Spreading speeds for one-dimensional monostable reaction-diffusion equations

International audienceWe establish in this article spreading properties for the solutions of equations of the type ∂ t u − a(x)∂ xx u − q(x)∂ x u = 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 sup-ported. Using homogenization techniques, we construct two speeds w ≤ w such that lim t→+∞ sup 0≤x≤wt |u(t, x)−1| = 0 for all w ∈ (0, w) and lim t→+∞ sup x≥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 particu-lar, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where w = w)