{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:20:18Z","timestamp":1760059218823,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T00:00:00Z","timestamp":1748563200000},"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>We investigate an inverse problem involving source and damping term with variable-exponent nonlinearities. We establish adequate conditions on the initial data for the decay of solutions as the integral overdetermination approaches zero over time within an acceptable range of variable exponents. This class of inverse problems, where internal terms such as source and damping are to be determined from indirect measurements, has significant relevance in real-world applications\u2014ranging from geophysical prospecting to biomedical engineering and materials science. The accurate identification of these internal mechanisms plays a crucial role in optimizing system performance, improving diagnostic accuracy, and constructing predictive models. Therefore, the results obtained in this study not only contribute to the theoretical understanding of nonlinear dynamic systems but also provide practical insights for reconstructive analysis and control in applied settings. The asymptotic behavior and decay conditions we derive are expected to be of particular interest to researchers dealing with stability, uniqueness, and identifiability in inverse problems governed by nonstandard growth conditions.<\/jats:p>","DOI":"10.3390\/axioms14060424","type":"journal-article","created":{"date-parts":[[2025,5,30]],"date-time":"2025-05-30T06:14:27Z","timestamp":1748585667000},"page":"424","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Decay Estimates for a Lam\u00e9 Inverse Problem Involving Source and Damping Term with Variable-Exponent Nonlinearities"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4849-9141","authenticated-orcid":false,"given":"Z\u00fclal","family":"M\u0131s\u0131r","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya 54050, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2208-5730","authenticated-orcid":false,"given":"Metin","family":"Yaman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya 54050, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1864","DOI":"10.1002\/mma.7891","article-title":"General decay and blow up of solutions for a class of inverse problem with elasticity term and variable-exponent nonlinearities","volume":"45","author":"Shahrouzi","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1902","DOI":"10.22436\/jnsa.009.04.44","article-title":"Finite time blow up of solutions to an inverse problem for a quasilinear parabolic equation with power nonlinearity","volume":"9","author":"Yaman","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Isakov, V. (1990). Inverse Source Problems, AMS.","DOI":"10.1090\/surv\/034"},{"key":"ref_4","unstructured":"Ramm, A.G. (2005). Inverse Problems, Springer."},{"key":"ref_5","first-page":"696","article-title":"Long-time behaviour of solutions to inverse problem for higher-order parabolic equation","volume":"23","author":"Yaman","year":"2019","journal-title":"Sak. Univ. J. Sci."},{"key":"ref_6","first-page":"73","article-title":"Do\u011frusal parabolik denklem i\u00e7in ters problemin \u00e7\u00f6z\u00fcm\u00fcn\u00fcn uzun zaman davran\u0131\u015f\u0131","volume":"7","author":"Yaman","year":"2003","journal-title":"SAU Fen Bilim. Enstit\u00fcS\u00fc Derg."},{"key":"ref_7","first-page":"106","article-title":"St.-Venant Type Estim. Wave Equation","volume":"6","author":"Yaman","year":"2002","journal-title":"SAU Fen Bilim. Enstit\u00fcs\u00fc Derg."},{"key":"ref_8","first-page":"69","article-title":"Asymptotic behaviour of the solutions of inverse problems for pseudo-parabolic equations","volume":"154","author":"Yaman","year":"2004","journal-title":"Appl. Math Comput."},{"key":"ref_9","unstructured":"Lagnese, J.E., and Lions, J.L. (1988). Modeling Analysis and Control of Thin Plates, Springer."},{"key":"ref_10","unstructured":"Isakov, V. (2006). On Inverse Problems for Partial Differential Equations, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.jmaa.2005.01.007","article-title":"On global behavior of solutions to an inverse problem for nonlinear parabolic equations","volume":"307","author":"Eden","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2368","DOI":"10.1002\/mma.3646","article-title":"Asymptotic stability and blowup of solutions for a class of viscoelastic inverse problem with boundary feedback","volume":"39","author":"Shahrouzi","year":"2016","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s10114-016-5081-7","article-title":"On behavior of solutions to a class of nonlinear hyperbolic inverse source problem","volume":"32","author":"Shahrouzi","year":"2016","journal-title":"Acta. Math. Sin. Engl. Ser."},{"key":"ref_14","first-page":"223","article-title":"A note on the unique solvability of an inverse problem with integral overdetermination","volume":"8","author":"Yaman","year":"2008","journal-title":"Appl. Math. E-Notes"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1006\/jmaa.1993.1012","article-title":"Existence and nonexistence of solutions for ut = div(|\u2207u|p-2\u2207u) + f(\u2207u,u,x,t)","volume":"172","author":"Zhao","year":"1993","journal-title":"J. Math. Anal. Appl."},{"key":"ref_16","unstructured":"Prilepko, A.I., Orlovskii, D.G., and Vasin, I.A. (2000). Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1080\/03605309408821066","article-title":"Solution of an inverse problem for the nonlinear heat equation","volume":"19","author":"Riganti","year":"1994","journal-title":"Commun. Partial. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1007\/BF02674572","article-title":"On the asymptotic behavior of solutions to inverse problems for parabolic equations","volume":"38","author":"Vasin","year":"1997","journal-title":"Sib. Math. