Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

Abstract

This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction–diffusion equations with monostable reaction terms f, including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u 0(x) are chiefly assumed to be exponentially bounded as x tends to + ∞ and separated away from the unstable steady state 0 as x tends to − ∞. On the one hand, we give some conditions on u 0 which guarantee that, for some λ > 0, the quantity c λ = λ +f′(0)/λ is the asymptotic spreading speed, in the sense that lim t→+∞ u(t, ct) = 1 (the stable steady state) if c < c λ and lim t→+∞ u(t, ct) = 0 if c > c λ. These conditions are fulfilled in particular when u 0(x) e λx is asymptotically periodic as x → + ∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u 0 for which the ω-limit set of u(t, ct + x) as t tends to + ∞ is equal to the whole interval [0, 1] for all x ∈ ℝ and for all speeds c belonging to a given interval (γ1, γ2) with large enough γ1 < γ2 ≤ + ∞.