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Some dependence results between the spreading speed and the coefficients of the space–time periodic Fisher–KPP equation

By Grégoire Nadin

Abstract

International audienceWe investigate in this paper the dependence relation between the space-time periodic coefficients A, q and µ of the reaction-diffusion equation ∂ t u − ∇ · (A(t, x)∇u) + q(t, x) · ∇u = µ(t, x)u(1 − u), and the spreading speed of the solutions of the Cauchy problem associated with this equation and compactly supported initial data. We prove in particular that (1) taking the spatial or temporal average of µ decreases the minimal speed, (2) if the coefficients do not depend on t and q ≡ 0, then increasiong the amplitude of the diffusion matrix A increases the minimal speed, (3) if A = I N , µ is a constant, then the introduction of a space periodic drift term q = ∇Q increases the minimal speed. To prove these results, we use a variational characterization of the spreading speed that involves a family of periodic principal eigenvalues associated with the linearization of the equation near 0. We are thus back to the investigation of the dependence relation between this family of eigenvalues and the coefficients

Topics: eigenvalue optimization, reaction-diffusion equations, 47A75, 35B27, spreading speed AMS subject classification: 34L15, 35B40, 35K10, 35P15, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC]
Publisher: Cambridge University Press (CUP)
Year: 2011
DOI identifier: 10.1017/S0956792511000027
OAI identifier: oai:HAL:hal-01360579v1
Provided by: Hal-Diderot

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