Continuous dependence result for the higher-order conduction within the second gradient of type Ⅲ

Jincheng Shi; Yiwu Lin; Jincheng Shi; Yiwu Lin

doi:10.3934/era.2026015

Electronic Research Archive

2026, Volume 34, Issue 1: 318-335. doi: 10.3934/era.2026015

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Research article

Continuous dependence result for the higher-order conduction within the second gradient of type Ⅲ

Jincheng Shi ¹,
Yiwu Lin ^{2
,
,}

1.
Department of Applied Mathematics, Guangzhou Huashang College, Guangzhou 511300, China
2.
Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 511201, China

Received: 18 November 2025 Revised: 16 December 2025 Accepted: 05 January 2026 Published: 09 January 2026

This study investigated the spatial behavior and structural stability of solutions for a higher-order heat conduction model within the second gradient theory of type Ⅲ. By constructing a tailored energy functional and employing integral-differential inequality techniques, we derived a continuous dependence estimate for the solution with respect to the elastic coefficient $ \mu $. Our results demonstrate that the energy not only decays exponentially with respect to the spatial variable $ z $, but also diminishes as the parameter $ \mu $ tends to zero. This work extends the understanding of structural stability in unbounded domains and underscores the robustness of the model under perturbations of material parameters.
- integral-differential inequality,
- structural stability,
- sontinuous dependence,
- second gradient of type Ⅲ
Citation: Jincheng Shi, Yiwu Lin. Continuous dependence result for the higher-order conduction within the second gradient of type Ⅲ[J]. Electronic Research Archive, 2026, 34(1): 318-335. doi: 10.3934/era.2026015

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Abstract

This study investigated the spatial behavior and structural stability of solutions for a higher-order heat conduction model within the second gradient theory of type Ⅲ. By constructing a tailored energy functional and employing integral-differential inequality techniques, we derived a continuous dependence estimate for the solution with respect to the elastic coefficient $ \mu $. Our results demonstrate that the energy not only decays exponentially with respect to the spatial variable $ z $, but also diminishes as the parameter $ \mu $ tends to zero. This work extends the understanding of structural stability in unbounded domains and underscores the robustness of the model under perturbations of material parameters.

