Well-Posedness and decay analysis for a thermoelastic wave system with spatially variable-exponent damping and time-varying delay

Muhammad Zainul Abidin; Naeem Ullah; Muhammad Zainul Abidin; Naeem Ullah

doi:10.3934/cam.2026022

Communications in Analysis and Mechanics

2026, Volume 18, Issue 3: 535-552. doi: 10.3934/cam.2026022

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Research article

Well-Posedness and decay analysis for a thermoelastic wave system with spatially variable-exponent damping and time-varying delay

Muhammad Zainul Abidin ¹,
Naeem Ullah ^{2
,
,}

1.
School of Artificial Intelligence, Taizhou University, Taizhou 318000, Zhejiang, China
2.
School of Physical Science and Technology, Yangzhou University, Yangzhou 225009, Jiangsu, China

Received: 17 December 2025 Revised: 11 February 2026 Accepted: 26 March 2026 Published: 06 July 2026
35L05, 35B40, 35R10, 46E30

We study a thermoelastic wave–heat system on a smooth bounded domain in which the mechanical velocity is damped both by a spatially variable-exponent power and by a time-varying delay control. The exponent field $ s(\cdot) $ is log-Hölder with $ 2\le s_{-}\le s(x)\le s_{+} $, where $ s_{+} $ lies strictly below a scaling-critical bound; the delay is uniformly positive with derivative $ < 1 $; the delayed coefficient may change sign but is small relative to the primary damping profile. Encoding the delayed velocity through a first-order transport equation, we construct a Lyapunov functional that controls the elastic, thermal, and delay contributions and yields coercive dissipation. Using a Faedo–Galerkin approximation, compactness arguments in variable-exponent spaces, and the Minty–Browder method, we obtain global existence and uniqueness of strong solutions for admissible data. The same functional, together with a Komornik-type integral inequality, yields an explicit exponential decay rate determined by the time integral of the primary damping profile. The approach also adapts to Cattaneo heat conduction and to distributed delays.
Citation: Muhammad Zainul Abidin, Naeem Ullah. Well-Posedness and decay analysis for a thermoelastic wave system with spatially variable-exponent damping and time-varying delay[J]. Communications in Analysis and Mechanics, 2026, 18(3): 535-552. doi: 10.3934/cam.2026022

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Abstract

We study a thermoelastic wave–heat system on a smooth bounded domain in which the mechanical velocity is damped both by a spatially variable-exponent power and by a time-varying delay control. The exponent field $ s(\cdot) $ is log-Hölder with $ 2\le s_{-}\le s(x)\le s_{+} $, where $ s_{+} $ lies strictly below a scaling-critical bound; the delay is uniformly positive with derivative $ < 1 $; the delayed coefficient may change sign but is small relative to the primary damping profile. Encoding the delayed velocity through a first-order transport equation, we construct a Lyapunov functional that controls the elastic, thermal, and delay contributions and yields coercive dissipation. Using a Faedo–Galerkin approximation, compactness arguments in variable-exponent spaces, and the Minty–Browder method, we obtain global existence and uniqueness of strong solutions for admissible data. The same functional, together with a Komornik-type integral inequality, yields an explicit exponential decay rate determined by the time integral of the primary damping profile. The approach also adapts to Cattaneo heat conduction and to distributed delays.

