Vol. 8, No. 6, 2015

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ISSN: 1948-206X (e-only)
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Transition waves for Fisher–KPP equations with general time-heterogeneous and space-periodic coefficients

Grégoire Nadin and Luca Rossi

Vol. 8 (2015), No. 6, 1351–1377
DOI: 10.2140/apde.2015.8.1351

We study existence and nonexistence results for generalized transition wave solutions of space-time heterogeneous Fisher–KPP equations. When the coefficients of the equation are periodic in space but otherwise depend in a fairly general fashion on time, we prove that such waves exist as soon as their speed is sufficiently large in a sense. When this speed is too small, transition waves do not exist anymore; this result holds without assuming periodicity in space. These necessary and sufficient conditions are proved to be optimal when the coefficients are periodic both in space and time. Our method is quite robust and extends to general nonperiodic space-time heterogeneous coefficients, showing that transition wave solutions of the nonlinear equation exist as soon as one can construct appropriate solutions of a given linearized equation.

Fisher–KPP equation, reaction-diffusion, traveling waves, generalized transition waves, generalized principal eigenvalues
Mathematical Subject Classification 2010
Primary: 35B40, 35K10, 35K57, 35P05, 35B51
Received: 24 March 2014
Revised: 15 March 2015
Accepted: 11 May 2015
Published: 5 September 2015
Grégoire Nadin
Sorbonne Universités
UPMC Univ Paris 06, UMR 7598
Laboratoire Jacques-Louis Lions
75005 Paris
Luca Rossi
Dipartimento di Matematica
Università di Padova
via Trieste 63
I-35121 Padova