An Upper Bound for the Mean Speed of Transition Fronts and Unbounded Front Widths for Fisher KPP Equations in Almost Periodic Media

Abstract: In this paper, we investigate transition fronts and spreading solutions of Fisher--KPP equations in one-dimensional almost periodic media, both on the real line and on the lattice. Let $λ_1$ be the supremum of the spectrum of the linearized operator acting on $L<sup>2(\R)$ or $\ell<sup>2(\Z)$, respectively, and let (L(λ_1)) denote the spatial Lyapunov exponent at the spectral parameter (λ_1). We prove that, if $L(λ_1)&gt;0$, the global mean speed of any transition front is at most $λ_1/L(λ_1)$. Moreover, if a solution of the Cauchy problem spreads faster than this bound, its transition width is unbounded along a sequence of times. This occurs for a class of initial data with slowly decaying exponential tails. This contrasts with periodic media, where pulsating fronts exist at every speed above the minimal speed. As applications, we consider almost Mathieu coefficients and a continuous quasiperiodic coefficient with two frequencies. Together with the results of Nadin--Rossi and Liang--Wang--Zhou--Zhou \citep{LWZZ24}, our results provide an almost complete picture of the admissible speeds of generalized transition fronts, while leaving the critical cases open.