References

[1]	C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Ration. Mech. Anal., 29 (1968), 241–271. https://doi.org/10.1007/BF00276727 doi: 10.1007/BF00276727
[2]	M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity, Arch. Ration. Mech. Anal., 76 (1981), 97–133. https://doi.org/10.1007/BF00251248 doi: 10.1007/BF00251248
[3]	J. E. Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control Optim., 21 (1983), 968–984. https://doi.org/10.1137/0321059 doi: 10.1137/0321059
[4]	J. U. Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal., 23 (1992), 889–899. https://doi.org/10.1137/0523047 doi: 10.1137/0523047
[5]	I. Lasiecka, R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci., 27 (1998), 457–482.
[6]	G. Ponce, R. Racke, Global existence of small solutions to the initial value problem for nonlinear thermoelasticity, J. Differential Equations, 87 (1990), 70–83. https://doi.org/10.1016/0022-0396(90)90016-I doi: 10.1016/0022-0396(90)90016-I
[7]	S. Jiang, R. Racke, Evolution Equations in Thermoelasticity, Chapman & Hall/CRC, Boca Raton, FL, 2000. https://doi.org/10.1201/9781482285789
[8]	C. Cattaneo, Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena, 3 (1948), 83–101.
[9]	B. D. Coleman, M. E. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys., 47 (1967), 597–613. https://doi.org/10.1063/1.1711937 doi: 10.1063/1.1711937
[10]	M. I. A. Othman, Kh. Lotfy, R. M. Farouk, Transient disturbance in a half-space under generalized magneto-thermoelasticity with internal heat source, Acta Phys. Pol. A, 116 (2009), 185–192. https://doi.org/10.12693/APhysPolA.116.185 doi: 10.12693/APhysPolA.116.185
[11]	G. M. Ismail, K. A. Gepreel, Kh. Lotfy, A. M. S. Mahdy, A. El-Bary, A. M. Saeed, Influence of variable thermal conductivity on thermal-plasma-elastic waves of excited microelongated semiconductor, Alexandria Eng. J., 61 (2022), 12271–12282. https://doi.org/10.1016/j.aej.2022.06.024 doi: 10.1016/j.aej.2022.06.024
[12]	M. I. Marin, R. P. Agarwal, I. A. Abbas, Effect of intrinsic rotations, microstructural expansion and contractions in initial boundary value problem of thermoelastic bodies, Bound. Value Probl., 2014 (2014), 129. https://doi.org/10.1186/1687-2770-2014-129 doi: 10.1186/1687-2770-2014-129
[13]	A. E. Abouelregal, M. Marin, A. Öchsner, The influence of a non-local Moore–Gibson–Thompson heat transfer model on an underlying thermoelastic material under the model of memory-dependent derivatives, Contin. Mech. Thermodyn., 35 (2023), 545–562. https://doi.org/10.1007/s00161-023-01195-y doi: 10.1007/s00161-023-01195-y
[14]	I. Giorgio, L. Placidi, A variational formulation for three-dimensional linear thermoelasticity with "thermal inertia", Meccanica, 59 (2024), 1745–1756. https://doi.org/10.1007/s11012-024-01796-0 doi: 10.1007/s11012-024-01796-0
[15]	I. Giorgio, A variational formulation for one-dimensional linear thermoviscoelasticity, Math. Mech. Complex Syst., 9 (2021), 397–412. https://doi.org/10.2140/memocs.2021.9.397 doi: 10.2140/memocs.2021.9.397
[16]	J. E. Muñoz Rivera, R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26 (1995), 1547–1563. https://doi.org/10.1137/S0036142993255058 doi: 10.1137/S0036142993255058
[17]	B. Lazzari, R. Nibbi, On the energy decay of a linear hyperbolic thermoelastic system with dissipative boundary, J. Thermal Stresses, 30 (2007), 1159–1172. https://doi.org/10.1080/01495730701416101 doi: 10.1080/01495730701416101
[18]	M. I. Mustafa, Uniform stability for thermoelastic systems with boundary time-varying delay, J. Math. Anal. Appl., 383 (2011), 490–498. https://doi.org/10.1016/j.jmaa.2011.05.066 doi: 10.1016/j.jmaa.2011.05.066
[19]	S. Mansouri, L. Tebou, Stabilization of coupled thermoelastic Kirchhoff plate and wave equations, Electron. J. Differential Equations, 2020 (2020), 1–16. https://doi.org/10.58997/ejde.2020.121 doi: 10.58997/ejde.2020.121
[20]	V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley–Masson, Paris, 1994.
[21]	P. Martínez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419–444. https://doi.org/10.1051/cocv:1999116 doi: 10.1051/cocv:1999116
[22]	L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, In: Lecture Notes in Mathematics, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
[23]	A. Choucha, S. Boulaaras, B. Djafari-Rouhani, R. Guefaifia, A. Alharbi, Global existence and general decay for a nonlinear wave equation with acoustic and fractional boundary conditions coupling by source and delay terms, Results Appl. Math., 23 (2024), 100476. https://doi.org/10.1016/j.rinam.2024.100476 doi: 10.1016/j.rinam.2024.100476
[24]	M. Z. Abidin, J. C. Chen, Global well-posedness of generalized magnetohydrodynamics equations in variable exponent Fourier–Besov–Morrey spaces, Acta Math. Sin. (Engl. Ser.), 38 (2022), 2187–2198. https://doi.org/10.1007/s10114-022-0581-0 doi: 10.1007/s10114-022-0581-0
[25]	A. Fahad, S. M. Boulaaras, H. U. Rehman, I. Iqbal, D. Chou, Probing nonlinear wave dynamics: insights from the $(2+1)$-dimensional Konopelchenko–Dubrovsky system, Results Phys., 57 (2024), 107370. https://doi.org/10.1016/j.rinp.2024.107370 doi: 10.1016/j.rinp.2024.107370
[26]	A. Haraux, E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Ration. Mech. Anal., 100 (1988), 191–206. https://doi.org/10.1007/BF00282203 doi: 10.1007/BF00282203
[27]	H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138–146. https://doi.org/10.1137/0505015 doi: 10.1137/0505015
[28]	V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295–308. https://doi.org/10.1006/jdeq.1994.1051 doi: 10.1006/jdeq.1994.1051
[29]	S. A. Messaoudi, A. A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal., 96 (2017), 1509–1515. https://doi.org/10.1080/00036811.2016.1276170 doi: 10.1080/00036811.2016.1276170
[30]	S. A. Messaoudi, A. A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci., 40 (2017), 6976–6986. https://doi.org/10.1002/mma.4505 doi: 10.1002/mma.4505
[31]	S. Ghegal, I. Hamchi, S. A. Messaoudi, Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal., 99 (2020), 1333–1343. https://doi.org/10.1080/00036811.2018.1530760 doi: 10.1080/00036811.2018.1530760
[32]	M. Kafini, S. Messaoudi, On the decay and global nonexistence of solutions to a damped wave equation with variable-exponent nonlinearity and delay, Ann. Polon. Math., 122 (2019), 49–70.
[33]	A. Benaissa, A. Benguessoum, S. A. Messaoudi, Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 1–13. https://doi.org/10.14232/ejqtde.2014.1.11 doi: 10.14232/ejqtde.2014.1.11
[34]	A. Benguessoum, Global existence and energy decay of solutions for a wave equation with a time-varying delay term, Mathematica, 63 (2021), 32–46. https://doi.org/10.24193/MATHCLUJ.2021.1.04 doi: 10.24193/MATHCLUJ.2021.1.04
[35]	S. Antontsev, Wave equation with $p(x, t)$-Laplacian and damping term: existence and blow-up, Differ. Equ. Appl., 3 (2011), 503–525. https://doi.org/10.7153/dea-03-32 doi: 10.7153/dea-03-32
[36]	S. Antontsev, Wave equation with $p(x, t)$-Laplacian and damping term: blow-up of solutions, C. R. Mécanique, 339 (2011), 751–755. https://doi.org/10.1016/j.crme.2011.09.001 doi: 10.1016/j.crme.2011.09.001

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