Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions
The paper is concerned with the semilinear wave equations with time‐dependent damping γ(t)=α/(1+t) (α>0), under the effect of nonlinear source f behaving like a polynomial, and subject to Neumann boundary conditions. Constructing appropriate auxiliary functions, we obtain an explicit uniform dec...
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| Veröffentlicht in: | Mathematical methods in the applied sciences 2021-01, Vol.44 (1), p.303-314 |
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| creator | Luo, Jun‐Ren Xiao, Ti‐Jun |
| description | The paper is concerned with the semilinear wave equations with time‐dependent damping γ(t)=α/(1+t) (α>0), under the effect of nonlinear source f behaving like a polynomial, and subject to Neumann boundary conditions. Constructing appropriate auxiliary functions, we obtain an explicit uniform decay rate estimate for the solutions of the equation in terms of the exponent of f, when α is large enough. On the other hand, via a new hyperbolic version of Dirichlet quotients, we show that the upper estimate is optimal in some case, which implies the existence of slow solutions. |
| doi_str_mv | 10.1002/mma.6733 |
| format | Article |
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Constructing appropriate auxiliary functions, we obtain an explicit uniform decay rate estimate for the solutions of the equation in terms of the exponent of f, when α is large enough. On the other hand, via a new hyperbolic version of Dirichlet quotients, we show that the upper estimate is optimal in some case, which implies the existence of slow solutions.</description><identifier>ISSN: 0170-4214</identifier><identifier>EISSN: 1099-1476</identifier><identifier>DOI: 10.1002/mma.6733</identifier><language>eng</language><publisher>Freiburg: Wiley Subscription Services, Inc</publisher><subject>Boundary conditions ; Damping ; Decay rate ; Dirichlet problem ; Mathematical analysis ; Neumann boundary condition ; nonlinear source ; optimal decay rate ; Polynomials ; Quotients ; semilinear wave equation ; slow solution ; time‐dependent frictional dissipation ; Wave equations</subject><ispartof>Mathematical methods in the applied sciences, 2021-01, Vol.44 (1), p.303-314</ispartof><rights>2020 John Wiley & Sons, Ltd.</rights><rights>2021 John Wiley & Sons, Ltd.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c2933-88f89e44beaf5376129171b9cc48936db662f21c506d6065b063913a545b6ce83</citedby><cites>FETCH-LOGICAL-c2933-88f89e44beaf5376129171b9cc48936db662f21c506d6065b063913a545b6ce83</cites><orcidid>0000-0003-1961-5526</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1002%2Fmma.6733$$EPDF$$P50$$Gwiley$$H</linktopdf><linktohtml>$$Uhttps://onlinelibrary.wiley.com/doi/full/10.1002%2Fmma.6733$$EHTML$$P50$$Gwiley$$H</linktohtml><link.rule.ids>314,780,784,1417,27924,27925,45574,45575</link.rule.ids></links><search><creatorcontrib>Luo, Jun‐Ren</creatorcontrib><creatorcontrib>Xiao, Ti‐Jun</creatorcontrib><title>Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions</title><title>Mathematical methods in the applied sciences</title><description>The paper is concerned with the semilinear wave equations with time‐dependent damping γ(t)=α/(1+t) (α>0), under the effect of nonlinear source f behaving like a polynomial, and subject to Neumann boundary conditions. Constructing appropriate auxiliary functions, we obtain an explicit uniform decay rate estimate for the solutions of the equation in terms of the exponent of f, when α is large enough. On the other hand, via a new hyperbolic version of Dirichlet quotients, we show that the upper estimate is optimal in some case, which implies the existence of slow solutions.