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

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.