References

[1]	K. A. Ames, L. E. Payne, On stabilizing against modelling errors in a penetrative convection problem for a porous medium, Models Methods Appl. Sci., 4 (1990), 733–740. https://doi.org/10.1142/S0218202594000406 doi: 10.1142/S0218202594000406
[2]	A. O. Celebi, V. K. Kalantarov, D. Ugurlu, Continuous dependence for the convective Brinkman-Forchheimer equations, Appl. Anal., 84 (2005), 877–888. https://doi.org/10.1080/00036810500148911 doi: 10.1080/00036810500148911
[3]	A. O. Celebi, V. K. Kalantarov, D. Ugurlu, On continuous dependence on coefficients of the Brinkman-Forchheimer equations, Appl. Math. Lett., 19 (2006), 801–807. https://doi.org/10.1016/j.aml.2005.11.002 doi: 10.1016/j.aml.2005.11.002
[4]	F. Franchi, B. Straughan, Continuous dependence and decay for the Forchheimer equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 459 (2003), 3195–3202. https://doi.org/10.1098/rspa.2003.1169 doi: 10.1098/rspa.2003.1169
[5]	A. J. Harfash, Structural stability for two convection models in a reacting fluid with magnetic field effect, Ann. Henri Poincaré, 15 (2014), 2441–2465. https://doi.org/10.1007/s00023-013-0307-z doi: 10.1006/jmaa.1996.0428
[6]	P. N. Kaloni, J. Guo, Steady nonlinear double-diffusive convection in a porous medium based upon the Brinkman-Forchheimer Model, J. Math. Anal. Appl., 204 (1996), 138–155. https://doi.org/10.1006/jmaa.1996.0428 doi: 10.1006/jmaa.1996.0428
[7]	Y. F. Li, S. Z. Xiao, P. Zeng, The applications of some basic mathematical inequalities on the convergence of the primitive equations of moist atmosphere, J. Math. Inequal., 15 (2021), 293–304. https://doi.org/10.7153/jmi-2021-15-22 doi: 10.7153/jmi-2021-15-22
[8]	X. J. Chen, Y. F. Li, Spatial properties and the influence of the Soret coefficient on the solutions of time-dependent double-diffusive Darcy plane flow, Electron. Res. Arch., 31 (2022), 421–441. https://doi.org/10.3934/era.2023021 doi: 10.3934/era.2023021
[9]	Y. F. Li, X. J. Chen, Spatial decay bound and structural stability for the double-diffusion perturbation equations, Math. Biosci. Eng., 20 (2023), 2998–3022. https://doi.org/10.3934/mbe.2023142 doi: 10.3934/mbe.2023142
[10]	Y. Liu, S. Z. Xiao, Structural stability for the Brinkman fluid interfacing with a Darcy fluid in an unbounded domain, Nonlinear Anal-Real., 42 (2018), 308–333. https://doi.org/10.1016/j.nonrwa.2018.01.007 doi: 10.1016/j.nonrwa.2018.01.007
[11]	N. Q. Qin, J. C. Shi, Y. F. Wen, Spatial decay estimates for the moore–gibson–thompson heat equation based on an integral differential inequality, Axioms, 14 (2025), 265. https://doi.org/10.3390/axioms14040265 doi: 10.3390/axioms14040265
[12]	D. Iesan, R. Quintanilla, A second gradient theory of thermoelasticity, J. Elast., 154 (2023), 629–643. https://doi.org/10.1007/s10659-023-10020-1 doi: 10.1007/s10659-023-10020-1
[13]	D. Iesan, A. Magafia, R. Quintanilla, A second gradient theory of thermoviscoelasticity, J. Therm. Stress, 47 (2024), 1145–1158. https://doi.org/10.1080/01495739.2024.2365265 doi: 10.1080/01495739.2024.2365265
[14]	D. Iesan, Second gradient theory of thermopiezoelectricity, Acta Mech., 235 (2024), 5379–5391. https://doi.org/10.1007/s00707-024-03999-8 doi: 10.1007/s00707-024-03999-8
[15]	D. Iesan, R. Quintanilla, Second gradient thermoelasticity with microtemperatures, Electron. Res. Arch., 33 (2025), 537–555. https://doi.org/10.3934/era.2025025 doi: 10.3934/era.2025025
[16]	J. R. Fernandez, R. Quintanilla, Analysis of a higher order problem within the second gradient theory, Appl. Math. Lett., 154 (2024), 109086. https://doi.org/10.1016/j.aml.2024.109086 doi: 10.1016/j.aml.2024.109086
[17]	M. C. Leseduarte, R. Quintanilla, Phragmén-Lindelöf of alternative for the Laplace equation with dynamic boundary conditions, J. Appl. Anal. Comput., 7 (2017), 1323–1335. https://doi.org/10.11948/2017081 doi: 10.11948/2017081
[18]	Y. Liu, C. Lin, Phragmén-Lindelöf type alternative results for the stokes flow equation, Math. Inequal. Appl., 9 (2006), 671–694. https://doi.org/10.7153/mia-09-60 doi: 10.7153/mia-09-60
[19]	C. O. Horgan, L. E. Payne, Phragmén-Lindelöf type results for Harmonic functions with nonlinear boundary conditions, Arch. Rational Mech. Anal., 122 (1993), 123–144. https://doi.org/10.1007/BF00378164 doi: 10.1007/BF00378164
[20]	W. H. Chen, Cauchy problem for thermoelastic plate equations with different damping mechanisms, Commun. Math. Sci., 18 (2020), 429–457. https://doi.org/10.4310/cms.2020.v18.n2.a7 doi: 10.4310/cms.2020.v18.n2.a7
[21]	W. H. Chen, T. A. Dao, Asymptotic behaviors for the nonlinear viscous Boussinesq equation in the $L^q$ framework, J. Differ. Equations, 320 (2022), 558–597. https://doi.org/10.48550/arXiv.2109.09144 doi: 10.48550/arXiv.2109.09144
[22]	A. E. Green, P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elast., 31 (1993), 189–208. https://doi.org/10.1007/BF00044969 doi: 10.1007/BF00044969
[23]	Y. Liu, X. L. Qin, S. H. Zhang, Global existence and estimates for Blackstock's model of thermoviscous flow with second sound phenomena, J. Differ. Equations, 324 (2022), 76–101. https://doi.org/10.1016/j.jde.2022.04.001 doi: 10.1016/j.jde.2022.04.001
[24]	Y. Liu, X. L. Qin, J. C. Shi, W. J. Zhi, Structural stability of the Boussinesq fluid interfacing with a Darcy fluid in a bounded region in $R^2$, Appl. Math. Comput., 411 (2021), 126488. https://doi.org/10.1016/j.amc.2021.126488 doi: 10.1016/j.amc.2021.126488
[25]	Y. Liu, Continuous dependence for a thermal convection model with temperature-dependent solubility, Appl. Math. Comput., 308 (2017), 18–30. https://doi.org/10.1016/j.amc.2017.03.004 doi: 10.1016/j.amc.2017.03.004
[26]	Y. F. Li, X. J. Chen, J. C. Shi, Structural stability in resonant penetrative convection in a Brinkman-Forchheimer fluid interfacing with a Darcy fluid, Appl. Math. Opt., 84 (2021), 979–999. https://doi.org/10.1007/s00245-021-09791-7 doi: 10.1007/s00245-021-09791-7
[27]	J. C. Shi, S. M. Li, C. T. Xiao, Y. Liu, Spatial behavior for the quasi-static heat conduction within the second gradient of type Ⅲ, Electron. Res. Arch., 32 (2024), 6235–6257. https://doi.org/10.3934/era.2024290 doi: 10.3934/era.2024290
[28]	Y. Liu, W. H. Liao, C. Lin, Some alternative results for the nonlinear viscoelasticity equations, Appl. Anal., 89 (2010), 775–787. https://doi.org/10.1080/00036811003649082 doi: 10.1080/00036811003649082

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