Fast propagation for KPP equations with slowly decaying initial conditions

By François Hamel, Lionel Roques. 2009

133 citations recorded by the source corpus.

Abstract This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher–KPP reaction–diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time. This paper contains the first systematic study of the large-time behavior of solutions of KPP equations with slowly decaying initial conditions. Our results are in sharp contrast with the well-studied case of exponentially bounded initial conditions.

Subfields: Computer Networks and Communications, Mathematical Physics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/3d9cd97f66a9159c90a74e74e182fce463d59f93

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Citations in the corpus

The Influence of Fractional Diffusion in Fisher-KPP Equations (2012, background): “In fact, this behaviour is consubstantial to fast diffusion, and the model may be relevant to explain fast recolonisation events in ecology; see a discussion in [11].”
The Fisher–KPP problem with doubly nonlinear “fast” diffusion (2016, background): “In particular, in [35], Hamel and Roques found that the solutions of the Fisher-KPP problem with linear diffusion i.”
Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian (2015, background): “It is shown that the front propagation speed could be very fast depending on initial values (see [12,60], etc.”
Super-linear spreading in local and non-local cane toads equations (2015, background): “This is an addition to the growing list of “accelerating fronts” that have attracted some interest in recent years [9,13,14, 16,19,24,26,29].”
The Fisher-KPP problem with doubly nonlinear diffusion (2016, background): “In particular, in [30], Hamel and Roques found that the solutions of the Fisher-KPP problem with linear diffusion i.”
Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations (2012, background): “4), Hamel and Roques [9] also computed the position of the level sets of the function u(t, ·) as t→ +∞, according to the precise asymptotic behaviour of u0 at +∞.”
Generalized Transition Waves and Their Properties (2010, background): “hat f(0) = f(1) = 0, f(s) >0 and f0(s) f(s)=son (0;1), there are solutions u: [0;+1) R ![0;1]; (t;x) 7!u(t;x) such that u(t;1 ) = 1; u(t;+1) = 0 for all t0; and lim t!+1 ku x(t;)k L1(R) = 0; see [19]. At each time t, u(t;) connects 1 to 0, but since the solutions become uniformly atter and atter ”
Spreading speeds for a two-species competition-diffusion system (2018, background): “On an other hand, [18] exhibited a family of initial conditions for (2) such that the solution spreads with accelerated speed.”