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1186\/s13661-020-01467-5","article-title":"New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions","volume":"2020","author":"Messaoudi","year":"2020","journal-title":"Bound. Value Probl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1186\/s13661-022-01602-4","article-title":"General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities","volume":"23","author":"Park","year":"2022","journal-title":"Bound. Value Probl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1186\/s13661-022-01674-2","article-title":"Decay estimate in a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity","volume":"2022","author":"Kakumani","year":"2022","journal-title":"Bound. Value Probl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7479","DOI":"10.2298\/FIL2421479M","article-title":"Blow up solution of inverse problem for nonlinear hyperbolic equation with variable-exponents","volume":"38","author":"Yaman","year":"2024","journal-title":"Filomat"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"\u00c7etin, \u015e., Yakar, C., and Y\u0131lmaz, Y. (2024). Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale. Axioms, 13.","DOI":"10.3390\/axioms13090640"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"8389","DOI":"10.1002\/mma.7141","article-title":"Energy decay and blow-up of solutions for a class of system of generalized nonlinear Klein-Gordon equations with source and damping terms","volume":"45","author":"Messaoudi","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"605","DOI":"10.3934\/mcrf.2022010","article-title":"Theoretical and computational decay results for a memory type wave equation with variable- exponent nonlinearity","volume":"13","author":"Zahri","year":"2023","journal-title":"Math. Control Relat. Fields"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wang, C., Wang, C., Zhao, X., and Lv, Z. (2023). Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms. Axioms, 12.","DOI":"10.3390\/axioms12080780"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1601","DOI":"10.1007\/s40995-017-0471-y","article-title":"Energy Decay and Continuous Dependence for Damped Semilinear Wave Equation","volume":"43","author":"Yaman","year":"2019","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"ref_28","first-page":"1288","article-title":"On the decay of solutions of a viscoelastic wave equation with variable sources","volume":"47","year":"2023","journal-title":"Turk. J. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"334","DOI":"10.3906\/mat-1912-20","article-title":"Continuous dependence of solutions for damped improved Boussinesq equation","volume":"44","author":"Bayraktar","year":"2020","journal-title":"Turk. J. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"692","DOI":"10.15672\/hujms.590313","article-title":"Continuous dependence of solutions to double dispersive equation with dissipative term","volume":"50","author":"Uysal","year":"2021","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_31","first-page":"1648","article-title":"Continuous Dependence For Benjamin-Bona-Mahony-Burger Equation","volume":"20","year":"2018","journal-title":"Sak. Univ. J. Sci."},{"key":"ref_32","first-page":"1235","article-title":"Sturm-Liouville Problems with Polynomially Eigenparameter Dependent Boundary Conditions","volume":"27","year":"2023","journal-title":"Sak. Univ. J. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"30638","DOI":"10.3934\/math.20241479","article-title":"Asymptotic behavior of the wave equation solution with nonlinear boundary damping and source term of variable exponent-type","volume":"9","author":"Kafini","year":"2024","journal-title":"Mathematics"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"19971","DOI":"10.3934\/math.20231018","article-title":"On a nonlinear system of plate equations with variable exponent nonlinearity and logarithmic source terms: Existence and stability results","volume":"8","author":"Tatar","year":"2023","journal-title":"AIMS Math."},{"key":"ref_35","first-page":"1","article-title":"Lebesgue and Sobolev spaces with variable exponents","volume":"1","author":"Diening","year":"2011","journal-title":"Lect. Note Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"267","DOI":"10.4064\/sm-143-3-267-293","article-title":"Sobolev embeddings with variable exponent","volume":"143","author":"Edmunds","year":"2000","journal-title":"Stud Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1002\/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T","article-title":"Sobolev embeddings with variable exponent II","volume":"246","author":"Edmunds","year":"2002","journal-title":"Math Nachr."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1006\/jmaa.2000.7617","article-title":"On the spaces Lp(x) and Wm,p(x)(\u03a9)","volume":"263","author":"Fan","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1843","DOI":"10.1016\/S0362-546X(02)00150-5","article-title":"Existence of solutions for p(x)-Laplacian Dirichlet problem","volume":"52","author":"Fan","year":"2003","journal-title":"Nonlinear Anal."},{"key":"ref_40","unstructured":"Piskin, E. (2017). Sobolev Uzaylar\u0131, Se\u00e7kin."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/424\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:43:40Z","timestamp":1760031820000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/424"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,30]]},"references-count":40,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060424"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060424","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,30]]}}}