Spreading speeds for one-dimensional monostable reaction-diffusion equations

Abstract

We establish in this article spreading properties for the solutions of equations of the type ∂tu − a(x)∂xxu − q(x)∂xu = f(x, u), where a, q, f are only assumed to be uniformly continuous and bounded in x, the nonlinearity f is of monostable Kolmogorov, Petrovsky, and Piskunov type between two steady states 0 and 1 and the initial datum is compactly supported. Using homogenization techniques, we construct two speeds \documentclass[12pt]{minimal}\begin{document}$\underline{w}\le \overline{w}$\end{document}w̲≤w¯ such that \documentclass[12pt]{minimal}\begin{document}$\lim _{t\rightarrow +\infty }\sup _{0\le x\le wt} |u(t,x)-1| = 0$\end{document}limt→+∞sup0≤x≤wt|u(t,x)−1|=0 for all \documentclass[12pt]{minimal}\begin{document}$w\in (0,\underline{w})$\end{document}w∈(0,w̲) and \documentclass[12pt]{minimal}\begin{document}$\lim _{t\rightarrow +\infty } \sup _{x \ge wt} |u(t,x)| =0$\end{document}limt→+∞supx≥wt|u(t,x)|=0 for all \documentclass[12pt]{minimal}\begin{document}$w>\overline{w}$\end{document}w>w¯. These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear elliptic operators in unbounded domains. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where \documentclass[12pt]{minimal}\begin{document}$\overline{w}=\underline{w}$\end{document}w¯=w̲).