{"id":"https://openalex.org/W2892384501","doi":"https://doi.org/10.3934/nhm.2018025","title":"Long time behavior for the visco-elastic damped wave equation in <inline-formula><tex-math id=\"M1\">\\begin{document}$\\mathbb{R}^n_+$\\end{document}</tex-math></inline-formula> and the boundary effect","display_name":"Long time behavior for the visco-elastic damped wave equation in <inline-formula><tex-math id=\"M1\">\\begin{document}$\\mathbb{R}^n_+$\\end{document}</tex-math></inline-formula> and the boundary effect","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2892384501","doi":"https://doi.org/10.3934/nhm.2018025","mag":"2892384501"},"language":"en","primary_location":{"id":"doi:10.3934/nhm.2018025","is_oa":true,"landing_page_url":"https://doi.org/10.3934/nhm.2018025","pdf_url":"https://www.aimsciences.org/article/exportPdf?id=74aeefaf-431f-4cba-970c-9fcdbd536b12","source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks & Heterogeneous Media","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.aimsciences.org/article/exportPdf?id=74aeefaf-431f-4cba-970c-9fcdbd536b12","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101147932","display_name":"Linglong Du","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Linglong Du","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5101147932"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.10969579,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"13","issue":"4","first_page":"549","last_page":"565"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10940","display_name":"Navier-Stokes equation solutions","score":0.9905999898910522,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.6545127034187317},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.5678549408912659},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.5368542671203613},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.517333447933197},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5099190473556519},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.4960225522518158},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.48490244150161743},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.4451429843902588},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.42879486083984375},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.42811620235443115},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4228050708770752},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.42092639207839966},{"id":"https://openalex.org/keywords/hyperbolic-function","display_name":"Hyperbolic function","score":0.4185834228992462},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.4143509864807129},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.21960824728012085},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.1397954225540161}],"concepts":[{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6545127034187317},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.5678549408912659},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.5368542671203613},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.517333447933197},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5099190473556519},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.4960225522518158},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.48490244150161743},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.4451429843902588},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.42879486083984375},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.42811620235443115},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4228050708770752},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.42092639207839966},{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.4185834228992462},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.4143509864807129},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.21960824728012085},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.1397954225540161},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3934/nhm.2018025","is_oa":true,"landing_page_url":"https://doi.org/10.3934/nhm.2018025","pdf_url":"https://www.aimsciences.org/article/exportPdf?id=74aeefaf-431f-4cba-970c-9fcdbd536b12","source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks & Heterogeneous Media","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:8a216197dfaf4de1b145bd2dfaf643e1","is_oa":false,"landing_page_url":"https://doaj.org/article/8a216197dfaf4de1b145bd2dfaf643e1","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Networks and Heterogeneous Media, Vol 13, Iss 4, Pp 549-565 (2018)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3934/nhm.2018025","is_oa":true,"landing_page_url":"https://doi.org/10.3934/nhm.2018025","pdf_url":"https://www.aimsciences.org/article/exportPdf?id=74aeefaf-431f-4cba-970c-9fcdbd536b12","source":{"id":"https://openalex.org/S21752972","display_name":"Networks and Heterogeneous Media","issn_l":"1556-1801","issn":["1556-1801","1556-181X"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Networks & Heterogeneous Media","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2892384501.pdf","grobid_xml":"https://content.openalex.org/works/W2892384501.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4378879993","https://openalex.org/W4318472270","https://openalex.org/W4377141943","https://openalex.org/W1946342668","https://openalex.org/W3012219884","https://openalex.org/W2893539081","https://openalex.org/W3216434047","https://openalex.org/W1598394172","https://openalex.org/W2963341847","https://openalex.org/W2123573421"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,94],"investigate":[4],"the":[5,12,15,25,36,42,48,61,73,77,87,91,96,103],"existence":[6],"and":[7,53,81],"long":[8,37],"time":[9],"behavior":[10],"of":[11,72],"solution":[13,75,84,105],"for":[14,35,60,102,108],"nonlinear":[16,104],"visco-elastic":[17],"damped":[18,54],"wave":[19,50],"equation":[20],"in":[21,70],"$\\mathbb{R}^n_+$,":[22],"provided":[23],"that":[24,34],"initial":[26,63],"data":[27],"is":[28,32],"sufficiently":[29],"small.":[30],"It":[31],"shown":[33],"time,":[38],"one":[39],"can":[40,67],"use":[41],"convected":[43],"heat":[44],"kernel":[45],"to":[46],"describe":[47],"hyperbolic":[49],"transport":[51],"structure":[52],"diffusive":[55],"mechanism.":[56],"The":[57],"Green's":[58],"function":[59],"linear":[62],"boundary":[64,88],"value":[65],"problem":[66],"be":[68],"described":[69],"terms":[71],"fundamental":[74,83],"(for":[76],"full":[78],"space":[79],"problem)":[80],"reflected":[82],"coupled":[85],"with":[86],"operator.":[89],"Using":[90],"Duhamel's":[92],"principle,":[93],"get":[95],"$":[97,99],"L^p":[98],"decaying":[100],"rate":[101],"$\\partial_{{\\bf":[106],"x}}^{\\alpha}u$":[107],"$|\\alpha|\\le":[109],"1$.":[110]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2018-10-05T00:00:00"}