the reaction term does not depend on x or (in some cases) is small enough. Moreover, except for the threshold case, the fronts we construct and their speeds are almost periodic, in a sense. When our hypothesis is no longer satisfied, such generalized transition fronts still exist for an interval of average speeds, with explicit bounds. Our proof relies on the construction of sub and super solutions based on an accurate analysis of the properties of the generalized principal eigenvalues.