Opuscula Math. 43, no. 3 (2023), 409-428

Opuscula Mathematica

Existence and asymptotic stability for generalized elasticity equation with variable exponent

Mohamed Dilmi
Sadok Otmani

Abstract. In this paper we propose a new mathematical model describing the deformations of an isotropic nonlinear elastic body with variable exponent in dynamic regime. We assume that the stress tensor \(\sigma^{p(\cdot)}\) has the form \[\sigma^{p(\cdot)}(u)=(2\mu +|d(u)|^{p(\cdot)-2})d(u)+\lambda Tr(d(u)) I_{3},\] where \(u\) is the displacement field, \(\mu\), \(\lambda\) are the given coefficients \(d(\cdot)\) and \(I_{3}\) are the deformation tensor and the unit tensor, respectively. By using the Faedo-Galerkin techniques and a compactness result we prove the existence of the weak solutions, then we study the asymptotic behaviour stability of the solutions.

Keywords: asymptotic stability, variable exponent Lebesgue and Sobolev spaces, generalized elasticity equation.

Mathematics Subject Classification: 35B37, 35L55, 35L70, 46E30.

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  • Mohamed Dilmi (corresponding author)
  • ORCID iD https://orcid.org/0000-0003-2114-8891
  • University of Blida 1, Department of Mathematics, LAMDA-RO Laboratory, PO Box 270 Route de Soumaa, Blida, Algeria
  • Communicated by P.A. Cojuhari.
  • Received: 2022-02-21.
  • Revised: 2023-02-24.
  • Accepted: 2023-02-25.
  • Published online: 2023-05-17.
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Cite this article as:
Mohamed Dilmi, Sadok Otmani, Existence and asymptotic stability for generalized elasticity equation with variable exponent, Opuscula Math. 43, no. 3 (2023), 409-428, https://doi.org/10.7494/OpMath.2023.43.3.409

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