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A one-dimensional logistic like equation with nonlinear and nonlocal diffusion: strong convergence to equilibrium


Authors: Arnaud Ducrot and David Manceau
Journal: Proc. Amer. Math. Soc. 148 (2020), 3381-3392
MSC (2010): Primary 35B40, 35B35, 35K55, 35Q92
DOI: https://doi.org/10.1090/proc/14971
Published electronically: March 17, 2020
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Abstract: We consider a one-dimensional logistic like equation with a nonlinear and nonlocal diffusion term with periodic boundary conditions. In this note we focus on smooth nonlocal kernels exhibiting a repulsive effect, that is expressed in terms of the positivity of their Fourier transforms. Here we describe the large time behaviour of the solutions for a large class of initial data. We roughly prove that the nontrivial solutions converge strongly to the unique spatially homogeneous equilibrium, by providing a refined study of the properties of the characteristic curves associated to the problem.


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Arnaud Ducrot
Affiliation: Normandie Université, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France
Email: arnaud.ducrot@univ-lehavre.fr

David Manceau
Affiliation: Normandie Université, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, 76600 Le Havre, France
Email: david.manceau@univ-lehavre.fr

DOI: https://doi.org/10.1090/proc/14971
Received by editor(s): September 12, 2019
Received by editor(s) in revised form: December 12, 2019
Published electronically: March 17, 2020
Communicated by: Wenxian Shen
Article copyright: © Copyright 2020 American Mathematical Society