{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:07:05Z","timestamp":1760148425873,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,30]],"date-time":"2023-04-30T00:00:00Z","timestamp":1682812800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article is devoted to a study of the question of existence (in time) of weak solutions and the derivation of qualitative properties of such solutions for the nonlinear viscoelastic wave equation with variable exponents and minor damping terms. By using the energy method combined with the Faedo\u2013Galerkin method, the local and global existence of solutions are established. Then, the stability estimate of the solution is obtained by introducing a suitable Lyapunov function.<\/jats:p>","DOI":"10.3390\/axioms12050444","type":"journal-article","created":{"date-parts":[[2023,5,1]],"date-time":"2023-05-01T12:12:11Z","timestamp":1682943131000},"page":"444","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Existence and Qualitative Properties of Solution for a Class of Nonlinear Wave Equations with Delay Term and Variable-Exponents Nonlinearities"],"prefix":"10.3390","volume":"12","author":[{"given":"Mohamed","family":"Karek","sequence":"first","affiliation":[{"name":"Department of Earth Science, Faculty of Hydrocarbons and Earth Science and Renewable Energies, University of Kasdi Merbah Ouargla, Ouargla 30000, Algeria"}]},{"given":"Sadok","family":"Otmani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Kasdi Merbah Ouargla, Ouargla 30000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3561-6353","authenticated-orcid":false,"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, College of Sciences, Universit\u00e9 20 Ao\u00fbt 1955 Skikda Bp 26 Route El-Hadaiek, Skikda 21000, Algeria"}]},{"given":"Khaled","family":"Zennir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass 51452, Saudi Arabia"}]},{"given":"Hatim M.","family":"Elkhair","sequence":"additional","affiliation":[{"name":"Deanship of Scientific Research, Imam Mohammad Ibn Saud Islamic University, 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 13318, Saudi Arabia"}]},{"given":"Alnadhief H. A.","family":"Alfedeel","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia"}]},{"given":"Almonther","family":"Alarfaj","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2014.1.11","article-title":"Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback","volume":"2014","author":"Benaissa","year":"2014","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s00033-016-0641-3","article-title":"Energy decay for viscoelastic plates with distributed delay and source term","volume":"67","author":"Mustafa","year":"2016","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3024","DOI":"10.1016\/j.camwa.2017.07.048","article-title":"Nonlinear damped wave equation: Existence and blow-up","volume":"74","author":"Messaoudi","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1723","DOI":"10.1016\/j.camwa.2014.10.018","article-title":"A blow-up result for nonlinear generalized heat equation","volume":"68","author":"Kbiri","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_5","first-page":"503","article-title":"Wave equation with p(x, t)-laplacian and damping term: Existence and blow-up","volume":"3","author":"Antontsev","year":"2011","journal-title":"Differ. Equ. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1016\/j.crme.2011.09.001","article-title":"Wave equation with p(x,t)-laplacian and damping term: Blow-up of solutions","volume":"339","author":"Antontsev","year":"2011","journal-title":"C. R. Mec."},{"key":"ref_7","first-page":"592","article-title":"On spaces lp(x)(\u03c9), and wk,p(x)(\u03c9)","volume":"41","author":"Akosnfik","year":"1991","journal-title":"Czechoslov. Math. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"76","DOI":"10.30538\/psrp-oma2020.0054","article-title":"Exponential growth of solution with Lp-norm for class of non-linear viscoelastic wave equation with distributed delay term for large initial data","volume":"3","author":"Choucha","year":"2020","journal-title":"Open J. Math. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Diening, L., Harjulehto, P., Hasto, P., and Ruzicka, M. (2011). Lebesgue and Sobolev Spaces with Variable Exponents, Springer.","DOI":"10.1007\/978-3-642-18363-8"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1006\/jmaa.2001.7618","article-title":"Sobolev embedding theorems for spaces wk,p(x)(\u03a9)","volume":"262","author":"Fan","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","first-page":"935","article-title":"Stabilization of the wave equation with boundary or internal distributed delay","volume":"21","author":"Nicaise","year":"2008","journal-title":"Differ. Integral Equ."},{"key":"ref_12","unstructured":"Lions, J.L. (1969). Quelques Methodes de R\u00e9solution des Probl\u00e8mes aux Limites Non Lin\u00e9aires, Dunod Gauthier-Villars."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3400","DOI":"10.1016\/j.jksus.2020.09.027","article-title":"New solitary wave structures to the (2 + 1)-dimensional KD and KP equations with spatio-temporal dispersion","volume":"32","author":"Alam","year":"2020","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s13398-021-01058-8","article-title":"On the stability, integrability and boundedness analyses of systems of integro-differential equations with time-delay retardation","volume":"115","year":"2021","journal-title":"Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/444\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:27:02Z","timestamp":1760124422000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/444"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,30]]},"references-count":14,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["axioms12050444"],"URL":"https:\/\/doi.org\/10.3390\/axioms12050444","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,4,30]]}}}