</description><subject>Boundary conditions</subject><subject>Damping</subject><subject>Decay rate</subject><subject>Dirichlet problem</subject><subject>Mathematical analysis</subject><subject>Neumann boundary condition</subject><subject>nonlinear source</subject><subject>optimal decay rate</subject><subject>Polynomials</subject><subject>Quotients</subject><subject>semilinear wave equation</subject><subject>slow solution</subject><subject>time‐dependent frictional dissipation</subject><subject>Wave equations</subject><issn>0170-4214</issn><issn>1099-1476</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp10LtOwzAUBmALgUQpSDyCJRaWlONLnHisylVqYYHZOI5DXTVOayet8vakLSvTv3znoh-hWwITAkAf6lpPRMbYGRoRkDIhPBPnaAQkg4RTwi_RVYwrAMgJoSP0_WiN7nHQrY24agKOtnZr560OeK93Ftttp1vX-Ij3rl3infYuLp3_waWuN4fUvsTvtqu197hoOl_q0GPT-NIdx67RRaXX0d785Rh9PT99zl6T-cfL22w6TwyVjCV5XuXScl5YXaUsE4RKkpFCGsNzyURZCEErSkwKohQg0gIEk4TplKeFMDZnY3R32rsJzbazsVWrpgt-OKkoF6kEkoIc1P1JmdDEGGylNsHVw8eKgDr0p4b-1KG_gSYnundr2__r1GIxPfpf6dpxKw</recordid><startdate>20210115</startdate><enddate>20210115</enddate><creator>Luo, Jun‐Ren</creator><creator>Xiao, Ti‐Jun</creator><general>Wiley Subscription Services, Inc</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><orcidid>https://orcid.org/0000-0003-1961-5526</orcidid></search><sort><creationdate>20210115</creationdate><title>Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions</title><author>Luo, Jun‐Ren ; Xiao, Ti‐Jun</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c2933-88f89e44beaf5376129171b9cc48936db662f21c506d6065b063913a545b6ce83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Boundary conditions</topic><topic>Damping</topic><topic>Decay rate</topic><topic>Dirichlet problem</topic><topic>Mathematical analysis</topic><topic>Neumann boundary condition</topic><topic>nonlinear source</topic><topic>optimal decay rate</topic><topic>Polynomials</topic><topic>Quotients</topic><topic>semilinear wave equation</topic><topic>slow solution</topic><topic>time‐dependent frictional dissipation</topic><topic>Wave equations</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Luo, Jun‐Ren</creatorcontrib><creatorcontrib>Xiao, Ti‐Jun</creatorcontrib><collection>CrossRef</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><jtitle>Mathematical methods in the applied sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Luo, Jun‐Ren</au><au>Xiao, Ti‐Jun</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions</atitle><jtitle>Mathematical methods in the applied sciences</jtitle><date>2021-01-15</date><risdate>2021</risdate><volume>44</volume><issue>1</issue><spage>303</spage><epage>314</epage><pages>303-314</pages><issn>0170-4214</issn><eissn>1099-1476</eissn><abstract>The paper is concerned with the semilinear wave equations with time‐dependent damping γ(t)=α/(1+t) (α>0), under the effect of nonlinear source f behaving like a polynomial, and subject to Neumann boundary conditions. Constructing appropriate auxiliary functions, we obtain an explicit uniform decay rate estimate for the solutions of the equation in terms of the exponent of f, when α is large enough. On the other hand, via a new hyperbolic version of Dirichlet quotients, we show that the upper estimate is optimal in some case, which implies the existence of slow solutions.</abstract><cop>Freiburg</cop><pub>Wiley Subscription Services, Inc</pub><doi>10.1002/mma.6733</doi><tpages>12</tpages><orcidid>https://orcid.org/0000-0003-1961-5526</orcidid></addata></record> |
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| ispartof | Mathematical methods in the applied sciences, 2021-01, Vol.44 (1), p.303-314 |
| issn | 0170-4214 1099-1476 |
| language | eng |
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| source | Wiley Online Library Journals Frontfile Complete |
| subjects | Boundary conditions Damping Decay rate Dirichlet problem Mathematical analysis Neumann boundary condition nonlinear source optimal decay rate Polynomials Quotients semilinear wave equation slow solution time‐dependent frictional dissipation Wave equations |
| title | Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions |
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