Spreading speeds for one-dimensional monostable reaction-diffusion equations

Henri Berestycki (1) , Grégoire Nadin (2)

1 CAMS - Centre d'Analyse et de Mathématique sociales
2 LJLL - Laboratoire Jacques-Louis Lions

We establish in this article spreading properties for the solutions of equations of the type ∂ t u − a(x)∂ xx u − q(x)∂ 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 ≤ w such that lim t→+∞ sup 0≤x≤wt |u(t, x)−1| = 0 for all w ∈ (0, w) and lim t→+∞ sup x≥wt |u(t, x)| = 0 for all w > 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).

Mots clés: Key-words: Propagation and spreading properties; Heterogeneous reaction-diffusion equa-tions; Principal eigenvalues; Linear elliptic operator; Hamilton-Jacobi equations; Homoge-nization; Random stationarity and ergodicity; Almost periodicity AMS classification Primary: 35B40; 35B27; 35K57 Secondary: 35B50; 35P05; 47B65 Acknowledgements

Domaines: Equations aux dérivées partielles [math.AP]

Henri Berestycki, Grégoire Nadin. Spreading speeds for one-dimensional monostable reaction-diffusion equations. Journal of Mathematical Physics, 2012, 53, pp.115619. ⟨10.1063/1.4764932⟩. ⟨hal-01080135⟩