Spreading speeds for one-dimensional monostable reaction-diffusion equations

Abstract

International audienceWe 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)

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

“…For recent developments in asymptotic spreading of a single population in heterogeneous environments, we refer to [5,7,19] for the one-dimensional case, and to [6,8,45,56] for higher-dimensional case.…”

“…It was introduced by Freidlin [17], who employed probabilistic arguments to study the asymptotic behavior of solution to the Fisher-KPP equation modeling the population of a single species. Subsequently, the result was generalized by Evans and Souganidis using PDE arguments; see also [6,8,33,34,37,43]. The method was also applied by Barles, Evans and Souganidis [4] to study KPP systems, where several species spread at a common spreading speed.Finally, we also mention some related works on the Cauchy problem of interacting species spreading into open habitat.…”

Spreading speeds of KPP-type lattice systems in heterogeneous media

“…In this paper, we investigate the spreading properties for (2) in general heterogeneous media. Motivated by [8], we establish the theory of generalized principal eigenvalues of linear lattice systems to estimate the lower and upper spreading speeds ω * , ω * . Aiming to estimate the spreading speeds via the generalized principal eigenvalues, we also develop some new discrete Harnack-type inequalities, and homogenization techniques for lattice equations.…”

“…Then ε ln ψ ε (·) → 0 as ε → 0 locally uniform in (N, +∞). Using an argument similar to the proof of [8,Propsition 4.3], we can obtain a sequence (ε n , t n , x n ) → (0, t 0 , x 0 ) as n → +∞. Moreover, z εn (t n , x n ) → z * (t 0 , x 0 ) as n → +∞, and z εn − φ − ε n ln ψ εn reaches its strict minimum at (t n , x n ) over B r (t 0 , x 0 ) for n ≥ n 0 .…”

Speed-up of reaction-diffusion fronts by a line of fast diffusion

“…Due to its importance, several derivations have been given: Weinberger [37] with dynamical systems tools, BerestyckiHamel-Nadin [6] with PDE tools. If the x-dependence in f has no particular structure, many tools have been introduced to study the large time behaviour of the level sets of u: local spreading velocities (Berestycki-Hamel-Nadireashvili [7], [8]), transition fronts (Berestycki-Hamel [4]), generalised eigenvalues (Berestycki-Rossi [12]) with an application to a sharp description of the one-dimensional situation (Berestycki-Nadin [9]). System (1.1) presents a novel model of active heterogeneities, which will display new behaviours that we sum up as follows: even if there is no source term on the road, the overall propagation is always enhanced.…”