Acta Mathematica Sinica, English Series

, Volume 33, Issue 11, pp 1549–1564 | Cite as

On the existence, uniqueness and stability of solutions for semi-linear generalized elasticity equation with general damping term



In this paper, we consider a semi-linear generalized hyperbolic boundary value problem associated to the linear elastic equations with general damping term and nonlinearities of variable exponent type. Under suitable conditions, local and global existence theorems are proved. The uniqueness of the solution have been gotten by eliminating some hypotheses that have been imposed by other authors for different particular problems. We show that any solution with nontrivial initial datum becomes stable.


Generalized semi-linear elasticity equation nonlinear internal stabilization generalized Lebesgue space Sobolev spaces with variable exponents 

MR(2010) Subject Classification

58J45 35L53 35L71 46E30 46E35 


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We thank the referees for their time and comments.


  1. [1]
    Choquet-Bruhat, Y., Dewitt-Morette, C., Dillard-Bleick, M.: Analysis, Manifolds and Physics, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1977MATHGoogle Scholar
  2. [2]
    Diening, L., Hästö, P., Harjulehto, P., et al.: Lebesgue and Sobolev Spaces with Variable Exponents, Springer Lecture Notes, Springer-Verlag, Berlin, 2011Google Scholar
  3. [3]
    Diening, L., Ruzicka, M.: Calderon–Zygmund operators on generalized Lebesgue spaces L p(x) and problems related to fluid dynamics. J. Reine Angew. Math., 563, 197–220 (2003)MathSciNetMATHGoogle Scholar
  4. [4]
    Diening, L., Ruzicka, M.: Calderon–Zygmund operators on generalized Lebesgue spaces L p(x) and problems related to fluid dynamics. Preprint Mathematische Fakultät, Albert-Ludwigs-Universitä. Freiburg, 120, 197–220 (2002)MATHGoogle Scholar
  5. [5]
    Fan, X., Shen, J., Zhao, D.: Sobolev embedding theorems for spaces W k,p(x)(Ω). J. Math. Anal. Appl., 262, 749–760 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Fu, Y.: The existence of solutions for elliptic systems with nonuniform growth. Studi. Math., 151, 227–246 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Komornik, V.: Exact Controllability and Stabilization, The Multiplier Method, Masson-John Wiely, Paris, 1994Google Scholar
  8. [8]
    Kovacik, O., Rakosnik, J.: On spaces L p(x) and W 1,p(x)(Ω). Czechoslovak Math. J., 41, 592–618 (1991)MathSciNetMATHGoogle Scholar
  9. [9]
    Lions, L. J.: Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1966MATHGoogle Scholar
  10. [10]
    Ma, T. F., Soriano, J. A.: On weak solutions for an evolution equation with exponential nonlinearities. Nonlinear Analysis: Theory, Methods. Applications, 37, 1029–1038 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Rahmoune, A., Benabderrahmane, B.: Semilinear hyperbolic boundary value problem for linear elasticity equations. Applied Mathematics & Informatio. Sciences, 7(4), 1421–1428 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Rahmoune, A., Benabderrahmane, B., Nouiri, B.: A nonlinear hyperbolic problem for viscoelastic equations. Palestine Journal o. Mathematics, 7(4(1)), 1–11 (2015)MathSciNetMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Technical SciencesLaghouat UniversityLaghouatAlgeria

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