Maximal and minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media

This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, $$\partial_t u = \partial_{xx} u + f(x,u),$$ 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 $\mathcal{C}^1$ increasing function that satisfies $\lim_{x\to +\infty} \phi (x) = +\infty$ and $\lim_{x\to +\infty} \phi' (x) = 0$. 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 $w_\infty$, arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.