Maximal and Minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media
Résumé
This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, partial derivative(t)u = partial derivative(xx)u + f(x, u), associated with a compactly supported initial datum. A typical nonlinearity we consider is f(x, u) = mu(0)(phi(x))u(1 - u), where mu(0) is a 1-periodic function and phi is a C-1 increasing function that satisfies lim(x ->+infinity) phi(x) = +infinity and lim(x ->+infinity) phi'(x) = 0. Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for u for large times, depending on the speed of the convergence of phi at infinity. If phi grows sufficiently slowly, then we prove that the spreading speed of u oscillates between two distinct values. If phi grows rapidly, then we compute explicitly a unique and well determined speed of propagation w(infinity), arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.