Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain
Lifespan estimates for semilinear damped wave equations of the form ∂ t 2 u - Δ u + ∂ t u = | u | p in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation ∂ t v - Δ v = v 2 with the Dirichlet boundary c...
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| Veröffentlicht in: | Calculus of variations and partial differential equations 2024-12, Vol.63 (9), Article 240 |
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| container_title | Calculus of variations and partial differential equations |
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| creator | Ikeda, Masahiro Sobajima, Motohiro Taniguchi, Koichi Wakasugi, Yuta |
| description | Lifespan estimates for semilinear damped wave equations of the form
∂
t
2
u
-
Δ
u
+
∂
t
u
=
|
u
|
p
in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation
∂
t
v
-
Δ
v
=
v
2
with the Dirichlet boundary condition and the initial condition
v
(
0
)
=
ε
f
, the corresponding lifespan can be estimated from below and above by
exp
(
exp
(
C
ε
-
1
)
)
with different constants
C
. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control
L
1
-type norm and a new Gagliardo–Nirenberg type estimate with logarithmic weight are introduced. |
| doi_str_mv | 10.1007/s00526-024-02847-w |
| format | Article |
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∂
t
2
u
-
Δ
u
+
∂
t
u
=
|
u
|
p
in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation
∂
t
v
-
Δ
v
=
v
2
with the Dirichlet boundary condition and the initial condition
v
(
0
)
=
ε
f
, the corresponding lifespan can be estimated from below and above by
exp
(
exp
(
C
ε
-
1
)
)
with different constants
C
. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control
L
1
-type norm and a new Gagliardo–Nirenberg type estimate with logarithmic weight are introduced.</description><identifier>ISSN: 0944-2669</identifier><identifier>EISSN: 1432-0835</identifier><identifier>DOI: 10.1007/s00526-024-02847-w</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Analysis ; Boundary conditions ; Calculus of Variations and Optimal Control; Optimization ; Control ; Estimates ; Life span ; Mathematical and Computational Physics ; Mathematics ; Mathematics and Statistics ; Systems Theory ; Theoretical ; Thermodynamics ; Wave equations</subject><ispartof>Calculus of variations and partial differential equations, 2024-12, Vol.63 (9), Article 240</ispartof><rights>The Author(s) 2024</rights><rights>The Author(s) 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c244t-3aec25b64051c95df11b2feba15cfd3339456788454221895ed9a2db5ff7c4c63</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00526-024-02847-w$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00526-024-02847-w$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27903,27904,41467,42536,51298</link.rule.ids></links><search><creatorcontrib>Ikeda, Masahiro</creatorcontrib><creatorcontrib>Sobajima, Motohiro</creatorcontrib><creatorcontrib>Taniguchi, Koichi</creatorcontrib><creatorcontrib>Wakasugi, Yuta</creatorcontrib><title>Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain</title><title>Calculus of variations and partial differential equations</title><addtitle>Calc. Var</addtitle><description>Lifespan estimates for semilinear damped wave equations of the form
∂
t
2
u
-
Δ
u
+
∂
t
u
=
|
u
|
p
in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation
∂
t
v
-
Δ
v
=
v
2
with the Dirichlet boundary condition and the initial condition
v
(
0
)
=
ε
f
, the corresponding lifespan can be estimated from below and above by
exp
(
exp
(
C
ε
-
1
)
)
with different constants
C
. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control
L
1
-type norm and a new Gagliardo–Nirenberg type estimate with logarithmic weight are introduced.</description><subject>Analysis</subject><subject>Boundary conditions</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Control</subject><subject>Estimates</subject><subject>Life span</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Systems Theory</subject><subject>Theoretical</subject><subject>Thermodynamics</subject><subject>Wave equations</subject><issn>0944-2669</issn><issn>1432-0835</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kElLBDEQhYMoOI7-AU8Bz9GsvRxlcIMBL3ryENJJRTJML5P02PrvzdiCNw9FQfHeo96H0CWj14zS8iZRqnhBKJd5KlmS6QgtmBSc0EqoY7SgtZSEF0V9is5S2lDKVMXlAr2tg4c0mA5DGkNrRkjY9xEnaMM2dGAidqYdwOHJfACG3d6Moe9w6LDB49QTF1roUj6ZLYbPEWLIbte3JnTn6MSbbYKL371Er_d3L6tHsn5-eFrdronlUo5EGLBcNYWkitlaOc9Ywz00hinrnRCilqooq0oqyTmragWuNtw1yvvSSluIJbqac4fY7_a5h970-5gfSlowXkhGS3ZQ8VllY59SBK-HmAvHL82oPkDUM0SdIeofiHrKJjGbUhZ37xD_ov9xfQMAMHZM</recordid><startdate>20241201</startdate><enddate>20241201</enddate><creator>Ikeda, Masahiro</creator><creator>Sobajima, Motohiro</creator><creator>Taniguchi, Koichi</creator><creator>Wakasugi, Yuta</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope></search><sort><creationdate>20241201</creationdate><title>Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain</title><author>Ikeda, Masahiro ; Sobajima, Motohiro ; Taniguchi, Koichi ; Wakasugi, Yuta</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c244t-3aec25b64051c95df11b2feba15cfd3339456788454221895ed9a2db5ff7c4c63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Analysis</topic><topic>Boundary conditions</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Control</topic><topic>Estimates</topic><topic>Life span</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Systems Theory</topic><topic>Theoretical</topic><topic>Thermodynamics</topic><topic>Wave equations</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ikeda, Masahiro</creatorcontrib><creatorcontrib>Sobajima, Motohiro</creatorcontrib><creatorcontrib>Taniguchi, Koichi</creatorcontrib><creatorcontrib>Wakasugi, Yuta</creatorcontrib><collection>SpringerOpen</collection><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>Calculus of variations and partial differential equations</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ikeda, Masahiro</au><au>Sobajima, Motohiro</au><au>Taniguchi, Koichi</au><au>Wakasugi, Yuta</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain</atitle><jtitle>Calculus of variations and partial differential equations</jtitle><stitle>Calc. Var</stitle><date>2024-12-01</date><risdate>2024</risdate><volume>63</volume><issue>9</issue><artnum>240</artnum><issn>0944-2669</issn><eissn>1432-0835</eissn><abstract>Lifespan estimates for semilinear damped wave equations of the form
∂
t
2
u
-
Δ
u
+
∂
t
u
=
|
u
|
p
in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation
∂
t
v
-
Δ
v
=
v
2
with the Dirichlet boundary condition and the initial condition
v
(
0
)
=
ε
f
, the corresponding lifespan can be estimated from below and above by
exp
(
exp
(
C
ε
-
1
)
)
with different constants
C
. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control
L
1
-type norm and a new Gagliardo–Nirenberg type estimate with logarithmic weight are introduced.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00526-024-02847-w</doi><oa>free_for_read</oa></addata></record> |
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| ispartof | Calculus of variations and partial differential equations, 2024-12, Vol.63 (9), Article 240 |
| issn | 0944-2669 1432-0835 |
| language | eng |
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| source | SpringerLink (Online service) |
| subjects | Analysis Boundary conditions Calculus of Variations and Optimal Control Optimization Control Estimates Life span Mathematical and Computational Physics Mathematics Mathematics and Statistics Systems Theory Theoretical Thermodynamics Wave equations |
| title | Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain |
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