Generalized solutions to a semilinear wave equation
Semilinear wave equations in space dimension n ⩽ 9 with singular data and various types of nonlinearities are considered. We employ the framework of the algebra G L 2 of generalized functions. In the general case, the nonlinear term is regularized with respect to a small parameter ε such that it bec...
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| Veröffentlicht in: | Nonlinear analysis 2005-05, Vol.61 (3), p.461-475 |
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| creator | Nedeljkov, M. Oberguggenberger, M. Pilipović, S. |
| description | Semilinear wave equations in space dimension
n
⩽
9
with singular data and various types of nonlinearities are considered. We employ the framework of the algebra
G
L
2
of generalized functions. In the general case, the nonlinear term is regularized with respect to a small parameter
ε
such that it becomes globally Lipschitz for each such
ε
. This produces a family of Cauchy problems, to which we construct a family of solutions which in turn determines an element of
G
L
2
which serves as a generalized solution to the original equation. For cubic nonlinear terms the equation is directly solved in
G
L
2
without regularizations. Finally, in certain cases, it is shown that the solution to the regularized equation coincides with the solution to the nonregularized one. |
| doi_str_mv | 10.1016/j.na.2005.01.001 |
| format | Article |
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n
⩽
9
with singular data and various types of nonlinearities are considered. We employ the framework of the algebra
G
L
2
of generalized functions. In the general case, the nonlinear term is regularized with respect to a small parameter
ε
such that it becomes globally Lipschitz for each such
ε
. This produces a family of Cauchy problems, to which we construct a family of solutions which in turn determines an element of
G
L
2
which serves as a generalized solution to the original equation. For cubic nonlinear terms the equation is directly solved in
G
L
2
without regularizations. Finally, in certain cases, it is shown that the solution to the regularized equation coincides with the solution to the nonregularized one.</description><identifier>ISSN: 0362-546X</identifier><identifier>EISSN: 1873-5215</identifier><identifier>DOI: 10.1016/j.na.2005.01.001</identifier><identifier>CODEN: NOANDD</identifier><language>eng</language><publisher>Oxford: Elsevier Ltd</publisher><subject>Algebras of generalized functions ; Cutoff and regularization ; Exact sciences and technology ; Functional analysis ; Generalized solutions ; Mathematical analysis ; Mathematics ; Partial differential equations ; Sciences and techniques of general use ; Semilinear wave equations</subject><ispartof>Nonlinear analysis, 2005-05, Vol.61 (3), p.461-475</ispartof><rights>2005 Elsevier Ltd</rights><rights>2005 INIST-CNRS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c421t-a3e67c6bf95a218efb9a60d3feb65e2e7a8c7645998502f1dc75ce2af3eca7773</citedby><cites>FETCH-LOGICAL-c421t-a3e67c6bf95a218efb9a60d3feb65e2e7a8c7645998502f1dc75ce2af3eca7773</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.na.2005.01.001$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3541,27915,27916,45986</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=16576132$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Nedeljkov, M.</creatorcontrib><creatorcontrib>Oberguggenberger, M.</creatorcontrib><creatorcontrib>Pilipović, S.</creatorcontrib><title>Generalized solutions to a semilinear wave equation</title><title>Nonlinear analysis</title><description>Semilinear wave equations in space dimension
n
⩽
9
with singular data and various types of nonlinearities are considered. We employ the framework of the algebra
G
L
2
of generalized functions. In the general case, the nonlinear term is regularized with respect to a small parameter
ε
such that it becomes globally Lipschitz for each such
ε
. This produces a family of Cauchy problems, to which we construct a family of solutions which in turn determines an element of
G
L
2
which serves as a generalized solution to the original equation. For cubic nonlinear terms the equation is directly solved in
G
L
2
