\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}" name="description" /> Cambridge Journals Online - European Journal of Applied Mathematics - Abstract - Some dependence results between the spreading speed and the coefficients of the space–time periodic Fisher–KPP equation

European Journal of Applied Mathematics

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

GRÉGOIRE NADINa1

a1 CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France email: nadin@ann.jussieu.fr

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.

(Received April 22 2010)

(Revised January 06 2011)

(Accepted January 12 2011)

(Online publication February 28 2011)

Key words:

  • Eigenvalue optimization;
  • reaction-diffusion equations;
  • spreading properties;
  • space-time periodic parabolic equations