Unbounded mass radial solutions for the Keller–Segel equation in the disk
We consider the boundary value problem - Δ u + u - λ e u = 0 , u > 0 in B 1 ( 0 ) ∂ ν u = 0 on ∂ B 1 ( 0 ) , whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here B 1 ( 0 ) is the unit disk, ν the outer normal to ∂ B 1 ( 0 ) , and λ > 0 is a parameter. We...
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| Veröffentlicht in: | Calculus of variations and partial differential equations 2021-10, Vol.60 (5), Article 198 |
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| container_title | Calculus of variations and partial differential equations |
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| creator | Bonheure, Denis Casteras, Jean-Baptiste Román, Carlos |
| description | We consider the boundary value problem
-
Δ
u
+
u
-
λ
e
u
=
0
,
u
>
0
in
B
1
(
0
)
∂
ν
u
=
0
on
∂
B
1
(
0
)
,
whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here
B
1
(
0
)
is the unit disk,
ν
the outer normal to
∂
B
1
(
0
)
, and
λ
>
0
is a parameter. We show that, provided
λ
is sufficiently small, there exists a family of radial solutions
u
λ
to this system which blow up at the origin and concentrate on
∂
B
1
(
0
)
, as
λ
→
0
. These solutions satisfy
lim
λ
→
0
u
λ
(
0
)
|
ln
λ
|
=
0
and
0
<
lim
λ
→
0
1
|
ln
λ
|
∫
B
1
(
0
)
λ
e
u
λ
(
x
)
d
x
<
∞
,
having in particular unbounded mass, as
λ
→
0
. |
| doi_str_mv | 10.1007/s00526-021-02081-8 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2561092475</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2561092475</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-46a8747b4a998388557783994cd40db4cbfde27a733d33b938d774ee8f532b073</originalsourceid><addsrcrecordid>eNp9kLtOwzAUhi0EEqXwAkyWmAPHt9geUcVNrcQAnS0ndkpKGrd2MrDxDrwhT0LaILExHJ3hfP9_pA-hSwLXBEDeJABB8wwoGQYUydQRmhDOaAaKiWM0Ac15RvNcn6KzlNYARCjKJ2i-bIvQt847vLEp4WhdbRucQtN3dWgTrkLE3ZvHc980Pn5_fr34lW-w3_V2D-C6PZxdnd7P0Ullm-QvfvcULe_vXmeP2eL54Wl2u8hKRnSX8dwqyWXBrdaKKSWElIppzUvHwRW8LCrnqbSSMcdYoZlyUnLvVSUYLUCyKboae7cx7HqfOrMOfWyHl4aKnICmXIqBoiNVxpBS9JXZxnpj44chYPbSzCjNDNLMQZpRQ4iNoTTA7crHv-p_Uj_RDW8n</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2561092475</pqid></control><display><type>article</type><title>Unbounded mass radial solutions for the Keller–Segel equation in the disk</title><source>SpringerLink Journals - AutoHoldings</source><creator>Bonheure, Denis ; Casteras, Jean-Baptiste ; Román, Carlos</creator><creatorcontrib>Bonheure, Denis ; Casteras, Jean-Baptiste ; Román, Carlos</creatorcontrib><description>We consider the boundary value problem
-
Δ
u
+
u
-
λ
e
u
=
0
,
u
>
0
in
B
1
(
0
)
∂
ν
u
=
0
on
∂
B
1
(
0
)
,
whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here
B
1
(
0
)
is the unit disk,
ν
the outer normal to
∂
B
1
(
0
)
, and
λ
>
0
is a parameter. We show that, provided
λ
is sufficiently small, there exists a family of radial solutions
u
λ
to this system which blow up at the origin and concentrate on
∂
B
1
(
0
)
, as
λ
→
0
. These solutions satisfy
lim
λ
→
0
u
λ
(
0
)
|
ln
λ
|
=
0
and
0
<
lim
λ
→
0
1
|
ln
λ
|
∫
B
1
(
0
)
λ
e
u
λ
(
x
)
d
x
<
∞
,
having in particular unbounded mass, as
λ
→
0
.</description><identifier>ISSN: 0944-2669</identifier><identifier>EISSN: 1432-0835</identifier><identifier>DOI: 10.1007/s00526-021-02081-8</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Analysis ; Boundary value problems ; Calculus of Variations and Optimal Control; Optimization ; Control ; Mathematical and Computational Physics ; Mathematics ; Mathematics and Statistics ; Systems Theory ; Theoretical</subject><ispartof>Calculus of variations and partial differential equations, 2021-10, Vol.60 (5), Article 198</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-46a8747b4a998388557783994cd40db4cbfde27a733d33b938d774ee8f532b073</citedby><cites>FETCH-LOGICAL-c319t-46a8747b4a998388557783994cd40db4cbfde27a733d33b938d774ee8f532b073</cites><orcidid>0000-0002-9264-8159</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00526-021-02081-8$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00526-021-02081-8$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Bonheure, Denis</creatorcontrib><creatorcontrib>Casteras, Jean-Baptiste</creatorcontrib><creatorcontrib>Román, Carlos</creatorcontrib><title>Unbounded mass radial solutions for the Keller–Segel equation in the disk</title><title>Calculus of variations and partial differential equations</title><addtitle>Calc. Var</addtitle><description>We consider the boundary value problem
