{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:01:40Z","timestamp":1760148100904,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T00:00:00Z","timestamp":1680220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Imam Mohammad Ibn Saud Islamic University","award":["RG-21-09-51"],"award-info":[{"award-number":["RG-21-09-51"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The aim of this paper is to investigate the local weak existence and vacuum isolating of solutions, asymptotic behavior, and blow-up of the solutions for a wave equation involving the fractional Laplacian with nonlinear source. By means of the Galerkin approximations, we prove the local weak existence and finite time blow-up of the solutions and we give the upper and lower bounds for blow-up time.<\/jats:p>","DOI":"10.3390\/axioms12040343","type":"journal-article","created":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T08:27:27Z","timestamp":1680251247000},"page":"343","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Local Existence and Blow-Up of Solutions for Wave Equation Involving the Fractional Laplacian with Nonlinear Source Term"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9881-6806","authenticated-orcid":false,"given":"Younes","family":"Bidi","sequence":"first","affiliation":[{"name":"Laboratoire de Math\u00e9matiques Pures et Appliqu\u00e9es (LMPA), Universit\u00e9 Amar Telidji de Laghouat, Laghouat 03000, Algeria"},{"name":"Ecole Normale Sup\u00e9rieure de Laghouat, Laghouat 03000, Algeria"}]},{"given":"Abderrahmane","family":"Beniani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ain Temouchent Belhadj Bouchaib, Ain Temouchent 46000, Algeria"}]},{"given":"Keltoum","family":"Bouhali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass 51452, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Sciences, 20 Aout 1955 University, Skikda 21000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7895-4168","authenticated-orcid":false,"given":"Khaled","family":"Zennir","sequence":"additional","affiliation":[{"name":"Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 5701, Riyadh 11432, Saudi Arabia"}]},{"given":"Hatim M.","family":"ElKhair","sequence":"additional","affiliation":[{"name":"Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 5701, Riyadh 11432, Saudi Arabia"}]},{"given":"Eltegani I.","family":"Hassan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}]},{"given":"Almonther","family":"Alarfaj","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,31]]},"reference":[{"key":"ref_1","first-page":"435","article-title":"Traces for fractional Sobolev spaces with variable exponents","volume":"2","author":"Pezzo","year":"2017","journal-title":"Adv. Oper. 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