without regularizations. Finally, in certain cases, it is shown that the solution to the regularized equation coincides with the solution to the nonregularized one.</description><subject>Algebras of generalized functions</subject><subject>Cutoff and regularization</subject><subject>Exact sciences and technology</subject><subject>Functional analysis</subject><subject>Generalized solutions</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Partial differential equations</subject><subject>Sciences and techniques of general use</subject><subject>Semilinear wave equations</subject><issn>0362-546X</issn><issn>1873-5215</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2005</creationdate><recordtype>article</recordtype><recordid>eNp1kM1LAzEUxIMoWD_uHveit13zsUm23qRoFQpeFLyF1-wLpGyzbbJb0b_elBY8eXqH-c08Zgi5YbRilKn7VRWg4pTKirKKUnZCJqzRopScyVMyoULxUtbq85xcpLSimdBCTYiYY8AInf_Btkh9Nw6-D6kY-gKKhGvf-YAQiy_YYYHbEfbyFTlz0CW8Pt5L8vH89D57KRdv89fZ46K0NWdDCQKVtmrpphI4a9Atp6BoKxwulUSOGhqrVS2n00ZS7lhrtbTIwQm0oLUWl-TukLuJ_XbENJi1Txa7DgL2YzI8-2qpVQbpAbSxTymiM5vo1xC_DaNmv45ZmQBmv46hzOTu2XJ7zIZkoXMRgvXpz6dyLhM8cw8HDnPRncdokvUYLLY-oh1M2_v_n_wCQK15Pw</recordid><startdate>20050501</startdate><enddate>20050501</enddate><creator>Nedeljkov, M.</creator><creator>Oberguggenberger, M.</creator><creator>Pilipović, S.</creator><general>Elsevier Ltd</general><general>Elsevier</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>7U5</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20050501</creationdate><title>Generalized solutions to a semilinear wave equation</title><author>Nedeljkov, M. ; Oberguggenberger, M. ; Pilipović, S.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c421t-a3e67c6bf95a218efb9a60d3feb65e2e7a8c7645998502f1dc75ce2af3eca7773</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2005</creationdate><topic>Algebras of generalized functions</topic><topic>Cutoff and regularization</topic><topic>Exact sciences and technology</topic><topic>Functional analysis</topic><topic>Generalized solutions</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Partial differential equations</topic><topic>Sciences and techniques of general use</topic><topic>Semilinear wave equations</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Nedeljkov, M.</creatorcontrib><creatorcontrib>Oberguggenberger, M.</creatorcontrib><creatorcontrib>Pilipović, S.</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Nonlinear analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Nedeljkov, M.</au><au>Oberguggenberger, M.</au><au>Pilipović, S.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Generalized solutions to a semilinear wave equation</atitle><jtitle>Nonlinear analysis</jtitle><date>2005-05-01</date><risdate>2005</risdate><volume>61</volume><issue>3</issue><spage>461</spage><epage>475</epage><pages>461-475</pages><issn>0362-546X</issn><eissn>1873-5215</eissn><coden>NOANDD</coden><abstract>Semilinear wave equations in space dimension
n
⩽
9
with singular data and various types of nonlinearities are considered. We employ the framework of the algebra
G
L
2
of generalized functions. In the general case, the nonlinear term is regularized with respect to a small parameter
ε
such that it becomes globally Lipschitz for each such
ε
. This produces a family of Cauchy problems, to which we construct a family of solutions which in turn determines an element of
G
L
2
which serves as a generalized solution to the original equation. For cubic nonlinear terms the equation is directly solved in
G
L
2
without regularizations. Finally, in certain cases, it is shown that the solution to the regularized equation coincides with the solution to the nonregularized one.</abstract><cop>Oxford</cop><pub>Elsevier Ltd</pub><doi>10.1016/j.na.2005.01.001</doi><tpages>15</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0362-546X |
| ispartof | Nonlinear analysis, 2005-05, Vol.61 (3), p.461-475 |
| issn | 0362-546X 1873-5215 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_28504576 |
| source | ScienceDirect Journals (5 years ago - present) |
| subjects | Algebras of generalized functions Cutoff and regularization Exact sciences and technology Functional analysis Generalized solutions Mathematical analysis Mathematics Partial differential equations Sciences and techniques of general use Semilinear wave equations |
| title | Generalized solutions to a semilinear wave equation |
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