Article Dans Une Revue Journal of Dynamics and Differential Equations Année : 2012

Maximal and Minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media

Résumé

This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, partial derivative(t)u = partial derivative(xx)u + f(x, u), associated with a compactly supported initial datum. A typical nonlinearity we consider is f(x, u) = mu(0)(phi(x))u(1 - u), where mu(0) is a 1-periodic function and phi is a C-1 increasing function that satisfies lim(x ->+infinity) phi(x) = +infinity and lim(x ->+infinity) phi'(x) = 0. Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for u for large times, depending on the speed of the convergence of phi at infinity. If phi grows sufficiently slowly, then we prove that the spreading speed of u oscillates between two distinct values. If phi grows rapidly, then we compute explicitly a unique and well determined speed of propagation w(infinity), arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.

Dates et versions

hal-00939216 , version 1 (30-01-2014)

Identifiants

Citer

Jimmy Garnier, Thomas Giletti, Grégoire Nadin. Maximal and Minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media. Journal of Dynamics and Differential Equations, 2012, 24 (3), pp.521--538. ⟨10.1007/s10884-012-9254-5⟩. ⟨hal-00939216⟩
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