Unified stability analysis for a Volterra integro-differential equation under creation time perspective

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 73; no. 3
• Main Authors: Gomes Tavares, Eduardo H., Jorge Silva, Marcio A., Ma, To Fu
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2022; Springer Nature B.V
• Subjects: Differential equations; Engineering; Limit analysis; Mathematical Methods in Physics; Stability analysis; Theoretical and Applied Mechanics; Volterra integral equations; Wave equation; Viscoelasticity; Stability; 74D05; Energy; 35B35; Memory; 35B40; 35L53
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-022-01756-2

Many real-world applications are modeled by Volterra integral–differential equations of the form
u
tt
-
Δ
u
+
∫
α
t
g
(
t
-
s
)
Δ
u
(
s
)
d
s
=
0
in
Ω
×
(
0
,
∞
)
,
where
Ω
is a bounded domain of
R
N
and
g
is a memory kernel. Our main concern is with the concept of so-called
creation time
, the time
α
where past history begins. Separately, the cases
α
=
-
∞
(history) and
α
=
0
(null history) were extensively studied in the literature. However, as far as we know, there is no unified approach with respect to the intermediate case
-
∞
<
α
<
0
. Therefore we provide new stability results featuring (
i
) uniform and general stability when the creation time
α
varies over full range
(
-
∞
,
0
)
and (
ii
) connection between the history and the null history cases by means of a rigorous backward (
α
→
-
∞
) and forward (
α
→
0
-
) limit analysis.