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

Published online by Cambridge University Press:  28 February 2011

GRÉGOIRE NADIN
GRÉGOIRE NADIN*
Affiliation:
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France email: nadin@ann.jussieu.fr
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Abstract

We investigate in this paper the dependence relation between the space–time periodic coefficients A, q and μ of the reaction–diffusion equation\begin{linenomath}\[ \partial_t u -\nabla \cdot (A(t, x)\nabla u)+q(t, x)\cdot \nabla u = \mu(t, x)u(1-u), \] \end{linenomath}and the spreading speed of the solutions of the Cauchy problem associated with compactly supported initial data. We prove in particular that (1) taking the spatial or temporal average of μ decreases the minimal speed, (2) if μ is not constant with respect to x, then increasing the amplitude of the diffusion matrix A does not necessarily increase the minimal speed and (3) if A = IN, μ is a constant, then the introduction of a space periodic drift term q = ∇Q decreases the minimal speed. In order to prove these results, we use a variational characterisation of the spreading speed that involves a family of periodic principal eigenvalues associated with the linearisation of the equation near zero. We are thus back to the investigation of the dependence relation between this family of eigenvalues and the coefficients.

Keywords

Eigenvalue optimizationreaction-diffusion equationsspreading propertiesspace-time periodic parabolic equations

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 2 , April 2011 , pp. 169 - 185
Copyright
Copyright © Cambridge University Press 2011

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