Spreading speeds for one-dimensional monostable reaction-diffusion equations

We establish in this article spreading properties for the solutions of equations of the type ∂tu − a(x)∂xxu − q(x)∂xu = f(x, u), where a, q, f are only assumed to be uniformly continuous and bounded in x, the nonlinearity f is of monostable KPP type between two steady states 0 and 1 and the initial datum is compactly supported. Using homogenization techniques, we construct two speeds w ≤ w such that limt→+∞ sup0≤x≤wt |u(t, x)−1| = 0 for all w ∈ (0, w) and limt→+∞ supx≥wt |u(t, x)| = 0 for all w > w. These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear elliptic operators in unbounded domains. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where w = w). Key-words: Propagation and spreading properties, Heterogeneous reaction-diffusion equations, Principal eigenvalues, Linear elliptic operator, Hamilton-Jacobi equations, Homogenization, Random stationarity and ergodicity, Almost periodicity. AMS classification. Primary: 35B40, 35B27, 35K57. Secondary: 35B50, 35P05, 47B65. This study was supported in part by the French “Agence Nationale de la Recherche” within the project PREFERED. Henri Berestycki was partially supported by an NSF FRG grant DMS-1065979. Part of this work was carried out while the first author was visiting the Department of Mathematics of the University of Chicago.