A NEUTRAL VERSION OF THE DIFFUSIVE HUTCHINSON EQUATION: THE ADMISSIBLY STABLE MANIFOLDS AND ASYMPTOTIC BEHAVIOR

Abstract

Consider the neutral evolution equation d
= ( ) ( ) ( , ),
dt Fu B t u t t u t t + Φ under the conditions that the
family of linear partial differential operators (B t ( ))t≥0 generates the evolutionary process
(U t s ( , ))t s ≥ ≥0 having an exponential dichotomy on the half-line and the nonlinear delay operator
Φ ⋅ ( , ) t satisfes the ϕ -Lipschitz condition, where ϕ belongs to some admissible space on the halfline. The existence of admissibly stable manifolds of  -class is obtained in Nguyen - Trinh [Taiwan.
J. Math. 23 (2019), 897-923]. This short paper discusses the application of the theoretical result to
study asymptotic behavior of a neutral version of the diffusive Hutchinson equation..