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

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

2011-11-16

arxiv.org/abs/1111.3860v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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