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Transition waves for
Fisher–KPP equations with general time-heterogeneous and
space-periodic coefficients
Grégoire Nadin and Luca Rossi |
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Vol. 8 (2015), No. 6, 1351–1377
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DOI: 10.2140/apde.2015.8.1351
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Abstract |
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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. |
Keywords
Fisher–KPP equation, reaction-diffusion, traveling waves,
generalized transition waves, generalized principal
eigenvalues
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Mathematical Subject Classification 2010
Primary: 35B40, 35K10, 35K57, 35P05, 35B51
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Milestones
Received: 24 March 2014
Revised: 15 March 2015
Accepted: 11 May 2015
Published: 5 September 2015
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Authors |
| Grégoire Nadin | |
| Sorbonne Universités UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions 75005 Paris France |
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| Luca Rossi | |
| Dipartimento di Matematica Università di Padova via Trieste 63 I-35121 Padova Italy |
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