Opuscula Math. 43, no. 3 (2023), 409-428
https://doi.org/10.7494/OpMath.2023.43.3.409

 
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.

Full text (pdf)

  1. S. Antontsev, Wave equation with \(p(x,t)\)-Laplacian and damping term: existence and blow-up, J. Difference Equ. Appl. 3 (2011), 503-525.
  2. S.N. Antontsev, S.I. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity, J. of Math. Sciences 150 (2008), 2289-2301. https://doi.org/10.1007/s10958-008-0129-6
  3. M.M. Boureanu, Existence of solutions for anisotropic quasilinear elliptic equations with variable exponent, Adv. Pure Appl. Math. 1 (2010), no. 3, 387-411. https://doi.org/10.1515/apam.2010.025
  4. M.M. Boureanu, A. Matei, M. Sofonea, Nonlinear problems with \(p(\cdot)\)-growth conditions and applications to antiplane contact models, Advanced Nonlinear Studies 14 (2014), 295-313. https://doi.org/10.1515/ans-2014-0203
  5. Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406. https://doi.org/10.1137/050624522
  6. L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Berlin, Heidelberg, 2011.
  7. M. Dilmi, H. Benseridi, M. Dilmi, Asymptotic behavior for the elasticity system with a nonlinear dissipative term, Rend. Istit. Mat. Univ. Trieste 51 (2019), 41-60. https://doi.org/10.13137/2464-8728/27066
  8. G. Duvant, J.L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.
  9. X. Fan, D. Zhao, On the spaces \(L^{p(x)}(\Omega)\) and \(W^{k,p(x)}(\Omega)\), J. Math. Anal. Appl. 263 (2001), 424-446. https://doi.org/10.1006/jmaa.2000.7617
  10. X.L. Fan, D. Zhao, On the generalised Orlicz-Sobolev space \(W^{k,p(x)}(\Omega)\), Journal of Gansu Education College 12 (1998), no. 1, 1-6.
  11. M. Gaczkowski, P. Górka, D.J. Pons, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Mathematical Methods in the Applied Sciences 33 (2010), no. 2, 125-137.
  12. S. Ghegal, I. Hamchi, S.A. Messaoudi, Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal. 99 (2020), no. 8, 1333-1343. https://doi.org/10.1080/00036811.2018.1530760
  13. P. Gwiazda, F.Z. Klawe, A. Świerczewska-Gwiazda, Thermo-viscoelasticity for Norton-Hoff-type models, Nonlinear Analysis: Real World Applications 26 (2015), 199-228. https://doi.org/10.1016/j.nonrwa.2015.05.009
  14. V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Res. Appl. Math., vol. 36, Wiley-Masson, 1994.
  15. J.E. Lagnese, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary, Nonlinear Anal. 16 (1991), no. 1, 35-54. https://doi.org/10.1016/0362-546X(91)90129-O
  16. W. Lian, V.D. Rădulescu, R. Xu, Y. Yang, N. Zhao, Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var. 14 (2021), no. 4, 589-611. https://doi.org/10.1515/acv-2019-0039
  17. L.J. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1966.
  18. T.F. Ma, J.A. Soriano, On weak solutions for an evolution equation with exponential nonlinearities, Nonlinear Analysis: Theory, Methods & Applications 37 (1999), 1029-1038. https://doi.org/10.1016/S0362-546X(97)00714-1
  19. 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
  20. S.A. Messaoudi, J.H. Al-Smail, A.A. Talahmeh, Decay for solutions of a nonlinear damped wave equation with variable-exponent nonlinearities, Comput. Math. Appl. 76 (2018), 1863-1875. https://doi.org/10.1016/j.camwa.2018.07.035
  21. J.T. Oden, Existence theorems for a class of problems in nonlinear elasticity, J. Math. Anal. Appl. 69 (1979), 51-83. https://doi.org/10.1016/0022-247X(79)90178-1
  22. S. Otmani, S. Boulaaras, A. Allahem, The maximum norm analysis of a nonmatching grids method for a class of parabolic \(p(x)\)-Laplacian equation, Boletim da Sociedade Paranaense de Matemática 40 (2022), 1-13. https://doi.org/10.5269/bspm.45218
  23. A. Rahmoune, On the existence, uniqueness and stability of solutions for semi-linear generalized elasticity equation with general damping term, Acta Mathematica Sinica, English Series Nov. 33 (2017), no. 11, 1549-1564. https://doi.org/10.1007/s10114-017-6466-y
  24. V. Rădulescu, D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Quantitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015.
  25. M. Ružicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., Springer, Berlin, 2000.
  26. J. Simsen, M. Simsen, P. Wittbold, Reaction-diffusion coupled inclusions with variable exponents and large diffusion, Opuscula Math. 41 (2021), no. 4, 539-570. https://doi.org/10.7494/OpMath.2021.41.4.539
  27. R. Stegliński, Notes on applications of the dual fountain theorem to local and nonlocal elliptic equations with variable exponent, Opuscula Math. 42 (2022), no. 5, 751-761. https://doi.org/10.7494/OpMath.2022.42.5.751
  28. V.V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), 67-81.
  • 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.
Opuscula Mathematica - cover

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

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise that this website uses cookies to help us understand how the site is used. All data is anonymized. Recent versions of popular browsers provide users with control over cookies, allowing them to set their preferences to accept or reject all cookies or specific ones.