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Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weight
In this research work, we consider the second-order viscoelastic equation with a weak internal damping, a time-varying delay term and nonlinear weights u tt ( t ) + A u ( t ) - ∫ 0 t g ( t - s ) A u ( s ) d s + μ 1 ( t ) u t ( t ) + μ 2 ( t ) u t ( t - τ ( t ) ) = 0 ∀ t > 0 , together with suitab...
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| Published in: | Ricerche di matematica 2024-02, Vol.73 (1), p.433-469 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | In this research work, we consider the second-order viscoelastic equation with a weak internal damping, a time-varying delay term and nonlinear weights
u
tt
(
t
)
+
A
u
(
t
)
-
∫
0
t
g
(
t
-
s
)
A
u
(
s
)
d
s
+
μ
1
(
t
)
u
t
(
t
)
+
μ
2
(
t
)
u
t
(
t
-
τ
(
t
)
)
=
0
∀
t
>
0
,
together with suitable initial conditions. We first prove the existence of a unique global weak solution by means of the classical Faedo–Galerkin method. Then, by assuming the general condition:
g
′
(
t
)
≤
-
ξ
(
t
)
H
(
g
(
t
)
)
,
∀
t
≥
0
,
where
H
is a positive increasing and convex function and
ξ
is a positive function which is not necessarily monotone, we establish optimal explicit and general stability estimates which rely on the well-known multipliers method and some properties of convex functions. This study generalizes and improves many earlier ones in the existing literature. |
|---|---|
| ISSN: | 0035-5038 1827-3491 |
| DOI: | 10.1007/s11587-021-00617-w |