{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:04:34Z","timestamp":1760058274994,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,22]],"date-time":"2025-03-22T00:00:00Z","timestamp":1742601600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Qassim University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A system of nonlinear wave equations in viscoelasticity with variable exponents is considered. It is assumed that the kernel included in the integral term of the equations depends on both the time and the spatial variables. Using the Faedo\u2013Galerkin method and the contraction mapping principle, a theorem of unique solvability of the problem is proved. In addition, under appropriate variable assumptions, an estimate of the stability of the solution to the problem of determining the kernel is obtained. The study is based on Komornik\u2019s inequality. We expand the class of nonlinear boundary value problems that can be investigated by well-known methods.<\/jats:p>","DOI":"10.3390\/axioms14040243","type":"journal-article","created":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T06:21:38Z","timestamp":1742797298000},"page":"243","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stability Properties of Distributional Solutions for Nonlinear Viscoelastic Wave Equations with Variable Exponents"],"prefix":"10.3390","volume":"14","author":[{"given":"Mouhssin","family":"Bayoud","sequence":"first","affiliation":[{"name":"Department of Computer Science, University Center, El Cherif Bouchoucha-Aflou, Aflou-Laghouat P.O. Box 306, Algeria"}]},{"given":"Mohamed","family":"Karek","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Kasdi Merbah, Ouargla 30000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7895-4168","authenticated-orcid":false,"given":"Khaled","family":"Zennir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia"}]},{"given":"Keltoum","family":"Bouhali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5354-5448","authenticated-orcid":false,"given":"Loay","family":"Alkhalifa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1002\/mma.5919","article-title":"A sharper decay rate for a viscoelastic wave equation with power nonlinearity","volume":"43","author":"Miyasita","year":"2020","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"21","DOI":"10.3103\/S1066369X20090030","article-title":"Stabilization for Solutions of Plate Equation with Time-Varying Delay and Weak-Viscoelasticity in Rn","volume":"64","author":"Zennir","year":"2020","journal-title":"Russ. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1007\/s40840-018-0602-4","article-title":"Decay of an Extensible Viscoelastic Plate Equation with a Nonlinear Time Delay","volume":"42","author":"Feng","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1002\/(SICI)1099-1476(19970310)20:4<291::AID-MMA860>3.0.CO;2-W","article-title":"Inverse Problems for Identification of Memory Kernels in Viscoelasticity","volume":"20","author":"Janno","year":"1997","journal-title":"Math. Meth. Aool. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1002\/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I","article-title":"Blow up in a nonlinearly damped wave equation","volume":"231","author":"Messaoudi","year":"2001","journal-title":"Math. Nachr."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.nonrwa.2015.05.015","article-title":"Blow up of arbitrarily positive initial energy solutions for a viscoelastic wave equation","volume":"26","author":"Song","year":"2015","journal-title":"Nonlinear-Anal. Real World Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.na.2015.05.015","article-title":"Global nonexistence of positive initial energy solutions for a viscoelastic wave equation","volume":"125","author":"Song","year":"2015","journal-title":"Nonlinear Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1002\/mma.250","article-title":"Existence and uniform decay for a non-linear viscoelastic equation with strong damping","volume":"24","author":"Cavalcanti","year":"2001","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"103801","DOI":"10.1016\/j.nonrwa.2022.103801","article-title":"Blow-up for the Timoshenko-type equation with variable exponentss","volume":"71","author":"Ding","year":"2023","journal-title":"Nonlinear-Anal. Real World Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1137\/0505015","article-title":"Some additional remarks on the non existence of global solutions to non linear wave equations","volume":"5","author":"Levine","year":"1974","journal-title":"SIAM J. Math. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s002050050171","article-title":"Global non existence theorems for a class of evolution equations with dissipation","volume":"149","author":"Vitillaro","year":"1999","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_13","unstructured":"Lars, D., Harjulehto, P., Hasto, P., and Ruzicka, M. (2011). Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Matimatics, Springer."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1006\/jmaa.2000.7617","article-title":"On the spaces Lp(x) and Wm;p(x)","volume":"263","author":"Fan","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_15","unstructured":"Komornik, V. (1994). Exact Controllability and Stabilization the Multiplier Method, Masson-JohnWiley."},{"key":"ref_16","unstructured":"Lions, J.L. (1969). Quelques m\u2019ethodes de r\u2019esolution des problemes aux limites non lineaires. Dunod, Gaulthier-Villars."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/243\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:58:31Z","timestamp":1760029111000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/243"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,22]]},"references-count":16,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040243"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040243","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,3,22]]}}}