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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 |
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| container_title | Ricerche di matematica |
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| creator | Makheloufi, Hocine Bahlil, Mounir |
| description | 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. |
| doi_str_mv | 10.1007/s11587-021-00617-w |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2954219874</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2954219874</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-91a24e838cdc26ba300c91f0e30d11a4e0c4042d3f1816e8dadc9c5bceb615423</originalsourceid><addsrcrecordid>eNp9kEFuFDEQRa0IpAyBC7CyxNpQ1XZPdy-jCAJSJDbJ2nK7qxNHjj1xeRjNAbg3TgaJHata_PdfSV-IjwifEWD4woj9OCjoUAFscVCHM7HBsRuUNhO-ERsA3ase9Hgu3jE_ApihB7MRv69jnl2UB4pR7TLTkohZurRIrm4OMdSjLMT7WFmuubREuplrcb7KX4F9pui4Bi_pee9qyEkeQn2QTqaclM-pWVKVS6OOslJ5elW3LIZErrS_4f6hvhdvVxeZPvy9F-Lu29fbq-_q5uf1j6vLG-U1TlVN6DpDox794rvt7DSAn3AF0rAgOkPgDZhu0SuOuKVxcYuffD97mrfYm05fiE8n767k5z1xtY95X1J7abupATiNg2lUd6J8ycyFVrsr4cmVo0WwL3Pb09y2zW1f57aHVtKnEjc43VP5p_5P6w_JsoaG</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2954219874</pqid></control><display><type>article</type><title>Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weight</title><source>Springer Link</source><creator>Makheloufi, Hocine ; Bahlil, Mounir</creator><creatorcontrib>Makheloufi, Hocine ; Bahlil, Mounir</creatorcontrib><description>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.</description><identifier>ISSN: 0035-5038</identifier><identifier>EISSN: 1827-3491</identifier><identifier>DOI: 10.1007/s11587-021-00617-w</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Algebra ; Analysis ; Damping ; Galerkin method ; Geometry ; Initial conditions ; Mathematics ; Mathematics and Statistics ; Numerical Analysis ; Probability Theory and Stochastic Processes ; Stability ; Viscoelasticity</subject><ispartof>Ricerche di matematica, 2024-02, Vol.73 (1), p.433-469</ispartof><rights>Università degli Studi di Napoli "Federico II" 2021</rights><rights>Università degli Studi di Napoli "Federico II" 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-91a24e838cdc26ba300c91f0e30d11a4e0c4042d3f1816e8dadc9c5bceb615423</citedby><cites>FETCH-LOGICAL-c319t-91a24e838cdc26ba300c91f0e30d11a4e0c4042d3f1816e8dadc9c5bceb615423</cites><orcidid>0000-0001-7386-7623</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>Makheloufi, Hocine</creatorcontrib><creatorcontrib>Bahlil, Mounir</creatorcontrib><title>Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weight</title><title>Ricerche di matematica</title><addtitle>Ricerche mat</addtitle><description>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.</description><subject>Algebra</subject><subject>Analysis</subject><subject>Damping</subject><subject>Galerkin method</subject><subject>Geometry</subject><subject>Initial conditions</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Numerical Analysis</subject><subject>Probability Theory and Stochastic Processes</subject><subject>Stability</subject><subject>Viscoelasticity</subject><issn>0035-5038</issn><issn>1827-3491</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kEFuFDEQRa0IpAyBC7CyxNpQ1XZPdy-jCAJSJDbJ2nK7qxNHjj1xeRjNAbg3TgaJHata_PdfSV-IjwifEWD4woj9OCjoUAFscVCHM7HBsRuUNhO-ERsA3ase9Hgu3jE_ApihB7MRv69jnl2UB4pR7TLTkohZurRIrm4OMdSjLMT7WFmuubREuplrcb7KX4F9pui4Bi_pee9qyEkeQn2QTqaclM-pWVKVS6OOslJ5elW3LIZErrS_4f6hvhdvVxeZPvy9F-Lu29fbq-_q5uf1j6vLG-U1TlVN6DpDox794rvt7DSAn3AF0rAgOkPgDZhu0SuOuKVxcYuffD97mrfYm05fiE8n767k5z1xtY95X1J7abupATiNg2lUd6J8ycyFVrsr4cmVo0WwL3Pb09y2zW1f57aHVtKnEjc43VP5p_5P6w_JsoaG</recordid><startdate>20240201</startdate><enddate>20240201</enddate><creator>Makheloufi, Hocine</creator><creator>Bahlil, Mounir</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-7386-7623</orcidid></search><sort><creationdate>20240201</creationdate><title>Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weight</title><author>Makheloufi, Hocine ; Bahlil, Mounir</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-91a24e838cdc26ba300c91f0e30d11a4e0c4042d3f1816e8dadc9c5bceb615423</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Algebra</topic><topic>Analysis</topic><topic>Damping</topic><topic>Galerkin method</topic><topic>Geometry</topic><topic>Initial conditions</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Numerical Analysis</topic><topic>Probability Theory and Stochastic Processes</topic><topic>Stability</topic><topic>Viscoelasticity</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Makheloufi, Hocine</creatorcontrib><creatorcontrib>Bahlil, Mounir</creatorcontrib><collection>CrossRef</collection><jtitle>Ricerche di matematica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Makheloufi, Hocine</au><au>Bahlil, Mounir</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weight</atitle><jtitle>Ricerche di matematica</jtitle><stitle>Ricerche mat</stitle><date>2024-02-01</date><risdate>2024</risdate><volume>73</volume><issue>1</issue><spage>433</spage><epage>469</epage><pages>433-469</pages><issn>0035-5038</issn><eissn>1827-3491</eissn><abstract>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.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s11587-021-00617-w</doi><tpages>37</tpages><orcidid>https://orcid.org/0000-0001-7386-7623</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0035-5038 |
| ispartof | Ricerche di matematica, 2024-02, Vol.73 (1), p.433-469 |
| issn | 0035-5038 1827-3491 |
| language | eng |
| recordid | cdi_proquest_journals_2954219874 |
| source | Springer Link |
| subjects | Algebra Analysis Damping Galerkin method Geometry Initial conditions Mathematics Mathematics and Statistics Numerical Analysis Probability Theory and Stochastic Processes Stability Viscoelasticity |
| title | Global well-posedness and stability results for an abstract viscoelastic equation with a non-constant delay term and nonlinear weight |
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