Spreading speeds for one-dimensional monostable reaction-diffusion equations

TL;DR

This paper investigates the spreading speeds of solutions to one-dimensional monostable reaction-diffusion equations with heterogeneous coefficients, establishing conditions for the asymptotic behavior and deriving exact speeds in various random and periodic settings.

Contribution

It introduces new generalized principal eigenvalues to characterize spreading speeds in heterogeneous media, extending classical results to more general coefficient structures.

Findings

Established bounds for spreading speeds in heterogeneous media.

Derived exact spreading speeds for random stationary ergodic and almost periodic coefficients.

Connected spreading speeds to generalized principal eigenvalues.

Abstract

We establish in this article spreading properties for the solutions of equations of the type $\partial$ t u -- a(x)$\partial$ xx u -- q(x)$\partial$ 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 $\le$ w such that lim t$\rightarrow$+$\infty$ sup 0$\le$x$\le$wt |u(t, x)--1| = 0 for all w $\in$ (0, w) and lim t$\rightarrow$+$\infty$ sup x$\ge$wt |u(t, x)| = 0 for all w \textgreater{} 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).