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Transactions of the American Mathematical Society

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A center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the Gurtin-McCamy equation


Authors: Arnaud Ducrot and Pierre Magal
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37L10, 35J61, 35C07, 34C23, 47D62
DOI: https://doi.org/10.1090/tran/7780
Published electronically: April 4, 2019
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Abstract: This work is mainly motivated by the study of periodic wave train solutions for the so-called Gurtin-McCamy equation. To that aim we construct a smooth center manifold for a rather general class of abstract second order semilinear differential equations involving nondensely defined operators. We revisit results on commutative sums of linear operators using the integrated semigroup theory. These results are used to reformulate the notion of the weak solutions of the problem. We also derive a suitable fixed point formulation for the graph of the local center manifold that allows us to conclude the existence and smoothness of such a local invariant manifold. Then we derive a Hopf bifurcation theorem for second order semilinear equations. This result is applied to studying the existence of periodic wave trains for the Gurtin-McCamy problem, that is, for a class of nonlocal age-structured equations with diffusion.


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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

Pierre Magal
Affiliation: Université de Bordeaux, IMB, UMR 5251, F-33076 Bordeaux, France; and CNRS, IMB, UMR 5251, F-33400 Talence, France
Email: pierre.magal@u-bordeaux.fr

DOI: https://doi.org/10.1090/tran/7780
Keywords: Center manifold, second order semilinear equations, integrated semigroups, Hopf bifurcation, periodic wave trains, Gurtin--McCamy equation
Received by editor(s): December 20, 2017
Received by editor(s) in revised form: December 4, 2018
Published electronically: April 4, 2019
Article copyright: © Copyright 2019 American Mathematical Society