-
Δ
u
+
u
-
λ
e
u
=
0
,
u
>
0
in
B
1
(
0
)
∂
ν
u
=
0
on
∂
B
1
(
0
)
,
whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here
B
1
(
0
)
is the unit disk,
ν
the outer normal to
∂
B
1
(
0
)
, and
λ
>
0
is a parameter. We show that, provided
λ
is sufficiently small, there exists a family of radial solutions
u
λ
to this system which blow up at the origin and concentrate on
∂
B
1
(
0
)
, as
λ
→
0
. These solutions satisfy
lim
λ
→
0
u
λ
(
0
)
|
ln
λ
|
=
0
and
0
<
lim
λ
→
0
1
|
ln
λ
|
∫
B
1
(
0
)
λ
e
u
λ
(
x
)
d
x
<
∞
,
having in particular unbounded mass, as
λ
→
0
.</description><subject>Analysis</subject><subject>Boundary value problems</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Control</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Systems Theory</subject><subject>Theoretical</subject><issn>0944-2669</issn><issn>1432-0835</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp9kLtOwzAUhi0EEqXwAkyWmAPHt9geUcVNrcQAnS0ndkpKGrd2MrDxDrwhT0LaILExHJ3hfP9_pA-hSwLXBEDeJABB8wwoGQYUydQRmhDOaAaKiWM0Ac15RvNcn6KzlNYARCjKJ2i-bIvQt847vLEp4WhdbRucQtN3dWgTrkLE3ZvHc980Pn5_fr34lW-w3_V2D-C6PZxdnd7P0Ullm-QvfvcULe_vXmeP2eL54Wl2u8hKRnSX8dwqyWXBrdaKKSWElIppzUvHwRW8LCrnqbSSMcdYoZlyUnLvVSUYLUCyKboae7cx7HqfOrMOfWyHl4aKnICmXIqBoiNVxpBS9JXZxnpj44chYPbSzCjNDNLMQZpRQ4iNoTTA7crHv-p_Uj_RDW8n</recordid><startdate>20211001</startdate><enddate>20211001</enddate><creator>Bonheure, Denis</creator><creator>Casteras, Jean-Baptiste</creator><creator>Román, Carlos</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope><orcidid>https://orcid.org/0000-0002-9264-8159</orcidid></search><sort><creationdate>20211001</creationdate><title>Unbounded mass radial solutions for the Keller–Segel equation in the disk</title><author>Bonheure, Denis ; Casteras, Jean-Baptiste ; Román, Carlos</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-46a8747b4a998388557783994cd40db4cbfde27a733d33b938d774ee8f532b073</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Analysis</topic><topic>Boundary value problems</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Control</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Systems Theory</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bonheure, Denis</creatorcontrib><creatorcontrib>Casteras, Jean-Baptiste</creatorcontrib><creatorcontrib>Román, Carlos</creatorcontrib><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>Bonheure, Denis</au><au>Casteras, Jean-Baptiste</au><au>Román, Carlos</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Unbounded mass radial solutions for the Keller–Segel equation in the disk</atitle><jtitle>Calculus of variations and partial differential equations</jtitle><stitle>Calc. Var</stitle><date>2021-10-01</date><risdate>2021</risdate><volume>60</volume><issue>5</issue><artnum>198</artnum><issn>0944-2669</issn><eissn>1432-0835</eissn><abstract>We consider the boundary value problem
-
Δ
u
+
u
-
λ
e
u
=
0
,
u
>
0
in
B
1
(
0
)
∂
ν
u
=
0
on
∂
B
1
(
0
)
,
whose solutions correspond to steady states of the Keller–Segel system for chemotaxis. Here
B
1
(
0
)
is the unit disk,
ν
the outer normal to
∂
B
1
(
0
)
, and
λ
>
0
is a parameter. We show that, provided
λ
is sufficiently small, there exists a family of radial solutions
u
λ
to this system which blow up at the origin and concentrate on
∂
B
1
(
0
)
, as
λ
→
0
. These solutions satisfy
lim
λ
→
0
u
λ
(
0
)
|
ln
λ
|
=
0
and
0
<
lim
λ
→
0
1
|
ln
λ
|
∫
B
1
(
0
)
λ
e
u
λ
(
x
)
d
x
<
∞
,
having in particular unbounded mass, as
λ
→
0
.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00526-021-02081-8</doi><orcidid>https://orcid.org/0000-0002-9264-8159</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0944-2669 |
| ispartof | Calculus of variations and partial differential equations, 2021-10, Vol.60 (5), Article 198 |
| issn | 0944-2669 1432-0835 |
| language | eng |
| recordid | cdi_proquest_journals_2561092475 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Analysis Boundary value problems Calculus of Variations and Optimal Control Optimization Control Mathematical and Computational Physics Mathematics Mathematics and Statistics Systems Theory Theoretical |
| title | Unbounded mass radial solutions for the Keller–Segel equation in the disk |
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