In this research, we investigated the well-posedness of the Lord–Shulman model of generalized thermoelasticity for inhomogeneous anisotropic materials with variable coefficients. The presence of the differential thermoelastic coupling operator $ (1+\tau_0\partial_t) $ prevents the use of the standard energy method due to the lack of cancellation of the coupling terms in the weak formulation. To overcome this difficulty, we introduced a suitable transformation that absorbs the time derivative in the coupling term into new variables, which is an algebraic device to facilitate cancellation in energy estimates. The transformed problem was analyzed using the Faedo–Galerkin method and energy estimates, leading to the existence and uniqueness of weak solutions. The solution of the original system was then recovered through the inverse transformation.
Citation: Mogtaba Mohammed. Well-posedness of the Lord–Shulman thermoelastic model in general anisotropic materials[J]. Electronic Research Archive, 2026, 34(12): 9046-9065. doi: 10.3934/era.2026381
In this research, we investigated the well-posedness of the Lord–Shulman model of generalized thermoelasticity for inhomogeneous anisotropic materials with variable coefficients. The presence of the differential thermoelastic coupling operator $ (1+\tau_0\partial_t) $ prevents the use of the standard energy method due to the lack of cancellation of the coupling terms in the weak formulation. To overcome this difficulty, we introduced a suitable transformation that absorbs the time derivative in the coupling term into new variables, which is an algebraic device to facilitate cancellation in energy estimates. The transformed problem was analyzed using the Faedo–Galerkin method and energy estimates, leading to the existence and uniqueness of weak solutions. The solution of the original system was then recovered through the inverse transformation.
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Mogtaba Mohammed. Well-posedness of the Lord–Shulman thermoelastic model in general anisotropic materials[J]. Electronic Research Archive, 2026, 34(12): 9046-9065. doi: 10.3934/era.2026381
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