J. Evol. Equ.
© 2020 Springer Nature Switzerland AG
https://doi.org/10.1007/s00028-020-00588-9
Journal of Evolution
Equations
Strong and polynomial stability for delay semigroups
Shard Rastogi and Sachi Srivastava
Abstract. In this paper, we study strong and polynomial stability of the delay semigroup associated with
abstract delay differential equations. We also obtain new perturbation results for strong and polynomial
stability of C
0
-semigroups.
1. Introduction
In this article, we consider the abstract delay differential equations of the type
⎧
⎪
⎪
⎨
⎪
⎪
⎩
u
(t ) = Bu (t ) + u
t
, t > 0,
u (0) = x ,
u
0
= f ,
(DE)
where X is a Banach space, u
t
(θ) = u (t + θ), θ ∈ [−1, 0], x ∈ X , f lies in an appro-
priate space, ( B, D( B )) generates a C
0
-semigroup {T (t )}
t ≥0
on X , and is the delay
operator. The study of such abstract delay equations was pioneered by authors like
Hale [11], Webb [23] and Krasovski. Their approach was to associate a C
0
-semigroup
(defined on an appropriate state space) with the above delay equation such that the
semigroups’ orbits correspond to the mild solution of (DE). Investigation of qualita-
tive and asymptotic behaviour of this semigroup then provides detailed information
on the solution of the delay differential equation (see, for example, [1–3]).
Various notions of stability of C
0
-semigroups have been extensively studied in the
last few decades, particularly exponential stability (that is, convergence of all orbits
of {T (t )}
t ≥0
to zero with exponential rate) and strong stability (that is, convergence
of all orbits of {T (t )}
t ≥0
to zero). In recent years, a subclass of strongly stable C
0
-
semigroups, polynomially stable semigroups, has received much attention. Polynomial
stability, roughly speaking, corresponds to convergence of orbits of {T (t )}
t ≥0
to zero
at a ‘polynomial’ rate. A comprehensive survey of stability of C
0
-semigroups appears
in [9], while polynomial stability of C
0
-semigroups has been studied in [5, 8, 19, 20].
Mathematics Subject Classification: 47D06
Keywords: Delay differential equation, Polynomial stability, Strong stability, Uniformly bounded semi-
group, Delay semigroup.